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µ µ ÒØ À ÈÁÌÊ ¾ºÅ Ì ÊÁ Í ÇÊÊ Ä Ë ÇÅÈ Ê ÁËÇÆÌÀ ÇÊÁ» È ÊÁ Æ ÈË Ö Ö ÔÐ Ñ ÒØ º¾º ß Ù Ö Ø Ú Ø Ò Ð Ö Ñ ÓÖÖ Ð Í ¾ µ Ú Ð Ò Ø ³ Ø Ø Ò Ì Ì ÒØ ÙÜØÖÓ ÔÓ ÒØ Ó ÙÖÐ ÓÙÖ Ö Ø Ú Ø µº ÖÓ Ø Ö Ø Ú Ø ÔÓÙÖÍ Ø Ò Ø ³ Ø Ø ÓÐ ÒØ º Ò Öص Ó ÙÜÖ Ñ Ä ÕÙ ÖÑ Ñ ÙÚ Ñ Ø Ð Ø Ñ ÓÒ ÙØ ÙÖ ÓÖÖ ÔÓÒ¹ Ö Ñ ÓÒ ÙØ ÓÒ Ä ÓÑÔ Ö ÓÒ Ú Ð Ò Ø ³ Ø Ø Ô ÖÑ ØÙÒ ÓÑÔÖ Ò ÓÒ ÒØÙ Ø Ú ÕÙ ØÖ ßÄ ÕÙ ÖÑ Ð Ø ÑÔ Ú ØØÖ ÐÓÒ Ð Ö Ð Ö Ð ØÕÙ Ñ ÒØ Ø Ø µ ßÅ ÙÚ Ñ Ø Ð Ð Ô ÕÙ Ô ÖØ ÙÐ ØØÓÙ ÓÙÖ Ð Ñ Ø Ó ÙÒÐ Ö ³ ÑÔÙÖ Ø «Ø ³ Ø Ö Ò Ö ÙÖ Ð³ ÐÐ Ó Ö Ò ÙÔÖÓ Ð Ñ ÙÖÐ Ö Ùº ػ̾ºÄ Ø ÑÔ Ö ØÙÖ Ø Ò Ö ÙÖ Ð Ø ÑÔ Ö ØÙÖ ÃÓÒ ÓÌ ÙÑÓ Ð ßË Ñ ÓÒ ÙØ ÙÖ Ð Ò Ø ³ Ø ØÑÓÒØÖ ÙÒÔ Ù Ó Ô Ü Ø Ø ÓÒ Ø ÖÑ ÕÙ µ Ø ÙÒ ÖÓ Ò Ð ÒØ Ú Ì ÓÒ Ì Ì ÓÙÌ µº ÓÖÖ ÔÓÒ Ì²Ì µº Ô ÖÓÙÖ ÑÓÝ Ò Ð³ÓÖ Ö Ð Ñ ÐÐ Ð Ñ Ø ÅÓØصºÇÒ»Ì Ö Ñ ßÁ ÓÐ ÒØ Ð Ô ÅÓØØ ÒØÖ Ò ÙÒ Ö Ø Ú Ø Ø Ú» ÜÔ Ì ¾µ Ò Ð ÁÈÌÓÒØÖ ÆÊ Ö Ñ Ì º ÐÓÖ ÕÙ³ÓÒ Ö ÔÔÖÓ Ð ØÖ Ò Ø ÓÒ ÅÓØغ Ò «Ø ÒÕÙ ÒÓÑ Ö Ù ÕÙ ÒØ Ø ÁÈÌ ÓÒÒÓØ ÕÙ Ð ÓÑÔÓÖØ Ñ ÒØ ÙØ ÑÔ Ú ¼» ¼¼ ¼ ̵ ÔÓ ÕÙ ¹Ô ÖØ ÙÐ ÓÙ Ð ÓÙÔ Ø ÓÒ µ ÓÒØ ÓÒÒ Ú ÙÒ ÓÒÒ ÔÖ ÓÒ Ò Ä³ ØÙ ÔÐÙ ØØ ÒØ Ú Ö ÙÐØ Ø ÑÓÒØÖ Ô Ò ÒØ Ò ÙÆ Ò ÑÔÓÖØ ÒØ ¼» ¾ ÕÙ ÓÒÒ Ð Ö Ø Ú Ø Ø ÑÔ Ö ØÙÖ Ò «Ø ÔÖ ÙÔÓ ÒØÖ Ø Õ٠; ÓÒ ³ ØØ Ò ÕÙ Ð Ø ÑÔ Ú ³ ÒÒÙÐ ÓÑÑ ½ Ø ÒÓÖÖ Ø ½ º ̵»Ì¾ ¾ ¾ ¾º µ 4 3 2 1-4 -2 2 4.5.1.15.2.25.6.4.2 2 15 1 5-3 -2-1 1 2 3.2.4.6.8 1.6.4.2 44

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À ÈÁÌÊ ¾ºÅ Ì ÊÁ Í ÇÊÊ Ä Ë ÇÅÈ Ê ÁËÇÆÌÀ ÇÊÁ» È ÊÁ Æ ¾º¾º½ÌÖ Ò Ø ÓÒ ÅÓØØ Ò ÙÖ Ì Ë ¾ Ô ÓØÓ Ñ ÓÒ Ö ÔÓÒ Ý Ø Ñ ÖÑ ÓÒ ÓÖÖ Ð ÐÓÖ Õ٠гÓÒØÖ Ú Ö Ð ØÖ Ò Ø ÓÒ ÅÓØغÁÐ ¹ Ä Ø ÓÒÔÖ ÒØ ÐÐÙ ØÖ Ð Ò Ñ ÒØ ÑÔÓÖØ ÒØ ÕÙ ÔÖÓ Ù ÒØ Ò Ð ÈÖ ÒØ Ø ÓÒ Ð³ ÜÔ Ö Ò Ð ØÖÓÒ ÕÙ Ð ØÖ Ò Ø ÓÒµº ÙÜØ Ò ÕÙ Ô ØÖÓ ÓÔ ÑÓ ÖÒ ÓÒØ ÔÓÒ Ð ÔÓÙÖ Å Ì Ú Ò ÒÓÑ Ö Ù ÔÖ Ø ÓÒ ÒÓÒØÖ Ú Ð ÓÒ ÖÒ ÒØÐ Ô ØÖ Ü Ø Ø ÓÒ Ñ ÐÓ Ð ØÓ Ö ÒØ Ø ÜÔ Ö Ñ ÒØ ÙÜ Ø Ù ÔÓÙÖØ Ø ÖÐ Ø ÓÖ ÔÐÙ Ò Ø Ð Ö Ø Ö Ð ³ ÚÓ ÖÙÒ Ñ Ò Ú Ò ÔÐÙ Ö Ø Ô ÒÓÑ Ò ÔÓÙÖÔÖ ÒØ ÖÙÒ Ô ÓØÓ Ñ ÓÒºÇÒÔÖ ÒØ Ö Ò Ù Ø Ð ØÖ Ô ÖØ ÙÐ Ö ³ÙÒ ØÙ Ö ÒØ ÙØ Ð ÒØ ØØ Ø Ò ÕÙ ÙÖÐ Ð Ò Ø ÒØ Ð Ì Ë ¾º Ñ Ò Ö ØØ ØÙ ÊÈ Ë ØËÌŵ ØÒÓÙ ÐÐÓÒ ÙØ Ö Ð Ø ØÙØ ØÙ Ð ÜÔ Ö Ò ÈÖÓ Ð Ñ Ð Ô ÓØÓ Ñ ÓÒ Ò Ð Ý Ø Ñ ÖÑ ÓÒ ÓÖÖ Ð ÔÖÓ Ö ÓÖÑ Ð ÒÕÙ Ð ÔÖ Ñ Ö ØÙ ÙÖÐ ÓÜÝ Ñ Ø ÙÜ ØÖ Ò Ø ÓÒ Ö ÑÓÒØ ÒØ ÙÒ ØÖ ÒØ Ò ³ ÒÒ º Ò «Ø Ò Ð ØÖ Ú ÙÜ ÒØ Ö ÙÖ ¾¼¼½ Ð³Ó ÖÚ Ø ÓÒ ³ÙÒÔ ÕÙ Ô ÖØ ÙÐ Ö ÒÓÖÑ Ð ÙÖÐ Ô ØÖ Ô ÓØÓ Ñ ÓÒ Ø ÔÐÙØ ÓØÒ Ø Ú Ä ÜÔ Ö Ò Ô ÓØÓ Ñ ÓÒ ÙÖÐ Ñ Ø Ö ÙÜÓÖÖ Ð ÓÒØÓÒÒÙØÖ Ö ÑÑ ÒØ ÔÖÓ Ð Ñ Ò³ Ø ØÔ ÙÐ Ñ ÒØ Ñ ÖÖ ÒØÔÓÙÖÐ Ø ÓÖ Ò Ð Ø ØØÓÙØ ÑÔÐ Ñ ÒØ Ò ÓÒØÖ Ø ÓÒ Ú Ð³ Ò Ñ Ð Ñ ÙÖ Ø ÖÑÓ ÝÒ Ñ ÕÙ Ó Ö ÒØ Ú ÙÒ Ø ØÐ ÕÙ Ñ Ñ Ð³ÓÒ ÔÙÓÒ Ø Ø Ö ÓÖØ Ö ØÖ ÙØ ÓÒ ÔÓ Ô ØÖ Ð ÒØÖ Ð Ò Ú Ù ÖÑ ÓÖØ Ñ ÒØÓÖÖ Ð µºáð Ø Ö ÓÒÒÙ ÖÒ Ö Ñ ÒØÕÙ Ð ÔÐÙÔ ÖØ Ö ÙÐØ Ø ÖÑ ØÐ Ø ÐÐ Ø ÙØ Ò Ö Ð³ ÔÔÖÓ Ð ØÖ Ò Ø ÓÒÑ Ø Ð¹ ÓÐ ÒØ ½ º Ô ÓØÓ Ñ ÓÒ ÓÒØ Ö Ñ ØØÖ Ò Ù ÔÓÙÖ ÙÜÖ ÓÒ ºÄ ÔÖ Ñ Ö ØÐ Ð ÕÙ Ð Ø ÙÖ ÔÖÓ Ð Ñ Ð Ú ÙØ Ú Ö ÕÙ Ô Ò ÒØ ÙÑ Ø Ö ÙÓÒ Ö µ Ó ÙÙÒ ÓÐÙØ ÓÒÔÓ Ð Ò Ð ³ÙÒÔÖÓ Ð Ñ ÖÙ Ó Ø ÙÖ µ Ø ³ÙØ Ð ÖÙÒ ÙØÖ ÓÒ ÒØÖ Ô Ø Ð Ñ ÒØ ÔÓÙÖÐ Ñ Ø ÖÐ Ø ÐÐ Ð ÞÓÒ ³ Ò ÓÑÓ Ò Ø ºÄ ÓÒ ÔÖÓ Ð Ñ ØÕÙ³ Ú Ð Ò Ö ØÝÔ ÕÙ Ô ÓØÓÒ Ò ÒØ ÒÔ ÓØÓ Ñ ÓÒ ½¼¹½¼¼ ε ÓÒ Ò ÓÒ Ò Ò Ö ÐÔ Ð³ ÒØ Ö ÙÖ ÙÑ Ø Ö Ù Ñ ÙÒ ÓÙ ÔÐÙ ÓÙÑÓ Ò Ô Ô ÖØ ÒØ Ð ÙÖ Ù Ú ÒØг Ò Ö µºä Ò ÐÓ Ø ÒÙ Ø Ò Ð ÙÔ ÖÔÓ Ø ÓÒ ÓÒØÖ ÙØ ÓÒ ØÖ «Ö ÒØ Ð Ô Ý ÕÙ Ò ÙÖ Ò³ Ý ÒØ ÓÙÚ ÒØÖ Ò ÚÓ Ö Ú ÕÙ Ô Ù Ó ÙÖ ÙÑ Ø Ö ÙºÁÐ Ø ÓÒ Ù ÓÙÖ ³ Ù ÔÓ Ð ³ ÜØÖ Ö Ð Ò Ð ÒØ Ö ÒØ ÒØ Ö ÙÖ Ù Ý Ø Ñ µ ÒÑ Ò ÒØÙÒ Ò ÐÝ ÔÓÙÖÔÐÙ ÙÖ ÐÓÒ Ù ÙÖ ³ÓÒ «Ö ÒØ Ò ÒØ Ð ÔÐÙ ÙÖ Ý Ø Ñ ÓÖÖ Ð ÎÇ Î¾Ç ºººµ Ö ÓÒ Ð ÒØ Ò Ð Ñ ÙÖ Ô ØÖÓ ÓÔ ÕÙ Ø Ò ÕÙ Ø ÒØ Ñ ÒØ Ò ÖÙÒ ÓÒÒ Ö ÓÐÙØ ÓÒ Ö Ò Ò Ö Ô ÓØÓÒµº ÒÓÙÚ ÐÐ ÜÔ Ö Ò ½ ß½ ÓÒØ Ò Ñ Ò Ú Ò Ð³ Ü Ø Ò ³ÙÒÔ ØÖÓ Ø ÕÙ Ô ÖØ ÙÐ Ò Ú Ð Ö ÙÐØ Ø Ø ÖÑÓ ÝÒ Ñ ÕÙ ÙÖ Ý Ø Ñ ÙÖ ¾º½¼ºÄ Ö ÑÑ Ù ÑÓÖ ÙÖ ½º µò³ Ø ÓÒÔÐÙ ØÖÓÔ Ù Ó ÙØ Ù ÓÙÖº ØÖ Ò Ø ÓÒ ÅÓØØ ÙØ ØÖÙØÙÖ Ø ÓÖ Ö ÓÒØ Ó ØØ Ñ Ò Ö ÔÖÓ Öº ij Ð Ö Ø ³ÙØ Ð ÖÐ Ø Ò ÕÙ ÔÖ ÓÒ Þ ÓÒØÓÒ Ô ÖÐ Ù Ô ØÖ ÔÖ ÒØ Ñ Ò Ñ ÒØ ÔÖÓ Ù Ø Ô ÖÐ Ù Ø ØÙØ ÓÒ Ñ ÕÙ ÕÙ ÓÒØÙØ Ð ÔÓÙÖ Ð Ò ÖÐ ÍÒ Ð Ñ Ø Ø ÓÒ ÓÒØ Ð ÙØÑ Ð Ö ØÓÙØØ Ò ÖÓÑÔØ Ò ØÓÙØ ØÙ ØÐ ÓÑÔÐ Ü Ø 48

¾º¾ºÌÊ ÆËÁÌÁÇÆ ÅÇÌÌ ÆËÍÊ Ì Ë ¾ ÈÀÇÌÇ ÅÁËËÁÇÆ Ð Ñ Ð ÒÓÑÔ Ø Ð Ú ØÝÔ Ô ØÖÓ ÓÔ Ñ ÙÖ ³ÓÔØ ÕÙ Ò Ð³ Ò Ö ¹ ÖÓÙ ÓÒØØÓÙØ Ó ÔÓ Ð ÓÙ ÔÖ ÓÒ Þ ¾¼ µº 1486.6 ev 4.8 ev 21.2 ev LSDA CaVO 3 Intensity (arb. units) SrVO 3 Intensity (arb. unit) a: hν=7ev b: hν=5ev c: k-averaged d: hν=31ev e: Schramme et al. [14] hν=6ev a b c d e -3. -2.5-2. -1.5-1. -.5..5 Ò Ø ÓÖ Ô ÖÐ Ù Ø ØÙØ ÓÒ Ò Ö ÑÔÐ ÓÒ Ø Òصº ÖÓ Ø Ñ ÙÖ Ð³ Ò Ö Ô ÓØÓÒ Ò ÒØ ³ ÔÖ ½ º ÜËÖ½ º¾º½¼ß Ù Ô ÓØÓ Ñ ÓÒ ÒØ Ö Ò ÙÖ ÎÇ ØËÖÎÇ Ò ÓÒØ ÓÒ ÜÎÇ ØÙÒ Ý Ø Ñ ½ ÓÒØÐ Ð Ö ÙÖ 3 2 1 E-E F (ev) Binding energy (ev) Ñ Ð Ö ÙÖÎ¾Ç ½ Ð ÓÙÖ µôö ÒØ Ô Ö ÐÐ ÙÖ ³ ÑÔÓÖØ ÒØ «Ø ÙØ ÙÖ µº Ð Ò Ð ÙÖ Ø ÓÑ Ò ÒØ Ò Ð ÓÒ Ø ÓÒ ÒÓÖÑ Ð Ó ÙгÓÒÖ Ð Ð ÜÔ Ö Ò Ô ÓØÓ Ñ ÓÒ ÔÓÙÖÕÙÓ Ò Ô Ó ÖÚ ÖÐ ØÖ Ò Ø ÓÒ ÅÓØØ Ò ÙÖ Å ØØÖ ØØ Ò ÔÖ Ø ÕÙ Ò³ Ø Ô Ò ÒØÔ ÑÔÐ º ÓÑÑ Ò ÓÒ Ô ÖÖ Ñ ÖÕÙ ÖØÓÙØ ³ ÓÖ ÕÙ³ Ð ÙÖ ÍÒ ÙØÖ ÓÒ ³ ØØ ÕÙ ÖÐ ÔÖÓ Ð Ñ Ø Ù Ú Ô Öź Ö ÓÒ ØÓÐÐ ÓÖ Ø ÙÖ ºÈÙ ÕÙ ³ÙÒÑ Ø Ö ÙÓÖÖ Ð Ð ÒØ Ö Ø ÓÒ Ó Ú ÒØ ÓÙ ÖÙÒÖ ÓÐ ÔÐÙ ÑÔÓÖØ ÒØÕÙ³ Ò ÓÒ ÒØ Ö ÙÖº Ð ³ ÜÔÐ ÕÙ Ò «ØÔ ÖÐ Ñ ÒÙØ ÓÒÓÒ Ó ÒØ Ð³ Ö ÒØ Ø Ð ÓÒÒ Ø Ú Ø ÕÙ ÍÒØ ÐÑ Ø Ö Ù Ü ÒØÙÒ ØÖ Ò Ø ÓÒ ÅÓØØ Ò ÙÖ Ó Ø ÓÒÔÖ ÒØ ÖÙÒ Ô Ý ÕÙ ÞÖ ÙÐ Ö ÔÓÙÖÐ ÔÖÓÔÖ Ø Ñ ÖÓ ÓÔ ÕÙ ÓÒ ³ ØØ Ò ÙÒ Ö Ø Ú Ø Ñ Ø ÐÐ ÕÙ µº ÔÐÙ Ð ÙØØÖÓÙÚ ÖÙÒÑÓÝ Ò Ð Ý ÖÐ ØÖ Ò Ø ÓÒ ÅÓØØ ØÙØ Ð ÖÔÓÙÖ Ð Ð Ö Ô Ø Ú Ñ ÒØÖ Ò ÓÖ Ð³ ÒØ Ö Ø ÓÒÐÓ Ð ÒØÖ Ð ØÖÓÒ Ø «Ð ØÐ ÙÖ Ò Ö Ò Ø ÕÙ º ÔÖ ÓÒ Ò ÙÒ ÜÔ Ö Ò Ô ÓØÓ Ñ ÓÒ ØÑ Ð ÙÖ Ù Ñ ÒØ ÜÐÙغÁÐ Ü Ø Ñ Ð Ö ØÓÙØÙÒ Ò ØÕÙ ÔÓÙÖÖ Ø Ø Ö ØÓÙ Ö Ø Ö Ð ÐÓ Ò Ì Ë ¾º Ü Ñ ÒÓÒ ØÓÙØ ³ ÓÖ Ð ÔÖÓÔÖ Ø Ô Ý ÕÙ ÓÑÔÓ ÔÓÙÖ ÜÔÐ ÕÙ Ö ÕÙÓ ÐÖ ØÓÙÖÒ Ú ÒØ ÔÖ ÒØ ÖÐ Ö ÙÐØ Ø Ñ ÙÖ Ô ÓØÓ Ñ ÓÒº 49

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ÖØ Ð ½ºÅÓØØØÖ Ò Ø ÓÒ Ò ØÖ Ò ÔÓÖØÖÓ ÓÚ Ö ÒØ ÓÖ Ò ÓÑÔÓÙÒ ¹ Ì VOLUME 91, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S week ending 4JULY23 Mott Transition and Transport Crossovers in the Organic Compound - BEDT-TTF 2 Cu N CN 2 Cl P. Limelette, 1 P. Wzietek, 1 S. Florens, 2,1 A. Georges, 2,1 T. A. Costi, 3 C. Pasquier, 1 D. Jérome, 1 C. Mézière, 4 and P. Batail 4 1 Laboratoire de Physique des Solides (CNRS, U.R.A. 852), Bâtiment 51, Université de Paris-Sud, 9145 Orsay, France 2 LPT-Ecole Normale Supérieure (CNRS-UMR 8549), 24, rue Lhomond, 75231 Paris Cedex 5, France 3 Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany 4 Laboratoire Chimie Inorganique, Matériaux et Interfaces (CIMI), FRE 2447 CNRS, Université d Angers, 4945 Angers, France (Received 27 January 23; published 2 July 23) We have performed in-plane transport measurements on the two-dimensional organic salt - BEDT-TTF 2 Cu N CN 2 Cl. A variable (gas) pressure technique allows for a detailed study of the changes in conductivity through the insulator-to-metal transition. We identify four different transport regimes as a function of pressure and temperature (corresponding to insulating, semiconducting, bad metal, and strongly correlated Fermi-liquid behaviors). Marked hysteresis is found in the transition region, which displays complex physics that we attribute to strong spatial inhomogeneities. Away from the critical region, good agreement is found with a dynamical mean-field calculation of transport properties using the numerical renormalization group technique. DOI: 1.113/PhysRevLett.91.1641 PACS numbers: 71.3.+h, 74.7.Kn, 74.25.Fy The Mott metal-insulator transition (MIT) is a key phenomenon in the physics of strongly correlated electron materials. It has been the subject of extensive experimental studies in transition metal oxides, such as V 1 x Cr x 2 O 3 or chalcogenides such as NiS 2 x Se x (for a review, see Ref. [1]). In contrast to chemical composition, hydrostatic pressure allows in principle a continuous sweeping through the transition. For these materials, however, the appropriate range of pressure is several kilobars. For this reason, many fundamental aspects of the MIT are yet to be studied in detail. This issue is particularly important in view of recent theoretical predictions for, e.g., spectroscopy and transport close to the transition, which should be put to experimental test [2]. Layered charge-transfer salts of the - BEDT-TTF 2 X family (where X is a monoanion) offer a remarkable opportunity for such a study. Indeed, these compounds are known to display a great sensitivity to hydrostatic pressure [3]. The - BEDT-TTF 2 Cu N CN 2 Cl compound, in particular, (abbreviated -Cl below) displays a very rich phase diagram with paramagnetic insulating, antiferromagnetic insulating, superconducting, and metallic phases when pressure is varied over a range of a few hundred bars [3 5]. In this Letter, we report on an extensive experimental study of the in-plane resistivity of the -Cl compound for a range of pressure spanning both the insulating and the metallic phases. In contrast to previous studies [3], pressure is varied continuously using a helium gas cell, at constant temperature. By analyzing the pressure and temperature dependence of the measured resistivity, we have identified important crossover lines, which are summarized on the phase diagram of Fig. 1. These crossovers separate four different regimes of transport (to be described below) within the paramagnetic phase, corresponding to an insulator, a semiconductor, a bad metal, and a Fermi-liquid metallic regime. Our measurements were performed both with increasing and decreasing pressure sweeps, yielding a determination of the two spinodal lines P c 1 T and Pc 2 T shown in Fig. 1 (see also Fig. 3 below). A critical comparison is made to dynamical mean-field theory (DMFT [2]), which describes successfully the observed transport crossovers, while the critical region T(K) 8 7 6 5 4 3 2 Semiconductor Mott insulator AF insulator Bad metal Fermi liquid 1 1 2 3 4 5 6 7 8 P (bar) P c 1 (T) T * Met T * Ins Pc 2 (T) ρ(τ) max (dσ/dp) max FIG. 1 (color online). Pressure-temperature phase diagram of the -Cl salt. The crossover lines identified from our transport measurements delimit four regions, as described in the text. The spinodal lines defining the region of coexistence of insulating and metallic phases (hatched) are indicated, as well as the line where d =dp is maximum. The latter yields an estimate of the first-order transition line, ending in a critical end point. The transition line into an antiferromagnetic insulating phase has been taken from Ref. [5], while the superconducting phase [3,5] below 13 K has been omitted. 1641-1 31-97=3=91(1)=1641(4)$2. 23 The American Physical Society 1641-1 61

ÖØ Ð ½ºÅÓØØØÖ Ò Ø ÓÒ Ò ØÖ Ò ÔÓÖØÖÓ ÓÚ Ö ÒØ ÓÖ Ò ÓÑÔÓÙÒ ¹ Ì VOLUME 91, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S week ending 4JULY23 itself reveals more complex behavior. Numerical renormalization group (NRG) calculations of the resistivity within DMFT are reported here for the first time, and compare favorably to the experimental data. This paper is based mainly on the data set displayed in Fig. 2. The in-plane conductivity was measured using a standard four-terminal method as a function of pressure between 1 bar and 1 kbar. The temperature range 13 52 K was investigated, with pressure sweeps performed every Kelvin (for clarity, only selected temperatures are displayed in Fig. 2). The data clearly demonstrate how pressure drives the system from a low-conductivity insulating regime at low pressure (below 2 bar) to a highconductivity metallic regime at high pressure (above 35 bar), with a more complex transition region in the 2 35 bar range. The pressure interval in which the normalized difference between these two measurements exceeds our experimental precision (inset of Fig. 3) corresponds to the region of coexistence between the insulating and metallic phases. This interval is found to be rather large ( 1 bar) at 35 K, and still measurable (although the signal is close to our error bars) at 39 K. Note that a much narrower interval would be found from the width at halfmaximum. We emphasize that varying pressure rather than temperature is a much more accurate experimental technique for the investigation of the coexistence region, because of the shape of the spinodal lines in the P;T plane. This also ensures a better accuracy on pressure (of order 1 bar) as compared to a cooling experiment. Our measurements yield a determination of the critical point in the T c 39 K to T c 42 K range (P c 28 bar), somewhat higher than previously reported from NMR [5] or sound velocity [6] experiments. The data in Fig. 3 are far from the ideal situation of a sharp discontinuity in p for T<T c, and present rather a wide region of phase coexistence, in which the measured P displays sizable noise. This suggests a complex physics in this regime, with the likely appearance of insulating and metallic domains in the material, and pinning by disorder. We now focus on the low-pressure, insulating regime. At low temperature (below 5 K), an activation law exp =2T describes the data quite well. The temperature range over which this activation law applies is limited both from below and from above. Indeed, at the lowest temperatures a transition into the antiferromagnetic insulating phase is found (around 25 K at ambient pressure, see Fig. 1), while a crossover to a different insulating regime is observed at higher temperatures, as discussed below. Despite this limited range, a reasonable determination of the insulating gap in the paramagnetic insulator region of Fig. 1 can be achieved, since T. An approximately linear pressure dependence (see Fig. 4) is found: 74 2P bar (in Kelvin). Hence, the activation gap is still a large scale close to the coexistence region ( 4 K at 15 bar), of the order of 1 times the critical temperature associated with the transition. cannot be reliably determined above 15 bar, but a rough extrapolation would lead to a gap closure around 37 bar, on the high-pressure side of the coexistence region of Fig. 1, suggesting that the finitetemperature Mott transition in this material is not driven by the closure of the Mott gap. At higher temperature (above T 5 K at ambient pressure), the T dependence of the resistivity deviates from the above activation law (inset of Fig. 4). This crossover into a semiconducting regime, which extends over much of the high-t part of the phase diagram, is indicated in Fig. 1. Interestingly, the crossover scale is an order of magnitude smaller than the insulating gap. σ (Ω.cm) -1 18 15 12 9 6 3.15 dσ/dp (Ω.cm.bar) -1. 3 6 P (bar) 13 K 15 K 18 K 23 K 28 K 33 K 38 K 42 K 47 K 51 K 38 K 39 K 4 K 41 K 42 K 43 K 44 K 45 K 2 4 6 8 P (bar) σ/σ 4 bar 1.2 1..8.6 (σ dec -σ inc )/σ C.2 35 K 36 K 39 K. 1 2 3 4 P (bar).4 39 K P inc P dec 35 K P inc P dec.2 3 K P inc P dec 25 K P inc P dec 18 K P inc P dec 13 K P inc P dec. 1 2 3 4 5 P(bar) FIG. 2 (color online). Pressure dependence of the in-plane conductivity at different temperatures. The inset shows the derivative d =dp which reveals a sharp peak at low temperature. FIG. 3 (color online). Conductivity measured for increasing ( inc ) and decreasing ( dec ) pressure sweeps, at different temperatures up to 39 K. Hysteresis at 35 and 39 K is more apparent in the difference dec inc (inset). 1641-2 1641-2 62

ÖØ Ð ½ºÅÓØØØÖ Ò Ø ÓÒ Ò ØÖ Ò ÔÓÖØÖÓ ÓÚ Ö ÒØ ÓÖ Ò ÓÑÔÓÙÒ ¹ Ì VOLUME 91, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S week ending 4JULY23 (K) The high pressure regime above 3 bar is now considered. As demonstrated in the right inset of Fig. 5, the resistivity in this regime has a quadratic, Fermi-liquid dependence upon temperature at low temperatures: P A P T 2. The quality of the fit, obtained by replotting the data set of Fig. 2 as a function of T 2, illustrates the high precision of the variable pressure technique. The coherence temperature T above which this law is no longer valid (of the order of 35 K at 5 bar) defines the onset of a bad metal regime, as indicated in Fig. 1. The prefactor A P of the T 2 dependence is found to depend strongly on pressure, as displayed in Fig. 5, and the product A P T 2 remains approximately pressure independent. These findings correspond to a strongly correlated Fermi-liquid regime at ρ (mω.cm) 8 7 6 5 4 3 2 14 12 1 8 6 (K) 8 6 4 2 Mott Insulator 1 3 35 4 45 5 55 6 T(K) 5 1 15 ρ (Ω.cm).2.15.1.5. 1bar 8 bar 12 bar 14 bar semiconductor P (bar) 3 bar 4 bar 5 bar 6 bar 8 bar 1 2 3 4.1 3 4 5 6 7 8 9 1 P (bar) ~74-2 P bar FIG. 4 (color online). Pressure dependence of the paramagnetic gap. The inset displays 2dln =d 1=T vs T for a few pressures, from which the crossover from a Mott insulator to a semiconductor is identified. T 2 (K 2 ).3.25.2.15 A(mΩ.cm.K -2 ) FIG. 5 (color online). Pressure dependence of the T 2 coefficient A and residual resistivity. Inset: Plot of vs T 2, also showing the increase of T with pressure (straight lines are guides to the eyes). low temperature. We note that the residual resistivity has a weaker pressure dependence than A, but does increase close to the coexistence region. While A P cannot be determined precisely below 28 bar, the data suggest a divergency of A at P of order 2 bar, significantly smaller than the pressure at which the extrapolated insulating gap would vanish (of order 37 bar; see above). This suggests that the closure of the Mott-Hubbard gap and the loss of Fermi-liquid coherence are two distinct phenomena, associated with very different energy scales, as is also clear from the fact that the coherence scalet (a few tens of Kelvin) is much smaller than the insulating gap (several hundreds Kelvin). In order to better characterize the crossover into the bad metal as temperature is increased above T, we have performed measurements in a wider temperature range, up to 3 K, as displayed in Fig. 6.We confirm the general behavior reported by other authors [3]. At moderate pressures (a few hundred bars), the resistivity changes from a T 2 behavior below T to a regime characterized by very large values of the resistivity (exceeding the Ioffe-Regel- Mott limit by more than an order of magnitude) but still metalliclike (d =dt > ). For pressures in the transition region, this increase persists until a maximum is reached, beyond which a semiconducting regime is entered (d =dt <). This regime is continuously connected to that found on the insulating side, at temperatures above 5 K. Figure 1 summarizes the four regimes of transport which we have described so far. We now compare these experimental findings to the DMFT description of the Mott transition. Similarities between DMFT and some physical properties of BEDT organics have been emphasized previously [7,8]. One of ρ (Ω.cm).3.2.1 U=4 K. 5 1 15 2 25 3 T(K) 3 bar W=3543 K 6 bar W=3657 K 7 bar W=3691 K 15 bar W=3943 K 1 kbar W= 5 K FIG. 6 (color online). Temperature dependence of the resistivity at different pressures. The data (circles) are compared to a DMFT-NRG calculation (diamonds), with a pressure dependence of the bandwidth as indicated. The measured residual resistivity has been added to the theoretical curves. 1641-3 1641-3 63

ÖØ Ð ½ºÅÓØØØÖ Ò Ø ÓÒ Ò ØÖ Ò ÔÓÖØÖÓ ÓÚ Ö ÒØ ÓÖ Ò ÓÑÔÓÙÒ ¹ Ì VOLUME 91, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S week ending 4JULY23 the key qualitative outcomes of our measurements is the separation of energy scales found in the transition region. This is also a distinctive feature of DMFT, in which Hubbard bands are well separated from the quasiparticle peak in the correlated metal, and a coexistence region of insulating and metallic solutions delimited by two spinodal lines is found. The three transport regimes found on the metallic side of the transition are well accounted for within DMFT [2,8 1]. In this theory, a Fermi-liquid regime with long lived quasiparticles applies for T below the coherence scale T, with AT 2 and A / 1= T 2. AsT increases, an intermediate (bad metal) regime is reached, in which the low-energy peak is reduced and strongly temperature dependent, corresponding to a very short lifetime and a large resistivity increasing rapidly witht. AsT is further increased, the peak disappears, leaving a pseudogap in the density of states, and a semiconducting transport regime is found, with d =dt <. We have performed DMFT calculations of the resistivity for a simple Hubbard model at half-filling. The NRG technique was used [11], which is crucial to capture correctly the enhancement of A, and to determine accurately the resistivity maximum. The coupling was fixed at U 4 K, consistent with the measured critical temperature (within DMFT, T c U=1). A semicircular band was used, with a bandwidth W adjusted for each pressure, as indicated in Fig. 6. Comparison to the measured T involves also a global scale factor (the same for all curves). We find a good agreement between our calculations and the data, up to T 15 K, for pressures ranging from the transition (3 bar) to 1 kbar. The fitted bandwidth increases by about 4% in that range. This value, together with the magnitudes of U and W is consistent with theoretical [12] and experimental [13] estimates for these materials. We attribute the discrepancy observed above 15 K to the strong thermal dilatation of the materials, leading to a smaller effective bandwidth, an effect which could be taken into account by allowing for a T dependence of W in the calculation. The same effect might also contribute to the observed crossover on the insulating side at T 5 K, even though DMFT also leads to a similar prediction of a purely electronic crossover at T [2,9]. Finally, we observe that our data in the critical regime around T c ;P c do not appear to obey the critical behavior established within DMFT, namely, that of a liquid-gas transition in the Ising universality class [14]. The observed transition is much too smooth to be described in that manner. Besides the low dimensionality of these compounds, this could be attributed to strong spatial inhomogeneities or disorder in the transition region. To conclude, we have performed detailed transport measurements on the -Cl compound using a variable pressure technique. Four transport regimes have been identified as a function of temperature and pressure, separated by crossover lines. Hysteresis experiments have confirmed the first-order nature of the transition and allowed for the first detailed determination of the coexistence region. DMFT provides a good qualitative description of these crossovers, and new DMFT-NRG calculations of transport are in good agreement with the data. Further theoretical work is needed to understand the critical regime, however. We note that recent ultrasound velocity measurements [6] have revealed anomalies along crossover lines very similar to the two high-temperature lines reported here from transport measurements. This calls for further theoretical work aiming at connecting these two effects. We are grateful to R. Chitra, V. Dobrosavlevic, G. Kotliar, R. McKenzie, M. Poirier, M. Rozenberg, and A. M. Tremblay for useful discussions. T. A. C., S. F., and A. G. are grateful to KITP-UCSB for hospitality during the final stages of this work (under NSF Grant No. PHY99-7949). T. A. C. acknowledges support from the SFB 195 of the DFG. [1] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 7, 139 (1998). [2] For a review, see A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). [3] H. Ito, T. Ishiguro, M. Kubota, and G. Saito, J. Phys. Soc. Jpn. 65, 2987 (1996), and references therein. [4] R. H. McKenzie, Science 278, 82 (1997). [5] S. Lefebvre, P. Wzietek, S. Brown, C. Bourbonnais, D. Jérome, C. Mèziére, M. Fourmigué, and P. Batail, Phys. Rev. Lett. 85, 542 (2). [6] D. Fournier, M. Poirier, M. Castonguay, and K. Truong, Phys. Rev. Lett. 9, 1272 (23). [7] R. H. McKenzie, Comments Condens. Matter Phys. 18, 39 (1998). [8] J. Merino and R. H. McKenzie, Phys. Rev. B 61, 7996 (2). [9] M. J. Rozenberg, G. Kotliar, and X.Y. Zhang, Phys. Rev. B 49, 1181 (1994). [1] P. Majumdar and H. R. Krishnamurthy, Phys. Rev. B 52, R5479 (1995). [11] R. Bulla, T. A. Costi, and D. Vollhardt, Phys. Rev. B 64, 4513 (21). [12] M. Rahal, D. Chasseau, J. Gaultier, L. Ducasse, M. Kurmoo, and P. Day, Acta Crystallogr. Sect. B 53, 159 (1997). [13] A. K. Klehe, R. D. McDonald, A. F. Goncharov, V.V. Struzhkin, Ho-Kwang Mao, R. J. Hemley, T. Sasaki, W. Hayes, and J. Singleton, J. Phys. Condens. Matter 12, L247 (2). [14] G. Kotliar, E. Lange, and M. J. Rozenberg, Phys. Rev. Lett. 84, 518 (2). 1641-4 1641-4 64

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ÖØ Ð ¾ºËÔ ØÖÓ ÓÔ Ò ØÙÖ Ó ÅÓØØØÖ Ò Ø ÓÒ ØØ ÙÖ Ó ½Ì¹Ì Ë ¾ VOLUME 9, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S week ending 25 APRIL 23 Spectroscopic Signatures of a Bandwidth-Controlled Mott Transition at the Surface of1t-tase 2 L. Perfetti, 1 A. Georges, 2 S. Florens, 2 S. Biermann, 2,3 S. Mitrovic, 1 H. Berger, 1 Y. Tom m, 4 H. Höchst, 5 and M. Grioni 1 1 Institut de Physique des Nanostructures, Ecole Polytechnique Fédérale (EPFL), CH-115 Lausanne, Switzerland 2 Laboratoire de Physique Théorique de l Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris Cedex 5, France 3 Laboratoire de Physique des Solides, Université Paris-Sud, Batiment 51, 9145 Orsay, France 4 Department of Solar Energetics, Hahn-Meitner Institut, Glienicker Strasse 1, D-1419 Berlin, Germany 5 Synchrotron Radiation Center, University of Wisconsin, Stoughton, Wisconsin 53589-397, USA (Received 11 October 22; published 22 April 23) High-resolution angle-resolved photoemission data show that a metal-insulator Mott transition occurs at the surface of the quasi-two-dimensional compound 1T-TaSe 2. The transition is driven by the narrowing of the Ta 5d band induced by a temperature-dependent modulation of the atomic positions. A dynamical mean-field theory calculation of the spectral function of the half-filled Hubbard model captures the main qualitative feature of the data, namely, the rapid transfer of spectral weight from the observed quasiparticle peak at the Fermi surface to the Hubbard bands, as the correlation gap opens up. DOI: 1.113/PhysRevLett.9.16641 PACS numbers: 71.3.+h, 71.1.Fd, 71.45.Lr, 79.6.Bm Electronic correlations can modify the electronic structure of solids not only quantitatively, but also qualitatively, inducing new broken-symmetry phases which exhibit charge, spin, or orbital order, and more exotic states in low dimensions. One of the most notable consequences of electronic correlations is the much studied metal-insulator (M-I) Mott transition [1,2]. Recently, new theoretical approaches have considerably extended our understanding of this fundamental problem [2,3]. Many physical properties indirectly reflect the dramatic rearrangement of the electronic structure at the transition. Photoelectron spectroscopy, which probes the single-particle spectral function, can provide a direct view of such changes [4 6]. However, comparing samples with different compositions faces materials problems like stoichiometry, defects, and disorder. A quantitative analysis is further complicated by the known surface sensitivity of the technique [7,8]. An ideal experiment would record the energy and momentum-dependent spectrum, while tuning the crucial (W=U) parameter (U is the onsite Coulomb correlation energy; W is the bandwidth) in the same single crystal sample. Remarkably, it is possible to approach this ideal situation exploiting the occurrence of modulated structures [charge-density waves (CDWs)] in appropriate low-dimensional systems. There, the lattice distortion modulates the transfer integrals and therefore modifies the bandwidth. In materials that are close enough to a Mott transition, the reduced bandwidth may lead to an instability. There are strong indications for this scenario in the layered chalcogenide 1T-TaS 2, which presents a sharp order-of-magnitude increase of the resistivity att 18 K [9,1], with a strong rearrangement of the electronic states [11 15]. However, the complex phase diagram of the CDW in 1T-TaS 2 affects the electronic transition, which cannot be considered as a typical Mott transition. Isostructural and isoelectronic 1T-TaSe 2 exhibits a similar CDW, but only one phase below T C 475 K. Its electrical resistivity remains metallic albeit rather large to very low temperatures [9], suggesting that the Se compound lies farther from the instability than the S analog. Nevertheless, a transition could still occur at the crystal surface, where the U=W ratio is expected to be larger as a result of smaller screening and coordination. The surface sensitivity of angle-resolved photoemission (ARPES) is ideal to investigate such an inhomogeneous state. In this Letter we present evidence for a Mott transition at the surface of 1T-TaSe 2. High-resolution temperature-dependent ARPES shows for the first time the disappearance of the coherent quasiparticle signatures at the Fermi surface and the opening of a correlation gap. These results are qualitatively well described by a dynamical mean-field (DMFT) calculation and provide new insight into the spectral properties of the M-I transition. 1T-TaSe 2 has a layered structure, with the d 1 Ta atoms in a distorted octahedral environment. Adjacent layers interact weakly through van der Waals gaps, and all physical properties exhibit a strong anisotropy. A threefold CDW p develops below T C 475 K, with a commensurate 13 13 superstructure, analogous to the much studied low-temperature CDW phase of 1T-TaS 2 [9]. In real space the CDW corresponds to a modulation of the atomic positions, within a 13 Ta atoms unit, in the socalled star-of-david configuration. Extended Huckel calculations [16] suggest that the CDW splits the Ta d conduction band into subbands which contain a total of 13 electrons per unit cell. Two subbands, carrying 6 electrons each, are filled and lie below the Fermi level. The 16641-1 31-97=3=9(16)=16641(4)$2. 23 The American Physical Society 16641-1 67

ÖØ Ð ¾ºËÔ ØÖÓ ÓÔ Ò ØÙÖ Ó ÅÓØØØÖ Ò Ø ÓÒ ØØ ÙÖ Ó ½Ì¹Ì Ë ¾ VOLUME 9, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S week ending 25 APRIL 23 Binding Energy (ev) 1 k F2 k F1 3 K 7 K.5.5 Wave vector (fraction of ΓM) FIG. 1. ARPES intensity maps of 1T-TaSe 2 at T 3 K (left) and T 7 K (right) measured along the M highsymmetry direction (h 21 ev). The arrows mark Fermi level crossings by the Ta d band at 3 K. Fermi surface is formed by a half-filled subband carrying the 13th electron. The opening of a correlation gap in this subband is responsible for the resistivity jump in (bulk) 1T-TaS 2 [1]. We performed high-resolution ARPES measurements in Lausanne and at the Synchrotron Radiation Center, University of Wisconsin. The energy and momentum resolution were E 1 mev and k :2 A 1, and the Fermi level location was determined with an accuracy of 1 mev by measuring the metallic edge of a polycrystalline gold reference. Single crystal samples grown by the usual iodine transport technique were characterized by Laue diffraction and resistivity measurements, which confirmed the assignment to the 1T polytype. They were mounted on the tip of a closedcycle refrigerator and cleaved at a base pressure of 1 1 1 torr. We did not observe any sign of surface degradation or contamination over a typical 8 h run. Figure 1 shows ARPES intensity maps (h 21 ev) measured at 3 and 7 K along the high-symmetry M direction of the hexagonal Brillouin zone. A higher photon energy (h 5 ev) yields similar results. The maps correspond to the same CDW phase, as confirmed by low-energy electron diffraction (LEED), but exhibit remarkable differences. At 3 K the narrow topmost Ta d subband crosses the Fermi level at k F1;2 1=4 M, in good agreement with band structure calculations [17]. The filled CDW subbands are visible at :3eVand, with lower intensity at larger binding energies ( :8eV) and momenta. The overall dispersion of the Ta d band is influenced by the CDW superlattice, as will be discussed elsewhere. The parabolic band with a maximum at and :5eVis a Se p band. At 7 K the Ta d spectral weight is narrower and clearly removed from E F, and a gap has appeared. Bulk sensitive properties and surface sensitive LEED data rule out a structural phase transition between 3 and 7 K. The large spectral changes are therefore the consequence of an electronic surface transition. Signatures of a surface gap in 1T-TaSe 2 were previously observed at 7 K by scanning tunneling spectroscopy (STS) [18]. We characterized this transition by temperaturedependent measurements of the shallow Ta 4f core levels, which exhibit a CDW-induced fine structure [11,19]. We find (not shown) that between 3 and 7 K the CDWsplit components sharpen, and their energy separation increases by 4 mev, in agreement with lower-resolution data [2]. We conclude that the CDW amplitude and the corresponding lattice distortion are larger at the lower temperature. The increased distortion reduces the overlap between the cluster orbitals which form the basis of the band structure in the CDW phase. Calculations which explicitly account for this effect are not available, but the (W=U) ratio is certainly reduced, possibly below the critical value for the M-I transition. This is confirmed by an inspection of the ARPES signal at k k F1 [Fig. 2(a)], which reveals a sudden loss of intensity near E F below 26 K, and the appearance of a strong signal centered at.26 ev. All spectra were normalized to the same integrated area. The intensity at E F [Fig. 2(b)] exhibits a sharp step around 26 K, and a further linear decrease at lower temperature. The intensity and the spectral line shape (not shown) are recovered upon heating, but only at higher temperatures. The large hysteresis ( 8 K) suggests a first-order transition [21]. In Fig. 3, following common practice in ARPES work on the cuprates, we symmetrized the spectra of Fig. 2(a) around E F. This procedure removes the perturbing effect of the Fermi distribution on the intrinsic temperature dependence of the spectral function A k F ;!;T. The 3 K spectrum exhibits a broad ( 1eV) incoherent background and a weak quasiparticle (QP) feature at E F. The QP signal disappears at lower temperature, and spectral weight is transferred to the lower and as ARPES Intensity T (K) 7 22 26 3 a) b) E = E F k = k F.5 =E F 2 3 Binding Energy (ev) Temperature (K) FIG. 2 (color online). (a) ARPES spectra measured atk k F1 between 3 and 7 K; (b) temperature dependence of the ARPES signal at the Fermi surface, showing a sharp break and a large hysteresis. 16641-2 16641-2 68

ÖØ Ð ¾ºËÔ ØÖÓ ÓÔ Ò ØÙÖ Ó ÅÓØØØÖ Ò Ø ÓÒ ØØ ÙÖ Ó ½Ì¹Ì Ë ¾ VOLUME 9, NUMBER 16 ARPES Intensity T (K) 7 22 26 3 P H Y S I C A L R E V I E W L E T T E R S week ending 25 APRIL 23 A ( k,ω, T ) (arb. units) F 3 2.5 2 1.5 1.5.5 =E F -.5 Binding Energy (ev) FIG. 3. Temperature-dependent ARPES spectral function at k k F1 (h 21 ev). The spectra have been obtained from the raw spectra I E by symmetrization around E F : I E I E I E. inferred by symmetry upper sidebands, representing the lower (LHB) and upper (UHB) Hubbard subbands. The integrated intensity is conserved. We emphasize that the width ( 1 mev) of the central peak is much larger than expected for a coherent quasiparticle at the Fermi surface of a good metal. Clearly, in the 22 26 K temperature range, the corresponding excitations are heavily scattered, and their lifetime is short. This bad metal character is consistent with the broad maximum and large value of the electrical resistivity of bulk 1T-TaSe 2 at 25 3 K [9]. The spectra of Fig. 3 are qualitatively consistent with the changes expected at the Mott transition. In order to substantiate this, we have calculated A k F ;!;T for a one-band Hubbard model at half-filling, within the dynamical mean-field theory (DMFT) framework [3]. The iterated perturbation theory approximation [22] was used, and checks were made using quantum Monte Carlo and the maximum entropy method. The Coulomb term U was set at.52 ev, equal to the energy separation between the LHB and UHB features in Fig. 3. A semicircular density of states was used, with a bandwith W assumed to depend linearly on temperature between 3 K (W :5 ev) and 7 K (W :36 ev). These values are only indicative and are not the result of a specific attempt to find an optimum fit to the data, but we note that the overall magnitude of W is consistent with the dispersions observed in Fig. 1. The high-temperature (3 and 26 K) calculated spectra (Fig. 4) correspond to a correlated metal in the incoherent regime, with a broad low-energy peak and two intense Hubbard sidebands. The central peak is strongly reduced at 22 K, and at 7 K it has disappeared completely, leaving two sharp features centered at U=2 and separated by a real gap. The overall shape of the spectra is -.6 -.4 -.2.2.4.6 ω (ev) FIG. 4. Temperature-dependent spectral function A k F ;! of the half-filled Hubbard model calculated within DMFT for the temperatures of Fig. 4. U was fixed at U :52 ev, and the bandwidth is chosen as W :5 ev at 3 K (solid line);.48 ev at 26 K (dashed line);.44 ev at 22 K (dash-dotted line); and.36 ev at 7 K (dotted line). in good qualitative agreement with the data, as well as the dramatic transfer of spectral weight that takes place between the central peak and the Hubbard bands as the temperature is lowered. We observe that in both theory and experiment the spectral weight accumulated in the insulator near the maximum of the HB sidebands comes mainly from the QP central peak, but that a small fraction also comes from energies above.5 ev. This results in two energies at which all spectra approximately cross. The data provide the first direct momentum-resolved observation of two of the key predictions of DMFT regarding the one-particle spectrum through the Mott transition, namely, the three-peak structure in the metallic state [21] and the large transfers of spectral weight from the metal to the insulator [23 25]. The comparison of theory and experiment also reveals some differences. Unlike the calculated spectra, the intensity at E F is never completely suppressed in the experimental spectra. This signal could originate from the underlying metallic bulk or from surface inhomogeneities (metallic patches ) observed by STS [18]. It could also indicate that the tails of the LHB and UHB overlap slightly, a passibility which was considered in a different context [26]. Another quantitative difference is the sharper separation between the QP peak and Hubbard bands in the calculation. This is a known feature of DMFT which is likely to be weakened as dimensionality is lowered. Recent work in particular [27] suggests that long-wavelength charge modes partially fill the preformed gap. From a theoretical standpoint, the main discrepancy concerns the value of the critical temperature at which the 16641-3 16641-3 69

ÖØ Ð ¾ºËÔ ØÖÓ ÓÔ Ò ØÙÖ Ó ÅÓØØØÖ Ò Ø ÓÒ ØØ ÙÖ Ó ½Ì¹Ì Ë ¾ VOLUME 9, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S week ending 25 APRIL 23 first-order metal-insulator transition is observed. Indeed, with the observed value of U, the simple one-band Hubbard model treated within DMFT would have a first-order transition at T U=8 9 K, a factor of 3 smaller than observed experimentally. The trajectory in the [T; W=U ] space used in the theoretical calculation does not intersect the first-order transition line, so that theory would interpret the spectral changes as due to a rapid crossover between a bad metal and a Mott insulator. On the other hand, the experimental observation of a hysteresis does suggest a real transition. In a purely electronic model, it is known that orbital degeneracy does lead to increased T c values [28]. However, DMFT model calculations, including the whole d manifold, show a significant increase in T c only when the first subband is much closer to the Fermi level than in the experiment. Thus, it is unlikely that orbital degeneracy could explain the increased T c. The inadequacy of a purely electronic model is also suggested by the large width of the low-energy QP. In a purely electronic model, this can be accounted for only if the temperature is significantly larger than the critical temperature for the metalinsulator transition. The observation of a broad peak and a true transition strongly suggests that the coupling to lattice degrees of freedom plays an important role, as in fact expected in a CDW compound. Indeed, it has been shown in a simple model that the coupling to the lattice can lead to an increase in T c [29]. Also, it is possible that the observed hysteresis results from differences in the pinning of the CDW upon heating and cooling. Obviously, further investigation both theoretical and experimental is required to clarify these issues. In summary, we presented momentum-resolved high-resolution ARPES data which illustrate the spectral consequences of a bandwidth-controlled surface Mott transition in 1T-TaSe 2. The transition from a bad metal, characterized by a largely incoherent spectrum, to a correlated insulator is qualitatively captured by a DMFT calculation for the half-filled Hubbard model. Quantitative differences between theory and experiment suggest that the model should be extended to include the coupling to lattice degrees of freedom, in order to provide a more accurate description of electronic transitions in such CDW materials. We acknowledge R. Gaal for the resistivity measurements, correspondence with P. Aebi, E. Canadell, F. del Dongo, and the support of G. Margaritondo. This work has been supported by the Swiss NSF through the MaNEP NCCR. A. G. is grateful to F. Mila for discussions and hospitality at UNIL-Lausanne. A. G., S. F., and S. B. are most grateful to KITP-UCSB for the warm hospitality during the final stage of this work, where it was supported in part by NSF under Grant No. PHY99-7949, and also acknowledge computing time at IDRIS- CNRS under Project No. 21393. S. B has been supported by EC Grant No. HPMF-CT-2-658. The Synchrotron Radiation Center, University of Wisconsin Madison, is supported by NSF under Award No. DMR-8442. [1] N. F. Mott, Metal-Insulator Transitions (Taylor & Francis, London, 199). [2] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 7, 139 (1998). [3] For a review, see A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). [4] A. Fujimori et al., Phys. Rev. Lett. 69, 1796 (1992). [5] H. D. Kim et al., cond-mat-1844 (unpublished); J.W. Allen (private communication). [6] S. Shin et al., J. Phys. Soc. Jpn. 64, 123 (1995). [7] K. Maiti et al., Europhys. Lett. 55, 246 (21). [8] S. Suga and A. Sekiyama, J. Electron Spectrosc. Relat. Phenom. 124, 81 (22). [9] J. A. Wilson, F. J. DiSalvo, and S. Mahajan, Adv. Phys. 24, 117 (1975). [1] P. Fazekas and E. Tosatti, Philos. Mag. B 39, 229 (1979). [11] R. A. Pollak et al., Phys. Rev. B 24, 7435 (1981). [12] N.V. Smith, S. D. Kevan, and F. J. DiSalvo, J. Phys. C 18, 3175 (1985). [13] R. Manzke et al., Europhys. Lett. 8, 195 (1989). [14] R. Claessen et al., Phys. Rev. B 41, 827 (199). [15] F. Zwick et al., Phys. Rev. Lett. 81, 158 (1998). [16] M.-W. Whangbo and E. Canadell, J. Am. Chem. Soc. 114, 9587 (1992). [17] P. Aebi et al., J. Electron Spectrosc. Relat. Phenom. 117 118, 433 (21). [18] O. Shiino et al., Appl. Phys. A 66, S175 (1998). [19] H. P. Hughes and J. A. Scarfe, Phys. Rev. Lett. 74, 369 (1995). [2] H. P. Hughes, and R. A. Pollak, Philos. Mag. 34, 125 (1976). [21] A recent ARPES work on 1T-TaSe 2 [K. Horiba et al., Phys. Rev. B 66, 7316 (22)] reported metallic behavior at T 2 K. It is our experience that defects/ disorder rapidly destroy CDW transitions in layered materials. We speculate that subtle sample-related issues may have prevented the observation of the transition. [22] A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992). [23] A. Georges and W. Krauth, Phys. Rev. B 48, 7167 (1993). [24] M. J. Rozenberg, G. Kotliar, and X.Y. Zhang, Phys. Rev. B 49, 1181 (1994). [25] For an early discussion of spectral weight transfers in the cerium - transition, see L. Z. Liu et al., Phys. Rev. B 45, 8934 (1992). [26] D. J. Thouless, J. Phys. (Paris), Colloq. 37, C4-349 (1976). [27] S. Florens and A. Georges (unpublished). [28] S. Florens, A. Georges, G. Kotliar, and O. Parcollet, cond-mat 25263 [Phys. Rev. B (to be published)]. [29] P. Majumdar and H. R. Krishnamurthy, Phys. Rev. Lett. 73, 1525 (1994). 16641-4 16641-4 7

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ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ PHYSICAL REVIEW B 66, 165111 22 Quantum impurity solvers using a slave rotor representation Serge Florens 1,2 and Antoine Georges 1,2 1 Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 5, France 2 Laboratoire de Physique des Solides, Université Paris-Sud, Bât. 51, 9145 Orsay, France Received 1 July 22; revised manuscript received 7 August 22; published 29 October 22 We introduce a representation of electron operators as a product of a spin-carrying fermion and of a phase variable dual to the total charge slave quantum rotor. Based on this representation, a method is proposed for solving multiorbital Anderson quantum impurity models at finite interaction strength U. It consists in a set of coupled integral equations for the auxiliary field Green s functions, which can be derived from a controlled saddle point in the limit of a large number of field components. In contrast to some finite-u extensions of the noncrossing approximation, the method provides a smooth interpolation between the atomic limit and the weak-coupling limit, and does not display violation of causality at low frequency. We demonstrate that this impurity solver can be applied in the context of dynamical mean-field theory, at or close to half-filling. Good agreement with established results on the Mott transition is found, and large values of the orbital degeneracy can be investigated at low computational cost. DOI: 1.113/PhysRevB.66.165111 PACS number s : 71.27. a, 71.3. h I. INTRODUCTION The Anderson quantum impurity model AIM and its generalizations play a key role in several recent developments in the field of strongly correlated electron systems. In single-electron devices, it has been widely used as a simplified model for the competition between the Coulomb blockade and the effect of tunneling. 1 In a different context, the dynamical mean-field theory DMFT of strongly correlated electron systems replaces a spatially extended system by an Anderson impurity model with a self-consistently determined bath of conduction electrons. 2,3 Naturally, the AIM is also essential to our understanding of local moment formation in metals and to that of heavy-fermion materials, particularly in the mixed-valence regime. 4 It is therefore important to have at our disposal quantitative tools allowing for the calculation of physical quantities associated with the AIM. The quantity of interest depends on the specific context. Many recent applications require a calculation of the localized level Green s function or spectral function and possibly of some two-particle correlation functions. Many such impurity solvers have been developed over the years. Broadly speaking, these methods fall in two categories: numerical algorithms which attempt at a direct solution on the computer and analytical approximation schemes which may also require some numerical implementation. Among the former, the quantum Monte Carlo QMC approach based on the Hirsch-Fye algorithm and the numerical renormalization group play a prominent role. However, such methods become increasingly costly as the complexity of the model increases. In particular, the rapidly developing field of applications of DMFT to electronic structure calculations 5 requires methods that can handle impurity models involving orbital degeneracy. For example, materials involving f electrons are a real challenge to DMFT calculations when using the quantum Monte Carlo approach. Similarly, cluster extensions of DMFT require one to handle a large number of correlated local degrees of freedom. The dimension of the Hilbert space grows exponentially with the number of these local degrees of freedom, and this is a severe limitation to all methods, particularly the exact diagonalization and numerical renormalization group methods. The development of fast and accurate although approximate impurity solvers is therefore essential. 31 In the oneorbital case, the iterated perturbation theory IPT approximation 6 has played a key role in the development of DMFT, particularly in elucidating the nature of the Mott transition. 7 A key reason for the success of this method is that it becomes exact both in the limit of weak interactions and in the atomic limit. Unfortunately, however, extensions of this approach to the multiorbital case have not been as successful. 8 Another widely employed method is the noncrossing approximation NCA. 9,1 This method takes the atomic limit as a starting point and performs a self-consistent resummation of the perturbation theory in the hybridization to the conduction bath. It is thus intrinsically a strongcoupling approach, and indeed it is in the strong-coupling regime that the NCA has been most successfully applied. The NCA does suffer from some limitations, however, which can become severe for some specific applications. These limitations, are of two kinds. i The low-energy behavior of NCA integral equations is well known to display non-fermi-liquid power laws. This can be better understood when formulating the NCA approach in terms of slave bosons. 11,12 It becomes clear then that the NCA actually describes accurately the overscreened regime of multichannel models. This is in a sense a remarkable success of the NCA, but also calls for some care when applying the NCA to Fermi-liquid systems in the screened regime. It must be noted that recent progresses have been made to improve the low-energy behavior in the Fermi-liquid case. 13 ii A more important limitation for practical applications has to do with the finite-u extensions of NCA equations. Standard extensions do not reproduce correctly the noninteracting U limit. In contrast to IPT, they are not interpolative solvers between the weak-coupling and strong- 163-1829/22/66 16 /165111 16 /$2. 66 165111-1 22 The American Physical Society 113

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 coupling limits. Furthermore, the physical self-energy tends to develop noncausal behavior at low frequency i.e., d ( ) ] below some temperature the NCA pathology At half-filling and large U, this only happens at a rather low-energy scale, but away from half-filling or for smaller U, this scale can become comparable to the bandwidth, making the finite-u NCA of limited applicability. In this article, we introduce new impurity solvers which overcome some of these difficulties. Our method is based on a rather general representation of strongly correlated electron systems, which has potential applications to lattice models as well. 27 The general idea is to introduce a new slave particle representation of physical electron operators, which emphasizes the phase variable dual to the total charge on the impurity. This should be contrasted to slave boson approaches to a multiorbital AIM: there, one introduces as many auxiliary bosons as there are fermion states in the local Hilbert space. This is far from economical: when the local interaction depends on the total charge only, it should be possible to identify a collective variable which provides a minimal set of collective slave fields. We propose that the phase dual to the total charge precisely plays this role. This turns a correlated electron model at finite U) into a model of spin- and orbital- carrying fermions coupled to a quantum rotor degree of freedom. Various types of approximations can then be made on this model. In this article, we emphasize an approximate treatment based on a -model representation of the rotor degree of freedom, which is then solved in the limit of a large number of components. This results in coupled integral equations which share some similarities to those of the NCA, but do provide the following improvements. i The noninteracting (U ) limit is reproduced exactly. For a fully symmetric multiorbital model at half-filling and in the low-temperature range, the atomic limit is also captured exactly, so that the proposed impurity solver is an interpolative scheme between weak and strong coupling. The -model approximation does not treat as nicely the atomic limit far from half-filling, however though improvements are possible; see Sec. V E. ii The physical self-energy does not display any violation of causality even at low temperature or small U). This is guaranteed by the fact that our integral equations become exact in the formal limit of a large number of orbitals and components of the -model field. It should be emphasized, though, that the low-energy behavior of our equations is similar to the infinite-u) NCA and characterized by non-fermi-liquid power laws below some low-energy scale. As a testing ground for the new solver, we apply it in this paper to the DMFT treatment of the Mott transition in the multiorbital Hubbard model. We find an overall very good agreement with the general aspects of this problem, as known from numerical work and from some recently derived exact results. We also compare to other impurity solvers IPT, exact diagonalization, QMC, NCA, particularly regarding the one-electron spectral function. This article is organized as follows. In Sec. II, we introduce a representation of fermion operators in terms of the phase variable dual to the total charge taking the finite-u, multiorbital Anderson impurity model as an example. In Sec. III, a -model representation of this phase variable is introduced, along with a generalization from O(2) to O(2M). In the limit of a large number of components, a set of coupled integral equations is derived. In Sec. IV, we test this dynamical slave rotor DSR approach on the singleimpurity Anderson model. The rest of the paper Sec. V is devoted to applications of this approach in the context of DMFT, which puts in perspective the advantages and limitations of the new method. II. ROTORIZATION The present article emphasizes the role played by the total electron charge and its conjugate phase variable. We introduce a representation of the physical electron in terms of two auxiliary fields: a fermion field which carries spin and orbital degrees of freedom and the total charge raising and lowering operators which we represent in terms of a phase degree of freedom. The latter plays a role similar to a slave boson: here a slave O(2) quantum rotor is used rather than a conventional bosonic field. A. Atomic model We first explain this construction on the simple example of an atomic problem, consisting in an N-fold degenerate atomic level subject to a local SU(N)-symmetric Coulomb repulsion: H local d d U 2 d d N 2. 1 2 We use here d U/2 as a convenient redefinition of the impurity level, and we recast the spin and orbital degrees of freedom into a single index 1,...,N for N 2, we have a single orbital with two possible spin states, ). We note that is zero at half filling, due to particle-hole symmetry. The crucial point is that the spectrum of the atomic Hamiltonian 1 depends only on the total fermionic charge Q,...,N and has a simple quadratic dependence on Q: E Q Q U 2 Q N 2 2. 2 There are 2 N states, but only N 1 different energy levels, with degeneracies ( Q N ). In conventional slave boson methods, 14,15 a bosonic field is introduced for each atomic state 1 Q along with spin-carrying auxiliary fermions f ). Hence, these methods are not describing the atomic spectrum in a very economical manner. The spectrum of Eq. 1 can actually be reproduced by introducing, besides the set of auxiliary fermions f, a single additional variable namely, the angular momentum Lˆ i / associated with a quantum O(2) rotor i.e., an angular variable in,2 ). Indeed, the energy levels 2 can be obtained using the Hamiltonian 165111-2 114

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... PHYSICAL REVIEW B 66, 165111 22 H local f f U 2 Lˆ 2. 3 H f f U 2 Lˆ 2 k c k, k, c k, V k c k, f e i k, A constraint must be imposed which ensures that the total number of fermions is equal to the O(2) angular momentum up to a shift : f f 1 4 2. Lˆ This restricts the allowed values of the angular momentum to be l Q N/2 N/2, N/2 1,...,N/2 1, N/2, while in the absence of any constraint l can be an arbitrary positive or negative integer. The spectrum of Eq. 3 is Q Ul 2 /2, with l Q N/2 thanks to Eq. 4, so that it coincides with Eq. 2. To be complete, we must show that each state in the Hilbert space can be constructed in terms of these auxiliary degrees of freedom, in a way compatible with the Pauli principle. This is achieved by the following identification: 1 Q d 1 Q f l Q N/2, in which 1 Q d, f denotes the antisymmetric fermion state built out of d and f fermions, respectively, and l denotes the quantum rotor eigenstate with angular momentum l, i.e., l e il. The creation of a physical electron with spin corresponds to acting on such a state with f as well as raising the total charge angular momentum by one unit. Since the raising operator is e i, this leads to the representation d f e i, d f e i. Let us illustrate this for N 2 by writing the four possible states in the form d f, d f, d f 1, and d f 1 and showing that this structure is preserved by d f e i. Indeed, d d d f f e i f 1. The key advantage of the quantum rotor representation is that the original quartic interaction between fermions has been replaced in Eq. 3 by a simple kinetic term (Lˆ 2) for the phase field. B. Rotor representation of Anderson impurity models We now turn to an SU(N)-symmetric Anderson impurity model in which the atomic orbital is coupled to a conduction electron bath: H d d U 2 d d N 2 2 V k c k, d d c k,. k, k c k, k, c k, Using the representation 6, we can rewrite this Hamiltonian in terms of the ( f, ) fields only: 5 6 7 8 f c k, e i. We then set up a functional integral formalism for the f and degrees of freedom, and derive the action associated with Eq. 1. This is simply done by switching from phase and angular momentum operators (,Lˆ ) to fields (, ) depending on imaginary time,. The action is constructed from S d il H f f, and an integration over Lˆ is performed. It is also necessary to introduce a complex Lagrange multiplier h in order to implement the constraint Lˆ f f N/2. We note that, because of the charge conservation on the local impurity, h can be chosen to be independent of time, with ih,2 /. This leads to the following expression of the action: S d f h f ih 2 N 2U 2 h c k, k c k, V k c k, f e i H.c.. k, 9 1 We can recast this formula in a more compact form by introducing the hybridization function i k V k 2 i k 11 and integrating out the conduction electron bath. This leads to the final form of the action of the SU(N) Anderson impurity model in terms of the auxiliary fermions and phase field: S d f h f ih 2 N 2U 2 h d d f f e i ( ) i ( ). C. Slave rotors, Hubbard-Stratonovich fields, and gauge transformations 12 In this section, we present an alternative derivation of the expression 12 of the action which does not rely on the concept of slave particles. This has the merit of giving a more explicit interpretation of the phase variable introduced above by relating it to a Hubbard-Stratonovich decoupling field. This section is, however, not essential to the rest of the paper and can be skipped upon first reading. Let us start with the imaginary-time action of the Anderson impurity model in terms of the physical electron field for the impurity orbital: 165111-3 115

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES S d d d U 2 d d N 2 2 d d d d. 13 PHYSICAL REVIEW B 66, 165111 22 This is exactly expression 12, with the identification / ih. This, together with Eq. 17, provides an explicit relation between the quantum rotor and Lagrange multiplier fields on one side, and the Hubbard-Stratonovich field conjugate to the total charge, on the other. Because we have chosen an SU(N)-symmetric form for the Coulomb interaction, we can decouple it with only one bosonic Hubbard-Stratonovitch field ( ): S d d i d 2 2U in 2 d d d d. 14 Hence, a linear coupling of the field ( ) to the fermions has been introduced. The idea is now to eliminate this linear coupling for all the Fourier modes of the field, except that corresponding to zero frequency: 2. This can be achieved by performing the following gauge transformation: d f exp i exp i /. 15 The reason for the second phase factor in this expression is that it guarantees that the new fermion field f also obeys antiperiodic boundary conditions in the path integral. It is easy to check that this change of variables does not provide any Jacobian, so that the action simply reads S d f i f 2 2U i N 2 We now set d d exp i. 1 f f with 16 2 17 and notice that the field ( ) has the boundary condition ( ) () 2. It therefore corresponds to an O(2) quantum rotor, and the expression of the action finally reads S d f i f 2 2U i N 2 d d f f e i ( ) i ( ). 18 III. SIGMA-MODEL REPRESENTATION AND SOLUTION IN THE LIMIT OF MANY COMPONENTS A. From quantum rotors to a model Instead of using a phase field to represent the O(2) degree of freedom, one can use a constrained complex bosonic field X e i with X 2 1. 19 The action 12 can be rewritten in terms of this field, provided a Lagrange multiplier field ( ) is used to implement this constraint: S d f h f N 2 h2 X 2 h d 2U 2U h 2U X* X H.c. X 2 1 d d f f X X*. 2 Hence, the Anderson model has been written as a theory of auxiliary fermions coupled to a nonlinear O(2) model, with a constraint implemented by h) relating the fermions and the -model field X( ). A widely used limit in which models become solvable is the limit of a large number of components of the field. This motivates us to generalize Eq. 2 to a model with an O(2M) symmetry. The bosonic field X is thus extended to an M-component complex field X ( 1,...,M) with X 2 M. The corresponding action reads S d f h f N 2 h M h2 2U M d X 2 2U h 2U X* X H.c. X 2 d 1 d M, f f X X *. Let us note that this action corresponds to the Hamiltonian 165111-4 116

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... PHYSICAL REVIEW B 66, 165111 22 H f f U 2 Lˆ 2M, k c k,, k, k,, V k M c k, f X* f c k, X. c k, 21 In this expression, Lˆ, denotes the angular momentum tensor associated with the X vector. The Hamiltonian 21 is a generalization of the SU(N) O(2) U(N) Anderson impurity model to an SU(N) O(2M) model in which the total electronic charge is associated with a specific component of Lˆ. It reduces to the usual Anderson model for M 1. In the following, we consider the limit where both N and M become large, while keeping a fixed ratio N/M. We shall demonstrate that exact coupled integral equations can be derived in this limit, which determine the Green s functions of the fermionic and -model fields and the physical electron Green s function as well. The fact that these coupled integral equations do correspond to the exact solution of a welldefined Hamiltonian model Eq. 21 guarantees that no unphysical features like, e.g., violation of causality arise in the solution. Naturally, the generalized Hamiltonian 21 is a formal extension of the Anderson impurity model of physical interest. Extending the charge symmetry from O(2) to O(2M) is not entirely innocuous, even at the atomic level: as we shall see below, the energy levels of a single O(2M) quantum rotor have multiple degeneracies and depend on the charge angular momentum quantum number in a way which does not faithfully mimic the O(2) case. Nevertheless, the basic features defining the generalized model a localized orbital subject to a Coulomb charging energy and coupled to an electron bath by hybridization are similar to the original model of physical interest. B. Integral equations In this section, we derive coupled integral equations which become exact in the limit where both M and N are large with a fixed ratio: N N M N,M. 22 Following Ref. 12 see also Ref. 11, the two-body interaction between the auxiliary fermions and the -model fields is decoupled using bosonic bilocal fields Q(, ) and Q (, ) depending on two times. Hence, we consider the action S d f h f N 2 M h2 h M d 2U X 2 2U h 2U X* X H.c. X 2 d Q, Q, d M d d Q, X X* d d Q, f f. 23 In the limit 22, this action is controlled by a saddle point, at which the Lagrange multipliers take static expectation values h and, while the saddle-point values of the Q and Q fields are translation-invariant functions of time Q( ) and Q ( ). Introducing the imaginary-time Green s functions of the auxiliary fermion and -model fields as G f T f f, G X T X X *, 24 25 we differentiate the effective action 23 with respect to Q( ) and Q ( ), which leads to the following saddle-point equations Q ( ) N ( )G f ( ) and Q( ) ( )G X ( ). The functions Q(i n )( f ) and Q *(i n ) ( X ) define fermionic and bosonic self-energies: G f 1 i n i n h f i n, 26 G X 1 i n n 2 U 2ih n U X i n, 27 where n ( n ) is a fermionic bosonic Matsubara frequency. The saddle-point equations read X N G f, f G X, together with the constraints associated with h and : G f 1 2 2h NU G X 1, 1 NU G X G X. 28 29 3 31 165111-5 117

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 There is a clear similarity between the structure of these coupled integral equations and the infinite-u NCA equations. 9 We note also significant differences, such as the constraint equations. Furthermore, the finite value of the Coulomb repulsion U enters the bosonic propagator 27 in a quite novel manner. The two key ingredients on which the present method are based is the use of a slave rotor representation of fermion operators and the use of integral equations for the frequencydependent self-energies and Green s functions. For this reason, we shall denote the integral equations above under the name of dynamical slave rotor method in the following. C. Some remarks We make here some technical remarks concerning these integral equations. First, we clarify how the interaction parameter U was scaled in order to obtain the DSR equations above. This issue is related to the manner in which the atomic limit ( ) is treated in this method. In the original O(2) atomic Hamiltonian 1, the charge gap between the ground state and the first excited state is U/2 at half-filling ( ). In the DSR method, the charge gap is associated with the gap in the slave rotor spectrum. If the O(2M) generalization of Eq. 1 is written as in Eq. 21, H int U 2 Lˆ 2M,, 32, the spectrum reads E l Ul(l 2M 2)/(2M). As a result, the energy difference from the ground state to the first excited state is E 1 E U(2M 1)/(2M) U at large M, whereas it is U/2 at M 1. In order to use the DSR method in practice as an approximate impurity solver, the parameter U should thus be normalized in a different way than in Eq. 21, so that the gap is kept equal to U/2 in the large-m limit as well. Technically this can be enforced by choosing the normalization U 2 Lˆ 4M 2,, H int 33 instead of Eq. 32. Note that this scaling coincides with Eq. 32 for M 1, but does yield E 1 E U/2 for large M, as desired. This definition of U was actually used when writing the saddle-point integral equations 27, although we postponed the discussion of this point to the present section for reasons of simplicity. Let us elaborate further on the accuracy of the DSR integral equations in the atomic limit. In Appendix A, we show that the physical electron spectral function obtained within DSR in the atomic limit coincides with the exact O(2) result at half-filling and at T. This is a nontrivial result, given the fact that the constraint is treated on average and the above remark on the spectrum. In contrast to the NCA, the DSR method in its present form is not based by construction on a strong-coupling expansion around the exact atomic spectrum, so that this is a crucial check for the applicability of this method in practice. In the context of DMFT, for example, it is essential in order to describe correctly the Mott insulating state. 7 However, the DSR integral equations fail to reproduce exactly the O 2 atomic limit off half-filling, as explained in Appendix A. Deviations become severe for too high dopings, as discussed in Sec. V E. This makes the present form of DSR applicable only for systems in the vicinity of half-filling. We now discuss some general spectral properties of the DSR solver. From the representation of the physical electron field d f X and from the convention chosen for the pseudoparticle Green s functions 24 and 25, the oneelectron physical Green s function is simply expressed as G d G f G X. 34 Therefore Eq. 3, combined with the fact that f has a ( 1) discontinuity at which is obvious from Eq. 26, shows that d possesses also a ( 1) jump at zero imaginary time. This ensures that the physical spectral weight is unity in our theory and thus that physical spectral function are correctly normalized. Because the DSR integral equations result from a controlled large-n,m limit, it also ensures that the physical self-energy always has the correct sign i.e., Im d ( i ) ]. This is not the case 16 for the finite-u version of the NCA, 1 which is constructed as a resummation of the strong-coupling expansion in the hybridization function ( ). Strong-coupling resummations generically suffer from noncausality; see also Ref. 17 for an illustration. Finally, we comment on the noninteracting limit U. This is a major failure of the usual NCA, which limits its applicability in the weakly correlated regime. In the DSR formalism, this limit is exact as can be noticed from Eq. 27. Indeed, as U vanishes, only the zero-frequency component of G X (i n ) survives, so that G X ( ) simply becomes a constant. Because of the constraint 3, we get correctly G X ( ) 1 at U. From Eqs. 29 and 34, this proves that G d ( ) is the noninteracting Green s function: G U 1 d i n i n i n. 35 We finally acknowledge that an alternative dynamical approximation to the finite-u Anderson model 13 was recently developed as an extension of the NCA by Kroha and Wölfle a conventional slave boson representation was used in this work. Much progress has been made following this method, but to the authors knowledge, this technique has not yet been implemented in the context of DMFT one of the reasons is its computational cost. By developing the DSR approximation, we pursue a rather complementary goal: the aim here is not to improve the low-energy singularities usually encountered with integral equations, but rather to have a fast and efficient solver which reproduces correctly the main features of the spectral functions and interpolates between weak and strong coupling. In that sense, it is very well adapted to the DMFT context. 165111-6 118

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... IV. APPLICATION TO THE SINGLE-IMPURITY ANDERSON MODEL PHYSICAL REVIEW B 66, 165111 22 We now discuss the application of the DSR in the simplest setting: that of a single impurity hybridized with a fixed bath of conduction electrons. For simplicity, we focus on the half-filled, particle-hole-symmetric case, which implies h. The doped or mixed valence case will be addressed in the next section, in the context of DMFT. As the strength of the Coulomb interaction U is increased from weak to strong coupling, two well-known effects are expected see, e.g., Ref. 4. First, the width of the low-energy resonance is reduced from its noninteracting value (). As one enters the Kondo regime U with the conduction electrons bandwidth, this width becomes a very small energy scale, of the order of the Kondo temperature: T K 2U exp U/ 8. 36 A local Fermi-liquid description applies, with quasiparticles having a large effective mass and small weight: Z m/m* T K /. The impurity spin is screened for T T K. Second, the corresponding spectral weight is transferred to high energies, into Hubbard bands associated to the atomiclike transitions adding or removing an electron into the half-filled impurity orbital, broadened by the hybridization to the conduction electron bath. The suppression of the low-energy spectral weight corresponds to the suppression of the charge fluctuations on the local orbital. These satellites are already visible at moderate values of the coupling U/. As temperature is increased from T T K to T T K, the Kondo quasiparticle resonance is quickly destroyed, and the missing spectral weight is added to the Hubbard bands. The aim of this section is to investigate whether the integral equations introduced in this paper reproduce these physical effects in a satisfactory manner. A. Spectral functions We have solved numerically these integral equations by iteration, both on the imaginary axis and for real frequencies. Working on the imaginary axis is technically much easier. A discretization of the interval, is used with typically 8192 points and up to 32 768 for reaching the lowest temperatures, as well as fast fourier transforms for the Green s functions. Searching by dichotomy for the saddle-point value of the Lagrange multiplier Eq. 3 is conveniently implemented at each step of the iterative procedure. Technical details about the analytic continuation of Eqs. 28 31 to real frequencies, as well as their numerical solution, are given in Appendix B. In Fig. 1, we display our results for the impurity-orbital spectral function d ( ), at three different temperatures the density of states of the conduction electron bath is chosen as a semicircle with half-width 6, the resonant level width is.16, and we take U 2). The growth of the Kondo resonance as the temperature is lowered is clearly seen. The temperatures in Fig. 1 have been chosen such as to illustrate FIG. 1. d-level spectral function d ( ) for fixed U 2 and inverse temperatures 4,2,2, with the conduction electron bath described in the text. three different regimes: for T T K, no resonance is seen and the spectral density displays a pseudogap separating the two high-energy bands; for T T K, transfer of spectral weight to low energy is seen, resulting in a fully developed Kondo resonance for T T K. We have not obtained an analytical determination of the Kondo temperature within the present scheme. In the case of NCA equations, it is possible to derive a set of differential equations in the limit of infinite bandwidth which greatly facilitates this. This procedure cannot be applied here, because of the form 27 of the boson propagator. Nevertheless, we checked that the numerical estimates of the width of the Kondo peak are indeed exponentially small in U as in formula 36 ; see Fig. 2. However, because U is normalized as in Eq. 33 which gives the correct atomic limit, the prefactor inside the exponential appears to be twice too small. In Fig. 3, we display the spectral function for a fixed low temperature and increasing values of U. The strong reduction of the Kondo scale resonance width upon increasing U is clear in this figure. We note that the high-energy peaks have a width which remains of order, independently of U, which is satisfactory. However, we also note that they are not peaked exactly at the atomic value U/2, which might be an artifact of these integral equations. The shift is rather small, however. FIG. 2. Kondo temperature T K from the exact formula line and from the numerical solution of the DSR equations dots. Units of energy are such that.16. 165111-7 119

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 FIG. 3. d ( ) at low temperature ( 6) and for increasing U 1, 2, 3. Figure 4 illustrates how the high- and low-energy features in the d-level spectral functions are associated with corresponding features in the auxiliary particle spectral functions f and X. In particular, the -model boson slave rotor is entirely responsible for the Hubbard bands at high energy as expected, since it describes charge fluctuations. As stressed in the Introduction, an advantage of our scheme is that the d-level self-energy is always causal, even for small U or large doping. This is definitely an improvement as compared to the usual U-NCA approximation. This is illustrated by Fig. 5, from which it is also clear that d decreases and eventually vanishes as U goes to zero for a more detailed discussion and comparison to the NCA, see Sec. III. However, it is also clear from this figure that the low-energy behavior of the self-energy is not consistent with Fermi-liquid theory. This is a generic drawback of NCA-like integral equation approaches, which we now discuss in more detail. B. Low-energy behavior and Friedel sum rule We discuss here the low-energy behavior of the integral equations in the case of a featureless conduction electron bath, for both the d-level and auxiliary field Green s functions. As explained below, this low-energy behavior depends sensitively on the ratio N N/M which is kept fixed in the FIG. 4. Pseudoparticle spectral function at U 2 and 6. The Kondo resonance is visible in f ( ), dashed curve, whereas X ( ) displays higher-energy features, solid curve. FIG. 5. Imaginary part of the physical self-energy on the realfrequency axis, for U.5 upper curve and U 2 lower curve at 6. limit considered in this paper. The calculations are detailed in Appendix C, where we establish the following. i Friedel sum rule. The zero-frequency value of the d-level spectral function at T is independent of U and reads d T 1 /2 N 1 tan 2 N. N 1 37 This is to be contrasted with the exact value for the O(2) model (M 1), which is independent of N and reads exact d T 1 M 1, any N. 38 This is the Friedel sum rule, 4,18 which in this particle-holesymmetric case simply follows from the Fermi-liquid requirement that the inverse lifetime d ( ) should vanish at T since G d 1 i (i ) d (i )]. As a result, d () is pinned at its noninteracting value. The integral equations discussed here yield a nonvanishing d ( ) albeit always negative in order to satisfy causality and hence do not describe a Fermi-liquid at low energy. The result 37 is identical to that found in the NCA for U, but holds here for arbitrary U. There is actually no contradiction between this remark and the fact that our integral equations yield the exact spectral function in the U limit. Indeed, the limit U at finite T, does not commute with,t at finite U in which Eq. 37 holds. ii Low-frequency behavior. The auxiliary particle spectral functions have a low-frequency singularity characterized by exponents which depend continuously on N as in the U NCA : f ( ) 1/ f ( ) 1/ f, X ( ) 1/ X ( ) sgn( )/ X, with f 1 X 1/(N 1). These behaviors are characterized more precisely in Appendix C. A power-law behavior is also found for the physical self-energy d ( ) d () at low frequency as evident from Fig. 5. Let us comment on the origin of these low-energy features, as well as on their consequences for practical uses of the present method. First, it is clear from expression 21 that the Anderson impurity Hamiltonian generalized to SU(N) O(2M) actu- 165111-8 12

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... PHYSICAL REVIEW B 66, 165111 22 ally involves M channels of conduction electrons. Hence, the non-fermi-liquid behavior found when solving the integral equations associated with the N,M limit simply follows from the fact that multichannel models lead to overscreening of the impurity spin and correspond to a non-fermi-liquid fixed point. In that sense, these integral equations reproduce very accurately the expected low-energy physics, as previously studied for the simplest case of the Kondo model in Refs. 11 and 12. Naturally, this means that the use of such integral equations to describe the one-channel exactly screened case becomes problematic in the low-energy region. In particular, the exact Friedel sum rule is violated, the d-level lifetime remains finite at low energy, and non-fermi-liquid singularities are found. While the approach is reasonable in order to reproduce the overall features of the one-electron spectra, it should not be employed to calculate transport properties at low energy, for example. We note, however, that the deviation from the exact Friedel sum rule vanishes in the N limit. This is expected from the fact that in this limit the number of channels M is small as compared to orbital degeneracy (N). The violation of the sum rule remains rather small even for reasonable values of N. This parameter can actually be used as an adjustable parameter when using the present method as an approximate impurity solver. There is no fundamental reason for which N N should provide the best approximate description of the spectral functions of the one-channel case. We shall use this possibility when applying this method in the DMFT context in the next section: there, we choose N 3 in order to adjust to the known critical value of U for a single orbital. We also used this value in the calculations reported above. The zero-frequency value of the spectral density is thus d () 1.85, while the Friedel sum rule would yield d () 1.95 cf. Fig. 3. Hence the violation of the sum rule is a small effect of the order of 5%, comparable to the one found with numerically exact solvers, due to discretization errors. 19 Also, we point out that the pinning of d () at a value independent of U albeit not that of the Friedel sum rule is an important aspect of the present method, which will prove to be crucial in the context of DMFT in order to recover the correct scenario for the Mott transition. Finally, we emphasize that increasing the parameter N also corresponds to increasing the orbital degeneracy of the impurity level. This will be studied in more details in Sec. V D. In particular, we shall see that correlation effects become weaker as N is increased for a given value of U), due to enhanced orbital fluctuations. V. APPLICATIONS TO DYNAMICAL MEAN-FIELD THEORY AND THE MOTT TRANSITION A. One-orbital case: Mott transition, phase diagram Dynamical mean-field theory has led to significant progress in our understanding of the physics of a correlated metal close to the Mott transition. 2 The detailed description of this transition itself within DMFT is now well established. 2,7,15,19 22 In this section, we use these established results as a benchmark and test the applicability of the FIG. 6. Local spectral function at 4 and U 1,2,3 for the half-filled Hubbard model within DMFT, as obtained with the DSR solver. method introduced in this paper in the context of DMFT, with very encouraging results. As explained above, this is particularly relevant in view of the recent applications of DMFT to electronic structure calculations of correlated solids, 5 which call for efficient multiorbital impurity solvers. As is well known, 2,3,6 DMFT maps a lattice Hamiltonian onto a self-consistent quantum impurity model. We discuss first the half-filled Hubbard model and address later the doped case. We then have to solve a particle-hole-symmetric Anderson impurity model: S d d d U 2 d d 1 2 d d d d, 39 subject to the self-consistency condition t 2 G d. 4 In this expression, a semicircular density of states with halfbandwidth D 2t has been considered, corresponding to an infinite-connectivity Bethe lattice (z ) with hopping t ij t/ z. In the following, we shall generally express all energies in units of D(D 1). In practice, one must iterate numerically the DMFT loop : ( ) G d ( ) ( ) new t 2 G d, using some impurity solver. Here, we make use of the integral equations 28 31. The hybridization function ( ) being determined by the self-consistency condition 4, there are only two free parameters, the local Coulomb repulsion U and the temperature T normalized by D). We display in Fig. 6 the spectral functions obtained at low temperature, for increasing values of U, and in Fig. 7 the corresponding phase diagram. The value of the parameter N has been adapted to the description of the one-orbital case see below. The most important point is that we find a coexistence region at low enough temperature: for a range of couplings U c1 (T) U U c2 (T), both a metallic solution and an insulating solution of the paramagnetic DMFT equations exist. The Mott transition is thus first order at finite temperatures. This is in agreement with the results established for 165111-9 121

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 FIG. 7. Single-orbital paramagnetic phase diagram at halffilling. Squares indicate the exact diagonalization ED result, the DSR result is the solid line, and IPT the dashed line. this problem by solving the DMFT equations with controlled numerical methods, 19,21 as well as with analytical results. 2,22 In particular, the spectral functions that we obtain Fig. 6 display the well-known separation of energy scales found within DMFT: there is a gradual narrowing of the quasiparticle peak, together with a preformed Mott gap at the transition. In the next section, we compare these spectral functions to those obtained using other approximate solvers. As pointed out there, despite some formal similarity in the method, it is well known that the standard U-NCA does not reproduce correctly this separation of energy scales close to the transition. 23 Let us explain how the parameter N has been chosen in these calculations. As demonstrated below, the values of the critical couplings U c1 and U c2 and hence the whole phase diagram and coexistence window strongly depend on the value of this parameter. This is expected, since N is a measure of orbital degeneracy. What has been done in the calculations displayed above is to choose N in such a way that the known value 2 U c2 (T ) 2.9 of the critical coupling, at which the T metallic solution disappears in the singleorbital case, is accurately reproduced. We found that this requires N 3 note that N/M 2 in the one-orbital case, so that the best agreement is not found by a naive application of the large N,M limit. This value being fixed, we find a critical coupling U c1 (T ) 2.3 in good agreement with the value from adaptative exact diagonalizations U c1 2.4. The whole domain of coexistence in the (U,T) plane is also in good agreement with established results in particular we find the critical end point at T c 1/3, while QMC yields T c 1/4). These are very stringent tests of the applicability of the present method, since we have allowed ourselves to use only one adjustable parameter (N). In Sec. V D, we study how the Mott transition depends on the number of orbitals, which further validates the procedure followed here. B. Comparison to other impurity solvers: Spectral functions Let us now compare the spectral functions obtained by the present method with other impurity solvers commonly used for solving the DMFT equations. We start with the iterated perturbation theory approximation and the exact diagonalization method. Both methods have played a major role in the FIG. 8. Comparison between DSR dashed line, IPT solid line, and ED dotted line at U 2.4 and 6. early developments of the DMFT approach to the Mott transition. 7 A comparison of the spectral functions obtained by the present method to those obtained with IPT and ED is displayed in Fig. 8, for a value of U corresponding to a correlated metal close to the transition. The overall shape and characteristic features of the spectral function are quite similar for the three methods. A narrow quasiparticle peak is formed, together with Hubbard bands, and there is a clear separation of energy scales between the width of the central peak related to the quasiparticle weight and the preformed Mott pseudo gap associated with the Hubbard bands: this is a distinguishing aspect of DMFT. It is crucial for an approximate solver to reproduce this separation of energy scales in order to yield a correct description of the Mott transition and phase diagram. There are of course some differences between the three methods, on which we now comment. First, we note that the IPT approximation has a somewhat larger quasiparticle bandwidth. This is because the transition point U c2 is overestimated within IPT cf. Fig. 7, so that a more fair comparison should perhaps be made at fixed U/U c2. It is true, however, that the DSR method has a tendency to underestimate the quasiparticle bandwidth and particularly at smaller values of U. Accordingly, the Hubbard bands have a somewhat too large spectral weight, but are correctly located in first approximation. The detailed shape of the Hubbard bands is not very accurately known, in any case. The ED method involves a broadening of the -function peaks obtained by diagonalizing the impurity Hamiltonian with a limited number of effective orbitals, so that the high-energy behavior is not very accurate on the real axis. This is also true, actually, of the more sophisticated numerical renormalization group. We emphasize that, since the DSR method does not have the correct low-frequency Fermi-liquid behavior, the quasiparticle bandwidth should be interpreted as the width of the central peak in d ( ) while the quasiparticle weight Z cannot be defined formally. In Figs. 9 and 1, we display the spectral functions obtained by using the NCA method extended at finite U in the simplest manner. It is clear that the U-NCA underestimates considerably the quasiparticle bandwidth and thus yields a Mott transition at a rather low value of the coupling. This is not very surprising, since this method is based on a strongcoupling expansion around the atomic limit, and one could 165111-1 122

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... PHYSICAL REVIEW B 66, 165111 22 FIG. 9. U-NCA spectral function for U 1.5 at low temperature. think again of a comparison for fixed U/U c. More importantly, the U-NCA misses the important separation of energy scales between the central peak and the Mott gap. As a result, it does not reproduce correctly the phase diagram for the Mott transition within DMFT in particular regarding coexistence. A related key observation is that the U-NCA does not have the correct weak-coupling small-u) limit. To illustrate this point, we display in Fig. 1 the U-NCA and DSR spectral functions for a tiny value of the interaction U/D.1: the DSR result is shown to approach correctly the semicircular shape of the noninteracting density of states, while the U-NCA displays a characteristic inverted V shape: in this regime the violation of the Friedel sum rule becomes large and the negative lifetime pathology is encountered within U-NCA. In contrast, the DSR yields a pinning of the spectral density at a U-independent value and does not lead to a violation of causality. It should be emphasized, however, that, even though it yields the exact U density of states at T, the DSR method is not quantitatively very accurate in the weak-coupling regime. C. Double occupancy Finally, we demonstrate that two-particle correlators in the charge sector can also be reliably studied with the DSR method, taking the fraction of doubly occupied sites d n n as an example. FIG. 1. Results for U.1 showing how U-NCA overshoots the Friedel s sum rule. The slave rotor method is correctly converging towards the free density of states semicircular. FIG. 11. Double occupancy at 8 as a function of U. The ED result is indicated with squares. DSR overshoots the exact result on the left, whereas IPT is slightly displaced to the right. Using the constraint 4, we calculate the connected charge susceptibility in the following manner: c n 1 2 n 1 2 Lˆ Lˆ 1 U 2 2 U 2 G X 2 G X U G X 2. 41 The double occupancy is obtained by taking the equal-time value c 2 n n 2d. 42 In Fig. 11, we plot the double occupancy obtained in that manner, in comparison to the ED and IPT results. Within ED, d is calculated directly from the charge correlator. Within IPT, c ( ) is not approximated very reliably, 24 but the double occupancy can be accurately calculated by taking a derivative of the internal energy with respect to U. Figure 11 demonstrates that the DSR method is very accurate in the Mott transition region and in the insulator. In particular, the hysteretic behavior is well reproduced. In the weak-coupling regime, however, the approximation deteriorates. This issue actually depends on the quantity: the physical Green s function has the correct limit U, as emphasized above, but this is not true for the charge susceptibility hence for d ). The mathematical reason is that the constraint 4 is crucial for writing Eq. 41, but this constraint is only treated on average within our method. This shows the inherent limitations of slave boson techniques for evaluating two-particle properties. The frequency dependence of twoparticle correlators will be dealt with in a forthcoming publication. 165111-11 123

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 FIG. 12. Instability lines U c1 (T),U c2 (T) and coexistence region for one, two, three, and four orbitals (N 2, 4, 6, 8) as obtained by DMFT DSR solver. D. Multiorbital effects In this section, we apply the DSR method to study the dependence of the Mott transition on orbital degeneracy N. We emphasize that there are at this stage very few numerical methods which can reliably handle the multiorbital case, especially when N becomes large. Multiorbital extensions of IPT have been studied, 8 but the results are much less satisfactory than in the one-orbital case. The ED method is severely limited by the exponential growth of the size of the Hilbert space. The Mott transition has been studied in the multiorbital case using the QMC approach, with a recent study 25 going up to N 8 4 orbitals with spin. Furthermore, we have recently obtained 15 some analytical results on the values of the critical couplings in the limit of large N. Those, together with the QMC results, can be used as a benchmark of the DSR approximation presented here. As explained above, a value of N N 2 3 was found to describe best the single-orbital case (N 2). Hence, upon increasing N, we choose the parameter N such that N N /N N 2 N/2. However, for N large enough, it might be more accurate to use the naive value N N N. In Fig. 12, we display the coexistence region in the (U,T) plane for increasing values of N, as obtained with DSR. The values of the critical interactions grow with N. A fit of the transition lines U c1 (T) where the insulator disappears and U c2 (T) where the metal disappears yields U c1 N,T A 1 T N, U c2 N,T A 2 T N. 43 44 FIG. 13. Spectral function at U 2, 6 and for one solid line and two orbitals dashed line. These results are in good accordance with both the QMC data, 25 and the exact results established in Ref. 15. When increasing the number of orbitals, the coexistence region widens and the critical temperature associated with the end point of the Mott transition line also increases. We display in Fig. 13 the DSR result for the spectral function for N 2 and N 4, at a fixed value of U and T. Two main effects should be noted. First, correlations effects in the metal become weaker as N is increased for a fixed U), as clear, e.g., from the increase of the quasiparticle bandwidth. This is due to increasing orbital fluctuations. Second, the Hubbard bands also shift towards larger energies, an effect which can be understood in the way atomic states broadens in the insulator. 26 To conclude this section, we have found that the DSR yields quite satisfactory results when used in the DMFT context for multiorbital models. In the future, we plan to use this method in the context of realistic electronic structure calculations combined with DMFT, in situations where numerically exact solvers e.g., QMC become prohibitively heavy. There are, however, some limitations to the use of DSR at least in the present version of the approach, which are encountered when the occupancy is not close to N/2. We examine this issue in the next section. E. Effects of doping Up to now, we studied the half-filled problem i.e., containing exactly N/2 electrons, with exact particle-hole symmetry. In that case, d U/2 and the Lagrange multiplier h enforcing the constraint 31 could be set to h. In realistic cases, particle-hole symmetry will be broken so that ), and we need to consider fillings different from N/2. Hence, the Lagrange multiplier h must be determined in order to fulfill Eq. 31, which we rewrite more explicitly as n f 1 2 2h NU 1 1 NU n i n e i n e i n n 2 /U 2ih n /U X i n. 45 In this expression, n f (1/N) f f (1/N) d d is the average occupancy per orbital flavor. Note that the numbers of auxiliary fermions and physical fermions coincide, as clear from Eq. 34. Here n f is related to the d-level position or chemical potential, in the DMFT context by n f 1 n e i n i n h f i n. 46 We display in Fig. 14 the spectral function obtained with the DSR solver when doping the Mott insulator away from half-filling. A critical value of the chemical potential i.e., of ) is required to enter the metallic state. The spectral function displays the three expected features: lower and upper Hubbard bands, as well as a quasiparticle peak which, in 165111-12 124

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... PHYSICAL REVIEW B 66, 165111 22 FIG. 14. Doping the insulator (U 3 and 6) with,1 corresponding to n f.5,.4, respectively. this case where the doping is rather large, is located almost at the top of the lower Hubbard band. However, it is immediately apparent from this figure that the DSR method in the doped case overestimates the spectral weight of the upper Hubbard band. This can be confirmed by a comparison to other solvers e.g., ED. Note that the energy scale below which a causality violation appears within the U-NCA becomes rapidly large as the system is doped, while no such violation occurs within DSR. The DSR method encounters severe limitations however as the total occupancy becomes very different from N/2. This is best understood by studying the dependence of the occupancy upon, in the atomic limit. As explained in Appendix. A, the use of -model variables X in the large-m limit results in a poor description of the n f vs dependence Coulomb staircase. As a result, it is not possible to describe the Mott transitions occurring in the multiorbital model at integer fillings different from N/2 using the DSR approximation. It should be emphasized, however, that this pathology is only due to the approximation of the O(2) quantum rotor e i byao(m) -model field. A perfect description of the Coulomb staircase is found for all fillings when treating the constraint 4 on average while keeping a true quantum rotor. 27 Hence, it seems feasible to overcome this problem and extend the practical use of the dynamical slave rotor approach to all fillings. We intend to address this issue in a future work. VI. CONCLUSION We conclude by summarizing the strong points as well as the limitations of the new quantum impurity solver introduced in this paper, as well as possible extensions and applications. On the positive side, the DSR method provides an interpolating scheme between the weak coupling and atomic limits at half-filling. It is also free of some of the pathologies encountered in the simplest finite-u extensions of NCA negative lifetimes at low temperature. When applied in the context of DMFT, it is able to reproduce many of the qualitatively important features associated with the Mott transition, such as coexisting insulating and metallic solutions and the existence of two energy scales in the DMFT description of a correlated metal the quasiparticle coherence bandwidth and the preformed gap. Hence the DSR solver is quite useful in the DMFT context, at a low computational cost, and might be applicable to electronic structure calculations for systems close to half-filling when the orbital degeneracy becomes large. To incorporate more realistic modeling, one can introduce different energy levels for each correlated orbital, while the extension to nonsymmetric Coulomb interactions such as the Hund s coupling may require some additional work. The DSR method does not reproduce Fermi-liquid behavior at low energy, however, which makes it inadequate to address physical properties in the very-low-energy regime as is also the case with the NCA. The main limitation, however, is encountered when departing from half-filling i.e., from N/2 electrons in an N-fold degenerate orbital. While the DSR approximation can be used at small dopings, it fails to reproduce the correct atomic limit when the occupancy differs significantly from N/2 and in particular cannot deal with the Mott transition at other integer fillings in the multiorbital case. We would like to emphasize, however, that this results from extending the slave rotor variable to a field with a large number of components. It is possible to improve this feature of the DSR method by dealing directly with an O(2) phase variable, which does reproduce accurately the atomic limit even when the constraint is treated at the mean-field level. We intend to address this issue in a future work. Another possible direction is to examine systematic corrections beyond the saddle-point approximation in the large-n,m expansion. Finally, we would like to outline some other possible applications of the slave rotor representation introduced in this paper Sec. II. This representation is both physically natural and economical. In systems with strong Coulomb interactions, the phase variable dual to the local charge is an important collective field. Promoting this single field to the status of a slave particle avoids the redundancies of the usual slave boson representations. In forthcoming publications, we intend to use this representation for i constructing impurity solvers in the context of extended DMFT, 29 in which the frequency-dependent charge correlation function must be calculated, 28 ii constructing mean-field theories of lattice models of correlated electrons e.g., the Hubbard model, 27 and iii dealing with quantum effects on the Coulomb blockade in mesoscopic systems. ACKNOWLEDGMENTS We are grateful to B. Amadon and S. Biermann for sharing with us their QMC results on the multiorbital Hubbard model and to T. Pruschke and P. Lombardo for sharing with us their expertise of the NCA method. We also acknowledge discussions with G. Kotliar thanks to CNRS funding under Contract No. PICS-162 and with H.R. Krishnamurthy thanks to IFCPAR Contract No. 244-1, as well as with M. Devoret and D. Esteve. APPENDIX A: THE ATOMIC LIMIT In this appendix we prove the claim that the atomic limit of the model is exact at half-filling and at zero temperature 165111-13 125

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 Fig. 15. This discrepancy with the correct result even for one orbital finds its root in the large M treatment of the slave rotor X. APPENDIX B: NUMERICAL SOLUTION OF THE REAL-TIME EQUATIONS Here we show how the saddle-point equations 28 31 can be analytically continued along the real axis. We start with X ( ) N ( )G f ( ), which can be first Fourier transformed into FIG. 15. Impurity occupancy in function of the d-level position in the rotor description solid curve and the exact result dotted curve, in the two-orbital case. at finite temperature, deviations from the exact result are of order exp( U) and therefore negligible for pratical purposes. To do this, we first extract the values of the meanfield parameters and h from the saddle-point equations at zero temperature and ( ) : 1 d 2 2 U 1 2ih U h 1 2 2h NU 4h d NU 2 2 2, 2 U 2 2h 2. U A1 Performing the integrals shows that (U 2 4h 2 )/(4U) and (h ) 1/2, so that h.if U/2, the equations lead actually to a solution with an empty or full valence, which we show in Fig. 15. We can now compute the physical Green s function G d ( ) G f ( )G X ( ) from the pseudopropagators G f (i n ) 1/(i n ) and 1 G X i n 2 n /U U/4 2 /U 2i n /U 1 i n U/2 1 i n U/2. A2 Performing the convolution in imaginary frequency and taking the limit T leads to G d i n 1 i n G X i n G f i n i n A3 X i n d X e i n N d 1 G f 1 d 2 2 n F 1 n F 2 i n 1 2, where we used the spectral representation G z d G z B1 B2 for each Green s function. The notation here is quite standard: G ( ) Im G( i ), n F ( ) is the Fermi factor, and n B ( ) denotes the Bose factor. It is then immediate to continue i n i in Eq. B1, and using again the spectral decomposition B2, we derive an equation between retarded quantities: X N d G f n F N d n F G f. B3 A calculation along the same lines for the fermionic selfenergy f ( ) ( )G X ( ) leads to f d G X n B d n F G X. B4 The numerical implementation is then straightforward because Eqs. B3 and B4 can each be expressed as the convolution product of two quantities, so that they can be calculated rapidly using the fast Fourier transform FFT. The algorithm is looped back using the Dyson equations for real frequency G f 1 h f, B5 1/2 i n U/2 1/2 i n U/2. A4 G 1 X 2 2h U U X. B6 Because U/2, this is the correct atomic limit of the single-band model at half-filling. The result for the empty or full orbital, is however, not accurate, as shown in At each iteration, and h are determined using a bisection on Eqs. 3 and 31, which can be properly expressed in terms of retarded Green s functions: 165111-14 126

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ QUANTUM IMPURITY SOLVERS USING A SLAVE... PHYSICAL REVIEW B 66, 165111 22 1 d G X n B, n f 1 2 2h NU NU 2 d G X n B, where n f, the average number of physical fermions, is n f G f d G f n F. B7 B8 B9 We note here that solving these real-time integral equations can be quite difficult deep in the Kondo regime of the Anderson model or very close to the Mott transition for the full DMFT equations. The reason is that G f ( ) and G X ( ) develop low-energy singularities this is analytically shown in the next appendix, which make the numerical resolution very unprecise if one uses the FFT. In that case, it is necessary to introduce a logarithmic mesh of frequency losing the benefit of the FFT speed, but increasing the accuracy or to perform a Padé extrapolation of the imaginary-time solution. APPENDIX C: FRIEDEL S SUM RULE We present here for completeness the derivation of the slave rotor Friedel s sum rule, equation 37, at half-filling. The idea, motivated by the numerical analysis as well as theoretical arguments, 12,3 is that the pseudoparticles develop low-frequency singularities at zero temperature: G f A f f, G X A X Xsgn. C1 C2 Using the spectral representation G d e G, C3 we deduce the long-time behavior of the Green s functions we denote by (z) the gamma function : We have similarly G f A f 1 f G X A X 1 X sgn, C4 1 f 1. C5 1 X C6 if one assumes a regular bath density of states at zero frequency. The previous expressions allow to extract the longtime behavior of the pseudo-self-energies using the saddlepoint equations 28 and 29 : X NA f 1 f 2, C7 2 f f A X 1 X 2 sgn. C8 2 X The next step is to use Eq. C3 the other way around to get from Eqs. C7 and C8 the dependance of the selfenergies: X N A f 1 f sgn, C9 1 f 1 A X f 1 X. C1 1 X It is necessary at this point to calculate the real part of both self-energies. This can be done using the Kramers-Kronig relation, but analyticity provides a simpler route. Indeed, by noticing that (z) is an analytic function of z and must be univaluated above the real axis, we find X z N i( A f e f 1) /2 1 f sin f 1 /2 z 1 f, f z 1 A X e i X /2 1 X sin X /2 z 1 X. The same argument shows from Eqs. C1 and C2 that e i( f 1) /2 G f z A f sin f 1 /2 z f, G X z A X e i X /2 sin X /2 z X. C11 C12 We can therefore collect the previous expressions, using Dyson s formula for complex argument, G f 1 z z f z, G X 1 z z2 U X z, C13 C14 and this enables us to extract the leading exponents, as well as the product of the amplitudes: f 1 N 1, X N N 1, sin2 A f A X 1 N 1 2 N. N 1 C15 C16 C17 We finish by computing the long-time behavior of the 165111-15 127

ÖØ Ð ºÉÙ ÒØÙÑ ÑÔÙÖ ØÝ ÓÐÚ Ö Ù Ò Ð Ú ÖÓØÓÖÖ ÔÖ ÒØ Ø ÓÒ SERGE FLORENS AND ANTOINE GEORGES PHYSICAL REVIEW B 66, 165111 22 physical Green s function G d ( ) G f ( )G X ( ) together with Eqs. C4 and C5 : G d 2 N 1 tan 2 N 1 1 N 1. C18 This proves Eq. 37. In principle, next to leading order corrections can be computed by the same line of arguments, although this is much more involved. 12 Non-Fermi-liquid correlations in the physical Green s function would appear in this computation. 1 For an introduction, see L. Kouwenhoven and L. Glazman, Phys. World 14, 33 21. 2 For a review, see A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 1996. 3 For a review, see T. Pruschke, M. Jarrell, and J.K. Freericks, Adv. Phys. 42, 187 1995. 4 See, e.g. A. Hewson, The Kondo Problem to Heavy Fermions Cambridge University Press, Cambridge, England, 1997. 5 For recent reviews, see Strong Coulomb Correlations in Electronic Structure Calculations Advances in Condensed Matter Science, edited by V. Anisimov Taylor & Francis, London, 21 ; K. Held et al., in Quantum Simulations of Complex Many-Body Systems, edited by J. Grotendorst, D. Marx, and A. Muramatsu NIC Series Volume 1, Juelich, 22. 6 A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 1992. 7 X.Y. Zhang, M.J. Rozenberg, and G. Kotliar, Phys. Rev. Lett. 7, 1666 1993 ; A. Georges and W. Krauth, Phys. Rev. B 48, 7167 1993 ; M.J. Rozenberg, G. Kotliar, and X.Y. Zhang, ibid. 49, 1 181 1994. 8 G. Kotliar and H. Kajueter, Phys. Rev. B 54, R14 221 1996. H. Kajueter and G. Kotliar, Int. J. Mod. Phys. B 11, 729 1997. 9 N. Bikers, Rev. Mod. Phys. 59, 845 1993. 1 T. Pruschke and N. Grewe, Z. Phys. B: Condens. Matter 74, 439 1989. 11 D.L. Cox and A.E. Ruckenstein, Phys. Rev. Lett. 71, 1613 1993. 12 O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta, Phys. Rev. B 58, 3794 1998. O. Parcollet, Ph.D. thesis, Paris University, 1999. 13 J. Kroha and P. Wölfle, cond-mat/15491 unpublished. 14 G. Kotliar and A.E. Ruckenstein, Phys. Rev. Lett. 57, 1362 1986. 15 S. Florens, A. Georges, G. Kotliar, and O. Parcollet, Phys. Rev. B to be published. 16 T. Pruschke private communication. 17 S. Pairault, D. Senechal, and A-M. Tremblay, Eur. Phys. J. B 16, 85 2. 18 K. Yamada, Prog. Theor. Phys. 53, 97 1975. 19 R. Bulla, A.C. Hewson, and T. Pruschke, J. Phys.: Condens. Matter 1, 8365 1998 ; R. Bulla, T.A. Costi, and D. Vollhardt, Phys. Rev. B 64, 4513 21. 2 G. Moeller, Q. Si, G. Kotliar, M. Rozenberg, and D.S. Fisher, Phys. Rev. Lett. 74, 282 1995. 21 M.J. Rozenberg, R. Chitra, and G. Kotliar, Phys. Rev. Lett. 83, 3498 1999 ; W. Krauth, Phys. Rev. B 62, 686 2 ; J.Joo and V. Oudovenko, ibid. 64, 19312 21. 22 G. Kotliar, Eur. Phys. J. B 11, 27 1999 ; G. Kotliar, E. Lange, and M.J. Rozenberg, Phys. Rev. Lett. 84, 518 2. 23 T. Pruschke, D.L. Cox, and M. Jarrell, Phys. Rev. B 47, 3553 1993 ; P. Lombardo and G. Albinet, 65, 11511 22. 24 J.K. Freericks and M. Jarrell, Phys. Rev. B 5, 6939 1994. 25 B. Amadon and S. Biermann unpublished. 26 J.E. Han, M. Jarrell, and D.L. Cox, Phys. Rev. B 58, R4199 1998. 27 S. Florens and A. Georges unpublished. 28 S. Florens, L. de Medici, and A. Georges unpublished. 29 Q. Si and J.L. Smith, Phys. Rev. Lett. 77, 3391 1996 ; H.Kajueter Ph.D. thesis, Rutgers University, 1996; J.L. Smith and Q. Si, Phys. Rev. B 61, 5184 2 ; A.M. Sengupta and A. Georges, ibid. 52, 1295 1995. 3 E. Mueller-Hartmann, Z. Phys. B: Condens. Matter 57, 281 1984. 31 This is still an active field of research. See, e.g., the recently developed Local Moment Approach: D. E. Logan, M. P. Eastwood, and M. A. Tusch, J. Phys.: Condens. Matter 1, 2673 1998. 165111-16 128

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ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ PHYSICAL REVIEW B 66, 2512 22 Mott transition at large orbital degeneracy: Dynamical mean-field theory S. Florens, 1 A. Georges, 1 G. Kotliar, 2 and O. Parcollet 3 1 Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 5, France and Laboratoire de Physique des Solides, Université Paris-Sud, Bât. 51, 9145 Orsay, France 2 Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 8854 3 Service de Physique Théorique, CEA Saclay, 91191 Gif-Sur-Yvette, France Received 21 May 22; published 8 November 22 We study analytically the Mott transition of the N-orbital Hubbard model using dynamical mean-field theory and a low-energy projection onto an effective Kondo model. It is demonstrated that the critical interaction at which the insulator appears (U c1 ) and the one at which the metal becomes unstable (U c2 ) have a different dependence on the number of orbitals as the latter becomes large: U c1 N while U c2 N. An exact analytical determination of the critical coupling U c2 /N is obtained in the large-n limit. The metallic solution close to this critical coupling has many similarities at low-energy with the results of slave boson approximations, to which a comparison is made. We also discuss how the critical temperature associated with the Mott critical endpoint depends on the number of orbitals. DOI: 1.113/PhysRevB.66.2512 PACS number s : 71.1. w, 71.27. a I. INTRODUCTION The Mott transition is one of the central problems in the field of strongly correlated electronic systems. It is directly relevant to many compounds, such as V 2 O 3, NiS 2 x Se x, 1 fullerenes, 2 and two-dimensional organics of the -BEDT family. 3 Viewed from a more general perspective, the Mott transition confronts us with the need to describe electrons in a solid in an intermediate coupling regime, where perturbative methods around either the localized or the itinerant limit are of very limited use. The Mott phenomenon is also a key issue for the development of electronic structure calculations of strongly correlated materials, a subject of current intense investigation. 4 A breakthrough in the understanding of this problem resulted from the development of dynamical mean-field theory DMFT. 5 8 Following the observation that a Mott transition as a function of the interaction strength U takes place in a fully frustrated one-band Hubbard model at half-filling 9 within the DMFT, a rather surprising scenario was unraveled. 1 In the parameter space defined by temperature and interaction strength, two kinds of solutions of the DMFT equations were found, corresponding to an insulator and a metal, with a region U c1 (T) U U c2 (T) in which both types of solution coexist. The spectral function of the insulating solution has only high-energy features corresponding to Hubbard bands, while the metallic solution also has a low-energy quasiparticle resonance. Coexistence results in a first-order phase transition at finite temperature. The first order line ends into a second-order critical endpoint at T T MIT, U U MIT at which the coexistence domain closes (U c1 (T MIT ) U c2 (T MIT ) U MIT ). At T, the transition is also second-order, because the metallic solution always has lower energy than the insulating one. A qualitative picture of the evolution of the spectral function as a function of control parameters was obtained, and this resulted in successful experimental predictions. 11 Early work on these issues relied heavily on an accurate but approximate technique, the iterative perturbation theory. 6,1 Later on, exact results near U c2 (T ) were obtained, 12,13 thanks to the development of a renormalization method exploiting the separation of energy scales that holds true close to this critical coupling. This method performs a projection onto a low-energy effective theory after elimination of the degrees of freedom corresponding to Hubbard bands. A clear picture of the structure of the DMFT equations near the Mott transition and of the different critical points was finally obtained through the introduction of a Landau functional approach. 14 These established results for the one-band Hubbard model together with numerical methods for solving DMFT equations, such as Quantum Monte Carlo 9,15 and Numerical Renormalization Group, 16 provide now a consistent picture of the Mott transition within DMFT. In this paper, we investigate the Mott transition in the limit of large orbital degeneracy, within the DMFT. We establish analytically that the critical couplings U c1 (T ) and U c2 (T ) are not only different, but that they have very different scalings as a function of the number N of orbitals. Indeed, U c1 increases as N, while U c2 increases as N, so that the coexistence region widens. Close to U c2, the separation of scales which is at the heart of the projective method becomes asymptotically exact for large N. This allows us to determine exactly the value of the critical coupling U c2 /N at large N, and also to demonstrate that slave-boson-like approximations become asymptotically valid at low energy in this limit. Finally, we discuss how the finite temperature metal-insulator transition depends on the number of orbitals. Our findings on the critical couplings are in agreement with early QMC results for the two-orbital model (N 4), 17 and with recent results for higher values of N. 18 A widening of the coexistence window U c1,u c2 as N increases is clearly seen in these simulations. Also, for fixed N, the critical coupling required to enter the Mott state is largest 19 at the particle hole symmetric filling n 1/2. The present work also puts the results of Refs. 2 and 21 in a new perspective; there, a N scaling of the critical coupling was proposed. We find that this indeed applies to the coupling where the insu- 163-1829/22/66 2 /2512 13 /$2. 66 2512-1 22 The American Physical Society 131

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ S. FLORENS, A. GEORGES, G. KOTLIAR, AND O. PARCOLLET PHYSICAL REVIEW B 66, 2512 22 lator becomes unstable (U c1 ), while the true T Mott transition at which the quasiparticle residue vanishes takes place at U c2 N. Indeed, in Ref. 2, the N scaling was rationalized on the basis of a stability argument for the insulator. Finally, the rather high temperatures considered in Ref. 21 explain why only the N dependence was reported there: distinguishing U c2 from U c1 requires significantly lower temperatures. 18 The limit of large orbital degeneracy is relevant to the physics of systems with partially filled d- or f-shells, as well as to fullerenes. Direct numerical approaches become prohibitively difficult as the number of orbital increases QMC methods scale as a power law of N, while exact diagonalizations 22 scale exponentially. It is therefore important for future research to develop approximate but accurate impurity solvers which can handle many orbitals. The controlled results that we establish in this paper can be used as tests of these approximation methods. II. MULTI-ORBITAL HUBBARD MODEL We consider a generalized Hubbard model involving N species of electrons, with Hamiltonian, H i, j i N 1 N 1 t ij d i d j U 2 i N 1 2 d i d i n d i d i, 1 where i,j are sites indices, is the orbital index, is the chemical potential, and n is the average density of particles per species, n 1 N d i d i. 2 Introducing n in the Hamiltonian is just a convention for the chemical potential. In particular, it is convenient at halffilling where the particle hole symmetry implies n 1 2 and. For a single site atom, the spectrum consists of N 1 levels, with energies (U/2)(Q nn) 2 depending only on the total charge on the orbital, Q,...,N and with degeneracies ( N Q ). The usual single-orbital Hubbard model corresponds to N 2 (, ). The Hamiltonian considered here has a full SU(N) symmetry which includes both spin and orbital degrees of freedom. Starting from a more realistic model which assumes an interaction matrix U mm between opposite spins and U mm J mm between parallel spins, it corresponds to the limit of isotropic U mm U and vanishing Hund s coupling J mm. When U is large enough, we expect a Mott insulating state to exist at fillings n Q/N, corresponding to an integer occupancy of each site on average Q 1,...,N 1. In this paper, we investigate analytically the nature of these Mott transitions within the DMFT. We consider only phases with no magnetic ordering and study the transition between a paramagnetic metal and a paramagnetic Mott insulator. In Sec. III, we extract the large-n behavior of U c1 and U c2 using the low-energy projective technique analysis of the DMFT equations introduced in Ref. 12 for N 2 and extended in Ref. 23; we show that the equations for the U c s derived by this method are greatly simplified for N in the sense that an atomic limit becomes exact. In Sec. IV, we find quantitative agreement between these results and a multi-orbital slave boson method. Finally, in Sec. V, we consider the Mott transition at finite temperature. III. LARGE-N BEHAVIOR OF U c1 AND U c2 A. DMFT and the low-energy projective method Let us recall briefly the DMFT equations and their lowenergy projective analysis. 12,13 DMFT maps the lattice Hamiltonian above onto a multi-orbital Anderson impurity model with the same local interaction term than in 1 and a hybridization function (i n ). The local Green s function reads G d (i n ) 1 i n (i n ) d (i n ). In this expression, is the chemical potential shifted by the Hartree contribution and d (i n ) is a local self-energy. A selfconsistency requirement is imposed, which identifies G d (i n ) with the on-site Green s function of the lattice model with the same self-energy d (i n ). This reads G d i n d D i n d i n. In this expression, D( ) is the free electron density of state d.o.s. corresponding to the Fourier transform of the hopping matrix elements t ij. In the particular case of a semicircular d.o.s. with half-width D 2t, Eq. 3 simplifies to i n t 2 G d i n. Solving the impurity model subject to 3 determines both the hybridization function and the local Green s function in a self-consistent manner. In order to give an explicit Hamiltonian form to the Anderson impurity model, the function can be represented with an auxiliary bath of conduction electrons c k, with effective single-particle energies k and effective hybridizations V k to the local orbital such that H k i k V k 2 i k, k c k, c k, V k c k, d d c k, k, d d 3 4 5 U 2 d d n 2. 6 Within the DMFT, the Mott transition at U c2 and T is associated with a separation of energy scales. For U slightly smaller than U c2, the d-electron spectral function has a sharp quasiparticle peak, well separated from the high-energy Hubbard bands. Naturally, a small amount of spectral weight does connect these two features in the metal. Slightly above U c2, the Mott gap g separating the Hubbard bands of the insulating solution is finite. While the weight Z of the quasi- 2512-2 132

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ MOTT TRANSITION AT LARGE ORBITAL... PHYSICAL REVIEW B 66, 2512 22 particle peak vanishes at U c2, the Mott gap g disappears only at the smaller critical coupling U c1. Because of the self-consistency condition 3, the hybridization function of the effective conduction bath and hence the k s) also displays this separation of scales. This remark led the authors of Ref. 12 to use a projective method in order to study the critical behavior at U c2. Within this approach, the effective conduction electron bath is separated into a high energy sector (k H ) associated with the upper (H ) and lower (H ) Hubbard bands, and a lowenergy sector (k L) associated with the quasiparticle resonance. The impurity model Hamiltonian is thus written as for simplicity we take V L and V H independent of k H H k H, H H L H H H HL, H L k L, k c k, c k,, k c k, c k, V H c k, d h.c. 7a 7b U 2 d d n 2 d d, 7c H HL k L, V L c k, d h.c.. 7d We note that the self-consistency condition 3 implies that k L V 2 L /t 2 is proportional to the quasiparticle residue Z, so that it is clear that the low-energy hybridization vanishes at U c2. In order to obtain an effective theory close to U c2, one first diagonalizes the high-energy part H H of the Hamiltonian, and obtains an effective low-energy Hamiltonian by expanding in V L. From the presence of a finite gap in the high-energy bath k for k H ), H H describes an impurity in a semiconducting bath. The position of the d-level i.e., the chemical potential fixes the number of electrons in the ground-state. For a charge Q nn, this high-energy problem hasa( N Q ) degenerate ground-state, spanned by eigenvectors with eigenenergy E gs ). The detailed form of these states is complicated they mix with higher impurity charge states through V H ), but they form a representation of the SU(N) spin group, of dimension ( N Q ). Using a generalized Schrieffer-Wolff SW transformation to order V 2 L in order to project onto the low-energy Hilbert space defined by the selected charge sector of the impurity d, and the low-energy sector for the conduction bath k L), one obtains an effective Kondo Hamiltonian, 12,13,23 H eff k L, k c k, c k, V L 2 U J kk,, X c k, c k,, where X is a Hubbard operator connecting two states in the ground state manifold. The Hamiltonian contains potential scattering terms for and spin-flip 8 Kondo terms for, with the spin operator S X X. The matrix of dimensionless coupling constants reads 13,23 J U d d H H E gs U d d H H E gs. 9 We emphasize that the low-energy Hamiltonian has been derived to first order in V 2 L ; this is asymptotically justified exactly at the critical point and hence sufficient to determine the critical coupling, but deviations away from the critical point require to consider higher order terms in V L. Finally, we note that the bandwidth of this Kondo problem is of order Zt, while the coupling is of order V 2 L /U Zt 2 /U. Hence, the dimensionless coupling constant of the effective low-energy Kondo problem is of order one: we have an intermediate coupling Kondo problem. 14 The conditions that determine U c1 and U c2 have been derived within the projective method 12,23 and within the more general Landau functional approach 14 in which the stability of the insulating solution to different kinds of lowenergy perturbations is studied using a Landau functional of the hybridization function (i n )]. The condition for U c2 reads 2 U c2 t 2 nn N 1 J 2 X c c J J * J J *. 1 As explained in Refs. 14 and 23, an analogous equation is found for U c1, where off-diagonal terms in the average are taken to zero. For finite N, this equation is complicated to solve, since the average on the right-hand side is a function of J which we can only obtain by solving the low energy Kondo problem explicitly. However, we will show in the following that: 1 Using the SU(N) symmetry, the tensor structure of J can be parametrized with only two numbers A and B, which are dimensionless functions of N, n, and V H /U. This greatly simplifies 1 Sec. III B. 2 An explicit calculation of A and B can be performed in the atomic limit V H. This allows us to show that the two critical couplings have the following N-dependence at large N: U c1 NŨ c1, 11a U c2 NŨ c2. 11b In fact, the atomic limit evaluation of A and B becomes asymptotically exact when U is taken to be proportional to N. This is not the case when U N, but we show that corrections to the atomic limit do not modify the scaling 11a. 3 For U N, the effective Kondo problem can actually be solved explicitly at large N, using standard techniques. This allows for an explicit computation of the matrix element X c c involved in 1. Since, from the 2512-3 133

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ S. FLORENS, A. GEORGES, G. KOTLIAR, AND O. PARCOLLET PHYSICAL REVIEW B 66, 2512 22 above remark, the effective couplings entering the Kondo problem can be evaluated in the atomic limit in that case, this allows us to determine the prefactor Ũ c2 exactly Sec. III D. Because the atomic limit is not asymptotically exact for U N, we are not able to obtain an exact determination of Ũ c1. B. Simplication due to the SU N symmetry Computing the couplings 9 is a complicated task in general, since it requires a knowledge of the states,, and thus a diagonalization of H H. However, general considerations based on the SU(N) symmetry of the model allow one to reduce the unknown Kondo couplings J to two dimensionless numbers only. Let us consider the operator, O 1 1 d d H H E. 12 gs Since the Hamiltonian is SU(N) symmetric, this operator transforms under SU(N) with the representation F F* id ad: the tensor product of the fundamental F one box in the Young tableau language and its complex conjugate F* (N 1 boxes, which reduces to the sum of the identity id and the adjoint ad. If we denote by R the ground state representation a column with Q nn boxes, the tensor structure of J is then given by the Wigner-Eckart theorem; we have to find the occurrence of R in (id ad) R. We have a trivial term id R R and a single nontrivial one since ad R R..., where... denotes other irreducible representations. Then we have 1 d A 1 B 1 C, 13 d H H E gs where C are the Clebsch-Gordan coefficients R ad R. In order to avoid the computation of C, we can use the following argument. Let us consider the operator d d, which has the same symmetry as O 1, so the previous symmetry argument gives d d A 2 B 2 C 14 with the same C, but different coefficients A 2, B 2. Since B 2, we can eliminate C between the two equations. We finally obtain the general form, valid to first order in V L 2 for an SU(N) symmetric model, J A B d d. 15 In this expression, A(N,n;V H /U) and B(N,n;V H /U) are dimensionless coefficients which depend only on the dimensionless ratio V H /U associated with the high-energy problem and of course a priori also on orbital number N and density n. Inserting 15 into Eq. 1, we obtain the simplified form of the conditions which determine the critical couplings, 2 U c1 t 2 A nb 2 B 2 n 1 n N 1, 16a 2 U c2 t 2 A nb 2 B 2 n 1 n N 1 B 2 S c c. 16b C. The atomic limit The atomic limit approximation is defined by V H. In this limit, the states are just the impurity states 1 Q (Q,...,N), without mixing with the highenergy electron sector. In that case, A and B from 15 can be computed explicitly we choose here, i.e., the chemical potential is bound to be at the center of a step in the Coulomb staircase. We find A 2, B 4. 17 We note that these coefficients do not depend on N in the atomic limit. Moreover, it is clear that, in general for an explicit proof, see Sec. III D S c c N 2. 18 When using these results in 16a 16b, we obtain the N-dependence of the critical couplings quoted in 11a 11b. This derivation relies on the use of the atomic limit V H. An important question is therefore to check that the N-dependence of the critical couplings is not modified by the interaction V H that dresses the atomic states. We examine carefully this question in Appendix A, where we give an explicit proof that this is indeed the case. The problem is in fact quite different for U c1 and U c2. For the latter, U must be scaled as U NŨ. In that case, we show in the Appendix that the atomic limit calculation of A and B is in fact exact at large N. This can be expected since the band gap diverges with N, so that the Hubbard bands are well separated from the low energy degrees of freedom. However, one might worry that the growing number of orbitals N could counterbalance this effect. This is actually not the case, and we show that the corrections to J or A,B coming from V H are subdominant in large-n, order by order in perturbation theory. Moreover, the low-energy effective Kondo problem becomes exactly solvable when U N i.e., close to U c2 ). This is the subject of the next section Sec. III D, in which the exact value of the prefactor Ũ c2 is obtained. The problem is more involved for U c1, corresponding to the scaling U N. In that case, corrections in V H beyond the atomic limit do produce corrections to A and B hence to J. However, we are able to show Sec. 2 of the Appendix that these corrections are of order one, so that the scaling U c1 NŨ c1 does hold. 2512-4 134

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ MOTT TRANSITION AT LARGE ORBITAL... PHYSICAL REVIEW B 66, 2512 22 D. Explicitly large-n solution of the effective low energy Kondo problem at U c2 In this section, we explicitly solve the low energy effective Kondo problem in the large-n limit using standard methods, 24 in order to obtain the prefactor Ũ c2. The effective Hamiltonian 8 reduces at order (1/N 2 )to H eff k L, k c k, 4V L 2 NŨ c k, kk L, f f c k, c k, 2. 19 This model becomes exactly solvable at large N because the Kondo coupling scales as 1/N which is a result of U N). Introducing an auxiliary boson field b conjugate to k f c k, and a Lagrange multiplier field to enforce the constraint f f Nn, we see that, as N, the field b undergoes a Bose condensation, and the saddle-point values of the b, fields are determined by the equations at T ) G 1 f b 2 G cl, 2a n 1 d Im G f, 2b Ũ c2 2 4V L d G f G cl 1 d Im G f G cl. 2c In this expression, G cl denotes the Green s function of the low-energy part of the effective bath of conduction electrons, in the absence of Kondo coupling, namely, G 1 cl i 1 2 k L i k V L i. L 21 First we express the self-consistency condition 3 or more precisely its low-energy counterpart. We relate the lowenergy part of the on-site Green s function, G dl (i n ), to the auxiliary-fermion one, G f (i n ), by imposing that the lowenergy part of the interacting Green s function of the effective bath of conduction electrons, G cl, can be calculated either from the original Anderson model or from the lowenergy projected Kondo model, with identical results. G cl is related to G cl through the conduction electron T matrix, G cl G cl G cl 2 T. 22 The Anderson model T-matrix is V L 2 G dl, while it is b 2 G f for the low-energy Kondo model. Therefore G dl is directly related to G f by V L 2 G dl b 2 G f T. 23 Using this in Eq. 2a, one finds the expression of G dl in terms of the low-energy hybridization L, G 1 dl i n 1 Z i n L i n, where we have introduced the notation, Z b2 V L 2. 24 25 We can finally obtain G dl in closed form by making use of 24 into the self-consistency condition 3 restricted to the low-energy sector. This yields G dl i d D L i G 1 dl i 26 D d. 27 1 Z i The low-energy part of the local spectral density reads dl 1 Im G dl i D Z. 28 By comparing this form of G dl to its expression in terms of the local self-energy, we recognize that Z is the quasiparticle weight and yields the zero-frequency limit of d, d Z, Z 1 1 d. 29 The value of /Z can be determined by using the explicit form of d Z f into 2b, 1 n d Z D Z. 3 Changing variables, this leads to /Z (n), where (n) is the noninteracting value of the chemical potential, defined by n d D. 31 We note that this is precisely the result expected from Luttinger s theorem we have d () (n) which insures that the Fermi surface volume is unchanged by interactions. Hence, the low-energy part of the spectral function quasiparticle resonance is simply given by dl D Z n. 32 Its shape is simply given by the noninteracting d.o.s., centered at Z and with a width renormalized by Z. Second, we can now use 2c and Im L G f ( )/Z Im G f, to obtain Ũ c2 4 1 Z 2 d dl 33 2512-5 135

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ S. FLORENS, A. GEORGES, G. KOTLIAR, AND O. PARCOLLET PHYSICAL REVIEW B 66, 2512 22 d Z Z D Z. 34 Clearly, Z drops out of this equation, which has therefore a unique solution, the critical coupling Ũ c2 at which Z vanishes. This is expected: the low-energy effective Hamiltonian is valid only exactly at the critical point. A determination of the critical behavior of Z slightly away from the critical point would require considering higher-order terms in the effective low-energy Hamiltonian. We therefore obtain the large-n value of the critical coupling using the expression of /Z found above with Ũ c2 U c2 N 4 n n n d D. 35 36 This result is reminiscent of the critical coupling obtained in the Gutzwiller or slave-boson method. 25 A detailed comparison is made in Sec. IV. Taking for example a rectangular d.o.s. of half-width D, Eq. 35 yields U c2 N 4Dn 1 n. 37 Ũ c2 is largest at half-filling (n 1/2). This is because orbital fluctuations are maximum there, so that it is more difficult to destroy the metallic phase see however Ref. 19. Obviously, for finite N, a Mott transition is found only for rational values of n Q/N with integer Q these values are dense on the interval, 1 as N becomes large. Another way to obtain U c2 is to use the criterion of Sec. III C together with the present large-n limit in order to compute the matrix element S c c. This can be exactly evaluated at N using the effective action for the Kondo model 2, S eff d f f f f n d NŨ 4V L 2 b 2 bf c bc f d d c G cl 1 c. 38 Because this action is Gaussian, it is straightforward to get the following average here ): f f c c f f d GcL bf d GcL bf 2 b d GcL G f We find therefore, 2. 39 S c c N 2 b 2 d G f G cl 2 O N 4 which indeed scales as N 2. Inserting this into 16b, we have U c2 4t V Nb L Zt d G f G cl 41 which coincides with Eq. 2c since Z b/v L. Hence, we have checked that both methods explicit large-n solution and projective criterion of Sec. III C yield the same result for the critical interaction U c2. IV. COMPARISON TO A SLAVE-BOSON APPROACH In this section, we show that a multiorbital slave boson approximation reproduces the previous results close to the critical coupling U c2. This method is an extension to many orbitals of the Gutzwiller approximation, formulated in terms of slave bosons by Kotliar and Ruckenstein 26 and extended to many orbitals in Ref. 27. Indeed, with the scaling U NŨ, the Mott gap is of order N and quasiparticles dominate the physics over most of the energy range. Furthermore, expression 28 for the physical spectral function close to U c2 contains precisely the same ingredients than in slave boson methods, namely, a shift of the effective level which insures Luttinger s theorem, and a quasiparticle weight Z vanishing at the transition. The self-energy contains no further renormalizations in particular the quasiparticle lifetime is infinite. This bare-bone picture of a strongly correlated Fermi liquid is precisely that emphasized by slave boson approaches. Following Ref. 27, one introduces 2 N slave bosons 1 p associated with a given spin configuration 1 p on the impurity orbital. The probability that the orbital is in a given spin configuration is simply given by 1 p 1 p. The physical electron operator is represented as with d z f, 42 2512-6 136

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ MOTT TRANSITION AT LARGE ORBITAL... PHYSICAL REVIEW B 66, 2512 22 z g 1 p g 1 g 2 1 p 1 p 1,..., p 1,..., p 1,..., p 1 p 1 p 1 p 1 p g 2, 1/2, 1 p 1 p 1/2. This representation is supplemented by two constraints f f p 1 p 1,..., p 1 p 1 p 1 p, 1 p 1 p. 43 44a 44b At this stage, this is an exact representation of the Hilbert space, which can be used to rewrite the Hubbard Hamiltonian. In order to obtain an approximate solution, we shall assume that all auxiliary bosons undergo a Bosecondensation, i.e., become static c-numbers, and that furthermore the corresponding expectation value depends only on the total charge p and not on the specific spin configuration 1,..., p. We shall thus set, at each site, i 1 p p. 45 We emphasize that this approximation does not correspond to a controlled saddle-point, even in the large-n limit, since the number of auxiliary fields grows rapidly with N. Using this ansatz, the Hubbard Hamiltonian can be rewritten in the form; H i, j with N 1 and the constraints, t ij z 2 f i f j U 2 i p N 1 1 z n 1 n p N p 1 N p N N p p 2 p Nn 2 46 N 1 p p p 1 47 N p p 2 1, N 1 p 1 p 2 n f f. 48 49 The free-energy corresponding to this Hamiltonian must then be minimized with respect to the N 1 variational parameters p s, subject to the two constraints above. In the following, we briefly outline this procedure, focusing for simplicity on the half-filled case (n 1/2) at zero-temperature. In this case, particle hole symmetry makes the analysis simpler. It implies that p N p, and it can then be shown N that the constraint p 1 ( N 1 p 1 ) 2 p n 1/2 is automatically N satisfied when the first one p ( N p ) 2 p 1 is. Evaluating the ground-state energy of 46 yields the following expression to be minimized at T subject to just one constraint : 1 N 1 N E 4 p N Ũ 2 p N 1 p N p p 2 p p 1 2 N 2 2 p 5 with the absolute value of the noninteracting kinetic energy per orbital as defined in 36, and Ũ U/N, as defined above. It is clear from the Hamiltonian 46 that this slave-boson approximation leads to a picture of quasiparticles with no residual interactions. The quasiparticle weight reads Z z 2 1 N 1 n 1 n p N 1 2 p p p 1, 51 while the chemical potential shift in the previous section corresponds in the present context to the Lagrange multiplier associated with the constraint 44a it vanishes at halffilling. We also note that the noninteracting limit is correctly reproduced, thanks to the square-root normalization factors 26 introduced in 43. In that limit, Z 1 and E N. In order to analyze the Mott transition, we note that within this approach, an insulating solution can be found for arbitrary strength of the coupling U. It corresponds to the trivial solution. 2 N/2 N N/2 1, p p N 2 52 which yields the T occupancies in the atomic limit. This solution is always an extremum of the variational energy corresponding to a boundary, but not always a minimum. Indeed, a minimum with a lower energy exists for Ũ Ũ c2. To find this critical coupling, we perturb around the trivial solution. Close to U c2, only the occupancies of the 2 ( N N/2 ) 1 states with charge N/2 1 matter, and we set N/2, N/2 1, and p otherwise where denote terms that can be neglected. The energy difference with the trivial insulating solution reads N/2 1 2 N/2 N E 4 N 1 N/2 N 1 N N/2 1 2. 2 Ũ 2 N N/2 1 53 Hence, the critical coupling at which the metallic solution disappears reads 2512-7 137

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ S. FLORENS, A. GEORGES, G. KOTLIAR, AND O. PARCOLLET PHYSICAL REVIEW B 66, 2512 22 N 1 N/2 2 Ũ c2 16 N/2 N N N/2 1 4 N 2 N. 54 This value coincides with that obtained by Lu 25 using the Gutzwiller approximation. Remarkably, it agrees with our exact result in the N limit. Together with the considerations above, this suggests that such a slave boson approach becomes asymptotically exact, at low-energy and close to the critical coupling U c2, in this limit. This estimate of the critical coupling is already quite accurate for the smallest value of N 2 we take for reference a semicircular band with half-width D 1), U c N 2 8D 3 N 2 3.3 55 the NRG and the projective analysis of the one-orbital model indicate that U c is very close to 3. On the other hand, the large-n solution given by Eq. 35 obviously misses the subleading term of order N, and is therefore inaccurate at small N. The behavior of the quasiparticle weight close to the transition can be obtained by expanding the energy to next order 2 in, taking into account that N/2 q (1 Ũ/Ũ c2 ) q so that only the terms with q, 1 matter. This yields Z 1 U U c2. 56 We note that there are no 1/N prefactors in this expression when couplings are scaled properly. It is not entirely obvious to decide what the critical coupling U c1 is within this approach. Since the insulating solution exists for arbitrary U, one might be tempted to conclude that U c1. Another criterion would be to find at which coupling the optical gap of the insulator vanishes. Unfortunately, the above approximation condensed bosons is insufficient to discuss high-energy features Hubbard bands. Studies that go beyond this approximation 28 and incorporate fluctuations of the slave bosons suggest that the optical gap closes at U c2 ; in that sense the slave boson approximation does not yield a coexistence regime, in contrast to the exact results established above which are the generic situation within DMFT. V. FINITE TEMPERATURE TRANSITIONS We conclude this paper by addressing the finitetemperature aspects of the transitions discussed above. As in the one-band case, we expect a coexistence region U c1 (T) U U c2 (T) at finite temperature, which closes at a second-order liquid-gas critical point (U MIT,T MIT ), with U c1 (T MIT ) U c2 (T MIT ) U MIT. Comparison of freeenergies then yield a first-order transition line T T c (U) for T T MIT. This has been the subject of many studies in the single orbital case, and is now established on firm grounds. However, no analytical estimate of the critical temperature T MIT is available. This is highly desirable, since this scale appears to be strongly reduced in comparison to the electronic bandwidth. It would also be interesting to know how this scale depends on the orbital degeneracy. Let us start with a qualitative argument, which has two merits in our view: i it allows us to understand why T MIT /t is such a small energy scale for a small number of orbitals, and ii it provides a lower bound on T MIT. This argument is based on a comparison of the energy difference between the metallic and insulating solutions at T, to the corresponding entropy difference. For U NŨ close to U c2, the energy difference E E I (T ) E M (T ) is expected to behave as E N Ũ c2 Ũ 2 Ũ c2. 57 This is indeed supported by the slave-boson calculation of Sec. IV. The entropy difference S S I S M is dominated by the entropy of the insulating solution which has a degenerate ground-state since the entropy of the metal vanishes linearly with T at low temperature T. Therefore, S ln N/2 N N ln 2. 58 This allows us to estimate the behavior of the first-order critical line U c (T) or T c (U)] at low-temperature i.e., for U close to U c2 ). Indeed, the transition into the insulating state upon raising temperature occurs when the entropic gain overcomes the energy cost, leading to T c U Ũ c2 Ũ 2 Ũ c2. 59 This estimate is valid only at very low temperature, close to the T critical point U U c2. It indicates a quadratic dependence of the critical line, with a prefactor which is independent of N. 29 When U c1 is close to U c2, which is the case for a small number of orbitals in particular in the one-band case, one can roughly estimate the critical endpoint T MIT by using 59 in the regime of couplings where the insulating solution disappears, leading to T MIT (Ũ c2 U c1 /N) 2 /Ũ c2. It is clear from this formula that the ratio T MIT /t is small because of the small energy difference between the metallic and insulating solutions. For a larger number of orbitals however, this can only produce a lower bound on the critical endpoint T MIT. Indeed, the energy difference close to T MIT is certainly underestimated by 57, while the entropy difference is reduced as compared to N ln 2 since the entropy of the metal cannot be neglected at T MIT we note that it behaves as T with N). This argument shows that T MIT /t must either saturate to a constant, or increase with N, as N increases. We now turn to a more quantitative study of these finitetemperature issues, using the Landau functional formalism introduced in Ref. 14. A functional of the effective hybrid- 2512-8 138

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ MOTT TRANSITION AT LARGE ORBITAL... ization is introduced, which reads, for the N-orbital model on the Bethe lattice semicircular d.o.s., PHYSICAL REVIEW B 66, 2512 22 F N 2t 2 1 n i n 2 ln Z imp 6 in which Z imp is the partition function Z Tr e S imp of the effective impurity model, defined by the action, S imp d d d U 2 d d n 2 d d d d. 61 This functional is locally stationary for these hybridization functions which satisfy the DMFT equations, F i n t 2 G i n. 62 The local stability of these DMFT solutions is controlled by the matrix of second derivatives, 2 F nm i n i m N t 2 K nm, with, using 6, 61, K nm 1 2 d 1 d 1 d 2 d 2 e i n 1 1 i m 2 2 1 T d 1 1 d 1 1 d 2 2 d 2 2 2 T d 1 1 d 1 1 T d 2 2 d 2 2. 63 64 A solution of the DMFT equations is locally stable provided the nm matrix has no negative eigenvalues when evaluated for this solution. Hence, the couplings U c1 (T) resp. U c2 (T)] correspond to the instability line of the U,T plane along which a negative eigenvalue appears when nm is evaluated for the insulating resp. metallic solution. For T, these instability criteria should coincide with those derived from the projective method described above as verified below for U c1 (T )]. The critical endpoint T T MIT is such that, for T T MIT, no negative eigenvalue is found, at any coupling U, when the stability matrix is evaluated on the unique DMFT solution. It should thus be noted that, in order to determine (U MIT,T MIT ), it is not necessary to know both U c1 (T) and U c2 (T); either one of them is in principle sufficient. The practical difficulty in performing this stability analysis at finite temperature is that i it requires a knowledge of the finite-t solution of the DMFT equations or at least a reasonable approximation to it and ii the two-particle correlator 64 must be evaluated for this solution. Completing this program, for arbitrary orbital degeneracy, using numerical methods, is beyond the scope of this paper. However, we would like to present here a simpler calculation which gives some insight in the dependence of U c1 (T) and of T MIT )on FIG. 1. Instability temperature below which a negative eigenvalue is found when the stability matrix nm of Eq. 63 is evaluated in the atomic approximation. The various curves are for increasing number of orbitals (N 2,4,6,8,1,2), and an almost perfect scaling with T/ N 1 and U/ N 1 is found. the orbital degeneracy N. What we have done is to use the atomic limit as a very rough first approximation to the insulating solution. We have evaluated the two-particle correlator K nm and the stability matrix nm in this limit, for arbitrary N. This can be directly implemented on the computer, using the spectral decomposition of K nm onto atomic eigenstates some details are provided in Appendix B. The stability matrix is then diagonalized numerically a truncation to a large number of Matsubara frequencies is made, and we search for the line in the U,T plane where a negative eigenvalue is first found. The result of this calculation for the half-filled case n 1/2) is depicted in Fig. 1. There, the temperature below which a negative eigenvalue is found is plotted as a function of U. We first observe that no instability is found above a critical value of U, which does coincide with the value 16a ; U c1 (T )/t 4 N 1 estimated above from the projective method in the atomic approximation. This is a nontrivial consistency check on our calculations. Secondly, we find that the instability lines for different values of the orbital degeneracy N can all be collapsed on a single curve when both T and U are rescaled by N 1. This can be understood from the fact that, in the atomic limit, only the two scales U and T enter the two-particle correlator K nm not t, and that K is of order N 2, so that the instability criterion can be written from dimensionality considerations, Ũ/t ((T/ N 1)/Ũ). Naturally, we do not expect the curve in Fig. 1 to be a quantitatively reliable determination of U c1 (T), because of the atomic approximation involved. In particular, we see that, instead of terminating at a critical endpoint (U MIT,T MIT ), the instability line in Fig. 1 bends back, yielding a stability window at small U which is certainly a spurious aspect of the approximation. Hence only the branch of the curve corresponding to larger values of U has physical significance. It is tempting to conclude, from the observed scaling with N 1, that T MIT itself will grow in that manner as N is increased. Indeed, we note that recent QMC calculations by Amadon and Biermann 18 do show a marked increase of T MIT 2512-9 139

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ S. FLORENS, A. GEORGES, G. KOTLIAR, AND O. PARCOLLET PHYSICAL REVIEW B 66, 2512 22 with N, not inconsistent with a N scaling. However, we do not consider this issue to be entirely settled; rather, this atomic estimate provides an upper bound on the growth of T MIT with N. Indeed, it neglects the presence of the quasiparticle resonance which is reduced but still present as temperature is raised close to T MIT ). The resonance makes the solution less stable, so that the atomic calculation is likely to overestimate T MIT. Combined with the above entropy argument which provides a lower bound, we conclude that T MIT /t is an increasing function of N at moderate values of N, which either saturates at large-n or increases at most as N. Clearly, a more refined analysis of the above stability condition is required to settle this issue. VI. CONCLUSION In this paper, we have studied the Mott transition of the N-orbital Hubbard model in the fully symmetric case. The physical picture is qualitatively the same as for N 2, i.e., a coexistence region exists between an insulating and a metallic solution within two distincts critical couplings U c1 and U c2. We have shown that these critical couplings do not have the same dependence on the number of orbitals, asthe latter becomes large; U c1 N while U c2 N. We have obtained an exact analytical determination of the critical value Ũ c2 U c2 /N in this limit. These results explain the widening of the coexistence window U c1,u c2 observed in the numerical simulations of Ref. 17, and more recently in Ref. 18. Our findings also put the results of Refs. 2 and 21 in a new perspective, as discussed in the Introduction. We have shown that, at large N and close to the critical coupling U c2, the scaling U NŨ is appropriate, so that the separation of scale which is at the heart of the projective method becomes asymptotically exact. The low-energy physics close to U c2 is then well described by slave-boson-like approximations. This is the case in particular for the quasiparticle peak in the spectral function, which is separated from Hubbard bands by a large energy of order NŨ) and is accurately described by the simple form 32. In view of practical applications to electronic structure calculations with large orbital degeneracy e.g., f-electron systems, it would be highly desirable to have also an analytical determination of the high-energy part of the spectral function in the large-n limit or at least reasonable approximations to it. This is left for future investigations. Another issue that we have only partly addressed is the finite-temperature aspects of these transitions. The whole finite-temperature coexistence region widens as N is increased, and the critical temperature associated with the Mott critical endpoint does increase with the number of orbitals at intermediate values of N. The precise behavior of T MIT /t at large N deserves further studies, however. ACKNOWLEDGMENTS We thank B. Amadon and S. Biermann for useful discussions, and for sharing with us their recent QMC numerical results for this model. This work was supported by the NSF under Grant No. NSF DMR 96462 and by the PICS program of the CNRS under Contract No. PICS 162. APPENDIX A: LARGE-N SCALINGS AND THE ATOMIC LIMIT In this Appendix, we provide a detailed proof that the corrections in V H to the atomic limit (V H ) do not modify the large-n dependence of the critical couplings. This is done by an explicit investigation of the structure of these corrections. Moreover, we show that for U N i.e., when U c2 is considered, these corrections only produce subdominant terms which can be ignored at large N. 1. Corrections in V H for UÄNŨ For our purpose, the operator H H can be written, H H k H, H H H H H H 1, k c k, c k, U 2 d d n 2, H 1 H V H c k, d d c k,. k A1 First, let us consider the ground state energy E gs. As the lower Hubbard band H is filled in, we prefer to remove the constant contribution k H, k, so that the zeroth order energy is E H H. The next order term E 1 H H 1 is zero by conservation of the total charge Q on the d-level, so we need to look at the second order contribution, E 2 H H 1 2 E E N Q k H 2 V H Q U 2 k k H 2 V H A2 U 2 k since can only possess a charge Q 1 resp. Q 1) for the d-electron and a hole resp. particule excitation in the bath. Clearly as k is negative positive for k H (H ), the denominators are of order N since U NŨ) and E 2 is therefore scaling as N at large N. We would like to infer this to be true at all order in the development in V H. One can easily classify into two categories the intermediate states that appear at order V H 4 and beyond. There can either be a charge different form Q on the impurity as in the previous computation at order V H 2 ) and then the denominator coming from this state provides at least a factor N 1 by the presence of the Coulomb energy U NŨ. Or the impurity can be found in a charge Q state with possibly a reorganization of the spin configuration without a Coulomb energy cost. However this state possesses also many particle hole excitations in the bath, and this leads to a denominator which scales as the Mott gap g. For example, let us consider such a ket generated at order V H 4, like 1 Q d 2 N 1. This par- 2512-1 14

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ MOTT TRANSITION AT LARGE ORBITAL... PHYSICAL REVIEW B 66, 2512 22 ticular state but the argument is general, contributes a factor ( k k ) 1, where k H and k H. This quantity is always smaller than 1 g because the Mott gap g is also the gap of the c k, electrons through the self-consistency. As the gap g is expected to be of order U NŨ when U is close to U c2, each particle hole excitation also provides a factor N 1 in the perturbative expansion. This is enough to conclude that, at a given order in perturbation theory, each of the denominator gives a factor N 1 that balance the combinatorial factor coming from the string of charge excitation on the impurity level. Thus E gs is at most of order 1 when N is large. We finally need to compute J at large N. Let us consider first the case, so that and will differ by only one spin flip. We can now calculate at the lowest order in V H the desired matrix element, 1 d H H H 1 d H E gs, A3 with 1 1, k H, k H, V H d U c k, 2 k V H c U k, d. 2 k The contribution at order (V H ) is therefore 1 d H d H E gs 2 U A4 A5 because E gs O(1) and U NŨ. All terms coming at order V H cancel by conservation of the charge, so we examine now the next leading order for, 2, which is composed of three terms: one from the correction by V H to each of the two external kets, one from the development of the denominator in A5 at order V 2 H, and a mixed term between only one ket correction and the denominator developed at first order. Let us examine the first of these second order contributions, 1 2 1 1 d H d H E gs. A6 1 The state d 1 is composed of charge Q 2 excitations with energy 2 2 U/2) and of particle hole excitations in the bath with a charge Q on the impurity this has the energy k k ). Both energies scale like N and contribute an overall factor N 3 to (1) 2. There is also a combinatorial factor of order N coming from the choice in the spin-excitation in 1, but there is none from the state 1 as the spin configuration in d 1 is fixed by the one in d 1. Therefore 2 (1) O(N 2 ). The same argument works for the two other contribution at order V H 2. More generally, at a given order p in the development in V H, each of the p 1 denominators provides a factor of order N 1, because there is a finite gap g U c2 N between the two Hubbard bands. 3 However the string of p spin-flips generated by the H mix terms only contributes to an overall combinatorial factor at most N p/2 because one has to connect two fixed external configurations and. Hence in the limit of large N all terms after the first one are subdominant, so that O 1 N 2 2 U 2 NŨ A7 meaning that the atomic result for is exact in the large-n limit when U N. At this stage, we want to stress a posteriori that the hypotheses made in the preceding argument are consistent with this result. First it is correct to fix the chemical potential from the atomic limit (V H ), because perturbative corrections do not change the occupancy of the d-orbital, as can be checked from a direct expansion in V H of d d. Finally the existence of a large Mott gap g N can also be justified on the same grounds. 2. Some remarks about U c1 When U N on the other hand, the perturbative series for does not appear to stop after the first contribution, showing that U c1 is affected by V H in the large N limit. It is however straightforward to follow the line of arguments given in Sec. III C to see that each term in this development provides a leading contribution of order 1/ N using U g N). cannot be computed exactly in that case, because B(N,n;V H /( NŨ)) is now a nontrivial function of V H /Ũ at N. But this is enough to conclude that B(N,n;V H /U) is of order 1 at large N when U N, so that the result U c1 NŨ c1 holds. However, an exact evaluation of Ũ c1 appears to be a difficult task, even at N. APPENDIX B: NUMERICAL CALCULATION OF THE STABILITY MATRIX IN THE ATOMIC LIMIT In this appendix, we provide details on the computation of the stability matrix nm in the N-orbital atomic limit cf. Fig. 1. First, we use a spectral decomposition over the atomic eigenstates in Eq. 64. This yields 2 K nm A i n, i n,i n, i n /Z at B i n B i n /Z at with N Z at Q e E Q N Q, 2512-11 141

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ S. FLORENS, A. GEORGES, G. KOTLIAR, AND O. PARCOLLET PHYSICAL REVIEW B 66, 2512 22 A j, j 1 4 N Q N P S 4 e E Q P C 1 Q,P N 1 C 2 Q,P F 4 P j Q i 4 a j 1 P i,a P j, j 1 4, N B i n N Q e Q Q 1 E N 1 F 2 i n Q 1,1, i n Q, 1 N 1 Q F 2 i n Q 1, 1,i n Q,1, Q 2 P C 2 Q,P N 2 P N 2 P N 2 if P 1 1 P 1 2 and P 1 3 P 1 4 Q if P 1 1 P 1 2 and P 1 3 P 1 4, else Q 1 C 1 Q,P Q 1 N 1 if P 1 1,3 1,3 N 1 Q if P 1 1,3 2,4 F n j, j 1 n 1 n else n i 1 d i exp i 1 j j. In these expressions, E Q U(Q N/2) 2 /2 are the atomic energy levels, Q,a E Q a E Q, is the signature of the permutation P, and we used the notation f (x j, j 1 4) f (x 1,x 2,x 3,x 4 ). We compute F 4 and F 2, using the relations 1 2 3 4 and 1 2, respectively. The algorithm can be decomposed into three functions: i From U and T, compute nm for n, m 1 and then its lowest eigenvalue E (U,T); ii Find a point p below the instability temperature T i and a majoration T M of max(t i ) and U M of U c1 ; define a path in the U,T from, to (,T M )to(u M,T M )to(u M,); iii For about 5 points p on this path, solve for E (U,T) on the line defined by p and p using a dichotomy. To speed up the computation, the sums over permutations and Q in A and B are expanded automatically into C code, which is inlined in the main program. Diagonalizations are performed using the Fortran LAPACK library. Codes will be available at wwwspht.cea.fr/parcolle/. n 1 For a review, see, e.g., M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 7, 139 1998. 2 For a review, see, e.g., O. Gunnarsson, Rev. Mod. Phys. 69, 575 1996. 3 S. Lefebvre, P. Wzietek, S. Brown, C. Bourbonnais, D. Jérome, C. Mézière, M. Fourmigué, and P. Batail, Phys. Rev. Lett. 85, 542 2. 4 For a review, see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations, edited by V. Anisimov Gordon and Breach, New York 21. 5 W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 1989. 6 A. Georges and Kotliar, Phys. Rev. B 45, 6479 1992. 7 For a review, see A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 1996. 8 For a review, see, T. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys. 42, 187 1995. 9 M. Jarrell, Phys. Rev. Lett. 69, 168 1992 ; M. Rozenberg, X. Y. Zhang, and G. Kotliar, ibid. 69, 1236 1992 ; A. Georges and W. Krauth, ibid. 69, 124 1992. 1 X. Y. Zhang, M. J. Rozenberg, and G. Kotliar, Phys. Rev. Lett. 7, 1666 1993 ; A. Georges and W. Krauth, Phys. Rev. B 48, 7167 1993 ; M. Rozenberg, G. Kotliar, and X. Y. Zhang, ibid. 49, 1181 1994. 11 M. Rozenberg, G. Kotliar, H. Kajuter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf, Phys. Rev. Lett. 75, 15 1995. 12 G. Moeller, Q. Si, G. Kotliar, M. Rozenberg, and D. S. Fisher, Phys. Rev. Lett. 74, 282 1995. 13 G. Moeller, Ph.D. Rutgers unpublished ; http:// www.physics.rutgers.edu/ kotliar/thesis/moeller.ps.gz. 14 G. Kotliar, Eur. Phys. J. B 11, 27 1999 ; G. Kotliar, E. Lange, and M. J. Rozenberg, Phys. Rev. Lett. 84, 518 2. 15 J. Joo and V. Oudovenko, Phys. Rev. B 64, 19312 21. 16 R. Bulla, Adv. Solid State Phys. 4, 169 2 ; R. Bulla, 2512-12 142

ÖØ Ð ºÅÓØØØÖ Ò Ø ÓÒ ØÐ Ö ÓÖ Ø Ð Ò Ö Ý ÝÒ Ñ ÐÅ Ò Ð Ì ÓÖÝ MOTT TRANSITION AT LARGE ORBITAL... PHYSICAL REVIEW B 66, 2512 22 T. Costi, and D. Vollhardt, Phys. Rev. B 64, 4513 21. 17 M. J. Rozenberg, Phys. Rev. B 55, R4855 1997. 18 B. Amadon and S. Biermann unpublished. 19 This applies provided the bare density of states has a sufficiently smooth variation with energy, which is the case of the model d.o.s. considered in this paper. 2 O. Gunnarsson, E. Koch, and R. Martin, Phys. Rev. B 54, R1126 1996. 21 J. E. Han, M. Jarrell, and D. L. Cox, Phys. Rev. B 58, R4199 1998. 22 M. Caffarel and W. Krauth, Phys. Rev. Lett. 72, 1545 1994 ; Q. Si, M. Rozenberg, G. Kotliar, and A. Ruckenstein, ibid. 72, 2761 1994. 23 G. Kotliar and H. Kajueter, Phys. Rev. B 54, R14221 1996 ; H. Kajueter and G. Kotliar, Int. J. Mod. Phys. 11, 729 1997. 24 N. Read and D. M. Newns, Adv. Phys. 36, 799 1987. 25 P. Lu, Phys. Rev. B 49, 5687 1994. 26 G. Kotliar and A. Ruckenstein, Phys. Rev. Lett. 52, 1362 1986. 27 R. Fresard and G. Kotliar, Phys. Rev. B 56, 1299 1997 ; H. Hasegawa, J. Chem. Phys. 66, 3522 1997. 28 R. Raimondi and C. Castellani, Phys. Rev. B 48, 11453 1993. 29 One should note however that the large-n solution of Sec. III suggests that the temperature above which the metallic solution disappears spinodal has a different temperature dependence, T (Ũ Ũ c2 ) 3/2. 3 A. Georges and G. Kotliar, Phys. Rev. Lett. 84, 35 2. 2512-13 143

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ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú Coherence and Coulomb blockade in single electron devices: a unified treatment of interaction effects S. Florens, 1 P. San José, 2 F. Guinea, 2 and A. Georges 1 1 Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond 75231 Paris Cedex 5, France 2 Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-2849 Madrid, Spain We study the interplay between Coulomb blockade and the Kondo effect in quantum dots. We use a self-consistent scheme which describes mesoscopic devices in terms of a collective phase variable (slave rotor) and quasiparticle degrees of freedom. In the strong Coulomb blockade regime, we recover the description of metallic islands in terms of a phase only action. For a dot with well-separated levels, our method leads to the Kondo effect. We identify the regime where a crossover between the Coulomb blockade regime at high temperatures and the formation of a Kondo resonance at lower temperature takes place. In addition, we find that for a dot with many overlapping resonances, an inverse crossover can take place. A Kondo resonance which involves many levels within the dot is first formed, and this coherent state is suppressed by correlation effects at lower temperatures. A narrower Kondo resonance, due to a single level in the dot can emerge at even lower temperatures. I. INTRODUCTION Electron-electron interactions play a crucial role in single electron devices (quantum dots, metallic islands), see for instance [1 4]. The electrostatic repulsion between electrons in the dot can be described by the energy required to change the charge in the dot by one unit, E c. At temperatures, or frequencies, below this charging energy, but much greater than the spacing between the electronic levels in the dot, δe, Coulomb blockade effects dominate [5 8], and the conductance through the dot is suppressed, except at degeneracy points. At scales below the level splitting within the dot, and when the ground state of the dot is degenerate, electronic coherence can be restored by the Kondo effect [9, 1], leading to an increase in the conductance [11 15]. Hence, when the charging energy is much greater than the level spacing within the dot, E c δe, and the number of electrons in the dot is odd, one expects a crossover between a high temperature regime where Coulomb blockade effects dominate, and a low temperature one described by the Kondo effect [16]. The theoretical analysis of these two regimes, Coulomb blockade and the Kondo effect, has been carried out using rather different techniques. Coulomb blockade can be studied, when the coupling between the leads and the dot is weak, by describing electron transport in terms of sequential hopping processes [2], i.e. perturbatively in the tunneling matrix element. When the conductance is large, it is convenient to describe the internal degrees of freedom of the dot in terms of a collective variable, the phase conjugated to the total charge [17 19]. Using this representation, the renormalization of the effective charging energy by virtual fluctuations can be studied by a variety of methods [2 26]. This approach becomes exact when the level spacing within the dot tends to zero and the number of transmitting channels is large. In this limit the Kondo effect disappears. In the opposite limit, when the level spacing within the dot becomes comparable to other scales of interest, the standard procedure is to truncate the Hilbert space of the dot, and keep only the interacting electronic states closest to the Fermi level of the leads, so that the system is reduced to the Anderson model [27], see [11, 28, 29]. The situation when many levels within a magnetic impurity contribute to the Kondo effect was initially discussed in[3]. The same effect in quantum dots was analyzed in[31]. Other modifications of the Kondo effect in a quantum dot due the presence of many levels are discussed in [32 38]. Recent experiments[39] suggest that the regime where level spacing, level broadening and the charging energy are comparable can be achieved by present day techniques. The aim of this paper is to analyze the different regimes of a quantum dot characterized by the charging energy, E c, level spacing, δe, and coupling to the leads, Γ, where Γ is the typical width of each level. We introduce a new scheme which describes the charge excitations of the dot in terms of a collective phase, but also retains information about the electronic degrees of freedom on the dot. This approach allows to recover the Kondo effect in the case of quantum impurities with exactly degenerate levels (slave rotor representation introduced in [4] by two of us). Technically, the new scheme draws inspiration from self-consistent resummations used for describing the Kondo effect in single impurity models (the so-called Non Crossing Approximation, or NCA [41, 42]), but adapted in an economical way to the situation of mesoscopic structures (that can contain up to thousands of electrons). The outline of this paper is as follows. In section II we review basic properties of the single electron box, and present the model that we will study throughout. The 199

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 2 self-consistent approach used to analyze the model is presented in section III, and will be shown to describe the Coulomb blockade regime of metallic grains exactly (also for quantum dots); we will furthermore show that it is a reasonable approximation for dealing with the Kondo effect (in quantum dots). In section IV we will investigate in more detail intermediate regimes of conduction that were not previously attainable or considered in earlier works. In particular we will be interested in the different ways how one can cross over from coherent, Kondo assisted transport to the Coulomb blockade regime. Extensions of our approach to other systems such as quantum dot arrays will be briefly outlined in the conclusion. II. MODEL FOR SINGLE ELECTRON DEVICES behavior of quantum dots as well as metallic islands is the relatively large Coulomb energy E c = e 2 /(2C) 1K associated with small capacitances of the structure, C 1 15 F. We will take also into account the internal structure of the small charge droplet, with energy levels ǫ p (which are shape - and disorder - dependent), and an index σ = 1... N associated to transmitting channels to the outside circuit. The electrodes, to which the mesoscopic region is coupled, are described as Fermi Liquids with a spectrum ε k associated with a finite density of states D() at the Fermi level. Physical quantities, such as the conductance G, are also determined as a function of temperature T (we set β 1/T) and applied gate voltage V g. Summarizing all these energy scales, we arrive to the following hamiltonian: A. Phase representation and electronic degrees of freedom Let us first review the standard model for single electron devices. The basic ingredient that determines the H = kσ ε k a kσ a kσ + pσ ( ) 2 ǫ p d pσd pσ + E c d pσd pσ n g + t p ( k a kσ d pσ + d ) pσa kσ pσ kσp (1) We have introduced here the coupling t p k between the mode p in the electrode and mode k in the electron box. The index p = 1...N L corresponds to the N L energy levels of the dot (or metallic island), and σ = 1... N is the conserved channel index (including spin). We defined also n g as a dimensionless external gate voltage, n g ev g /(2E c ). Because of the large number of degrees of freedom involved with respect to a single impurity Anderson model (N L and/or N can be quite large), this hamiltonian is remarkably complicated. One way to make progress is to single out the relevant charge variable on the dot, and pilot its dynamics by means of a rotor degree of freedom: d pσ d pσ = L = i / θ (2) pσ This mapping is motivated by the fact that the hamiltonian H rotor = E c L2, with its quadratic spectrum, reproduces exactly the charging energy term in (1). Alternatively, one can argue, as in [17, 18], that a metallic grain can be studied with the same formalism used for a superconducting grain, except that the gap vanishes. In this case, the phase is the collective variable conjugate to the number of particles, as implied by equation (2). In order to fulfill the Pauli principle, one must however keep fermionic degrees of freedom, so that the creation of a physical electron on the dot is represented as: d pσ = f pσ eiθ (3) using a channel - and momentum - carrying quasiparticle f pσ. According to (2), a constraint L = pσ f pσ f pσ must also be implemented [4], introducing for this purpose a Lagrange multiplier δµ. Integrating explicitly the electrode in (1), one gets the following expression for the action of the original model in this new representation: β S = dτ f pσ ( τ +ǫ p δµ)f pσ + ( τθ + iδµ) 2 +in g τ θ+ 4E pσ c β β dτ dτ 1 pp (τ τ )f pσ N (τ)f p σ(τ )e iθ(τ) iθ(τ ) (4) L pp σ In principle, δµ should be integrated over [, 2π/β] in order to preserve the correct structure of the Hilbert space. 2

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 3 One can notice in this expression how the indices are positioned in the last term (σ is conserved by the charge transfer procedure, whereas p is not). We also have defined the bath function: pp (iω) k N L t p k tp k = iω ε k dε D(ε) N Lt p (ε)t p (ε) iω ε (5) introducing the lead density of states D(ε) = k δ(ε ε k) (we have extracted here a factor 1/N L for future convenience). B. Metallic grains: phase only approach In the case of small metallic islands [6, 7], the number of transverse channels N 1 4 1 6 1 is quite large, so that the fermionic degrees of freedom can be integrated out explicitely from (4). Technically, this is done by expanding the initial action (4) in the coupling function pp (τ), leading to an exact expression of the effective action: S[θ] = β dτ ( τθ + iδµ) 2 + in g τ θ (6) 4E c ( 1) n n β β log dτ m dτ 1 m m (τm τ n! m)e iθ(τm) iθ(τ m ) f p mσ m (τ m )f p m σ m (τ m) n= m=1 pmp N L p mp m σm The average over the fermionic variables... in the preceding expression is taken with respect to the free fermionic action S = β dτ pσ f pσ ( τ + ǫ p δµ)f pσ, and simplifies considerably in the large N limit because Wick contractions at leading order occur between pair of fermions holding the same channel index σ m. We can then re-exponentiate the last term in (6), and, if one notes that the multiplier δµ can be treated in the grand canonical ensemble, i.e. reabsorbed in n g (due to small charge fluctuations over many accessible states), one gets the effective phase-only action: S[θ] = β dτ ( τθ) 2 4E c + in g τ θ β β dτ dτ α (τ τ )e iθ(τ) iθ(τ ) (7) The kernel α (τ) in the last equation is given by the expression: α (τ) = 1 pp (τ) f pσ N (τ)f p σ() L pp σ which can be evaluated: (8) α (τ) α t (π/β) 2 [sin(πτ/β)] 2 (9) with α t = NN L t 2 D()ρ () (1) We have defined here the grain density of states, ρ (ǫ) = p δ(ǫ ǫ p), and have taken the large bandwidth limit to obtain equation (9). We have also assumed for simplicity that the coupling to the lead is a constant, t p k = t (point contact). The parameter α t is interpreted as the (dimensionless) high temperature conductance between the dot and the island, to lowest order in the hopping t, namely α t = G(T )/(4π 2 e 2 /h) [17, 18, 22]. Physically, the use of the phase-only action [22] is vindicated by the fact that the charge fluctuations are suppressed by the Coulomb blockade phenomenon, allowing one to forget about the fermionic statistics of the charge carriers (incoherent transport through the island). This phase-only action (7) can be handled using a variety of techniques [2 26]. An important question, which is very relevant for experiments on small metallic grains strongly coupled to the environment [8, 43], is the destruction of Coulomb blockade in the strong tunneling regime α t >.1. This implies a generic renormalization of the Coulomb energy E c towards smaller values, denoted here by Ec [21, 23]. In the following, we will be more interested in this question in the context of quantum dots, for which the model (7) is inadequate. However, the phase-only approach will be considered as an important benchmark for any approach that intends to describe the Coulomb blockade regime. 21

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 4 III. UNIFIED APPROACH TO SINGLE ELECTRON DEVICES A. Self-consistent scheme Our purpose here is to develop a simple approximation scheme (avoiding direct numerical solution of the model (1)) that could encompass the main interaction effects in single electron devices: the Coulomb blockade and the Kondo effect. One can note that one is usually forced to employ different techniques to handle the Coulomb interaction, depending on the regime one is interested in. Since the details of the dynamics of the electrons are not usually involved in the Coulomb blockade phenomenon, introducing the total charge as the physically relevant variable and using the phase-only action (7) seems mandatory to deal with this regime. On the other hand, dealing with the Kondo effect requires a rather sophisticated treatment of the correlation effects among electrons, and a simplification of the initial model (1) is unavoidable. We will see in this paper that one can reconcile these two paradigms if one treats on an equal footing fermionic degrees of freedom and the phase variable conjugated to the total charge. Drawing inspiration from strong coupling approaches for single impurity models (NCA [41] and its generalization to the phase representation (2) as introduced by two of us [4]), we will perform a self-consistent decoupling of the fermion-rotor coupling in (4): S + β dτ pσ β f pσ ( τ + ǫ p δµ)f pσ + dτ ( τθ + iδµ) 2 4E c + in g τ θ + β β dτ β β dτ dτ 1 pp (τ τ ) e iθ(τ) iθ(τ ) f pσ N (τ)f p σ(τ ) (11) L pp σ dτ 1 pp (τ τ ) f pσ N (τ)f p σ(τ ) e iθ(τ) iθ(τ ) L pp σ (12) This approximation goes far beyond the lowest order perturbative result that led to the phase-only action (7), and is the central starting point of the present work. Its distinctive feature is the decoupling of fermionic and phase degrees of freedom, whose joint dynamics is nevertheless determined self-consistently. Indeed, the fermionic selfenergy obviously depends on the phase-phase correlator, and reciprocally. Moreover, the bosonic part of the effective action (12) is similar in structure to the phase-only approach (7), allowing to use the large body of work on this particular model. In order to detail the method of solution, we start here by simplifying the kernels appearing in the previous selfconsistent action. The bosonic kernel can be expressed in terms of the full propagator G pp f of the f pσ fermions: α(τ) 1 pp (τ) f N pσ(τ)f p σ() L pp σ = N 1 pp (τ)g pp f ( τ) (13) N L pp where the Green s function Ĝf has been introduced in a matrix notation: [ Ĝ f = Ĝ 1 1 ] 1 [ Σf = N Ĝ 1 1 ] 1 Σf Ĝ (14) L N L with the free propagator in the electron box: G pp (iω) = δ pp (15) iω ǫ p + δµ We also have introduced the self-energy of the quasiparticles: Σ pp f (τ) = pp (τ) e iθ(τ) iθ() (16) The self-consistent phase action that one needs to solve finally reads: S = β dτ ( τθ + iδµ) 2 4E c + in g τ θ β β dτ dτ α(τ τ )e iθ(τ) iθ(τ ) (17) with α(τ) given previously. The set of equations (13-17) is the main technical result of this paper. In practice, 22

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 5 this is solved by an iterative procedure which starts with a given kernel α(τ) as input to the action (17), from which a new correlator e iθ(τ) iθ(τ ) is computed. It is then fed back to the self-energy (16), allowing to compute the quasiparticle propagator (14), and then a new kernel α(τ) from equation (13). This full cycle is repeated until convergence is reached. In all further calculations, we will assume a point contact between lead and box, t p k = t, so that pp (τ) = (τ) and Σ pp f (τ) = Σ f(τ) = (τ) e iθ(τ) iθ(τ ), which allows to simplify expressions (13) and (14) into: α(τ) = N (τ)g loc f ( τ) (18) G loc f (iω) 1 N L G pp f pp [ G loc f (iω) ] 1 = 1 1 N L p (iω) (19) 1 iω ǫ p + δµ Σ f (iω) (2) In the following two paragraphs, we will sketch how this novel scheme allows to capture both Coulomb blockade and Kondo effect in single electron devices. B. Metallic islands: Coulomb blockade We consider here the case of a metallic island, for which the energy spectrum is taken as a continuous density of states ρ (ǫ). Due to the smallness of the Fermi wavelength with respect to the typical transverse size of the island, the number of transmitting channels is usually quite large, N 1, as discussed before in section II B. One interesting remark is that our self-consistent phase model (11-12) exactly reproduces the phase-only approach (7) in this case. Indeed, we note that the fermionic self-energy (16) is suppressed with respect to the kernel (18) by a relative factor of order 1/N. The self-consistency between fermions and the phase variable can therefore be ignored at N 1 (one has to scale t 1/ N), leading to a free fermionic propagator: G pp (τ) f pσ ()f p σ(τ) (21) G pp (iω) = δ pp iω ǫ p (22) Putting this expression back into (13), one recovers indeed the phase-only action (7-8). This test case implies that our self-consistent scheme allows to deal correctly with the Coulomb blockade phenomenon in metallic grains. C. Quantum dots: interaction effects In quantum dots, the discreteness of the energy spectrum ǫ p and the fact that N is generally of order one (unless the point contacts are quite open) invalidate the phase-only approach (7). This is clear by the fact that coherent transport through the dot can be restored due to the Kondo effect at low temperature. We now sketch how the self-consistent action (17) is able to describe this phenomenon. 1 Coulomb blockade regime in quantum dots We consider first temperatures smaller than the charging energy E c, but still greater than the interlevel spacing δe. We can therefore take a continuous limit for the dot (due to thermal smearing of the energy levels), which leads crudly to the simplification: 1 1 N L (iω ǫ p + δµ) iπρ () (23) p From (2), this gives the f pσ fermion propagator in the dot: [ ] 1 G loc f (iω) i πρ () Σ f(iω) (24) The constant imaginary part in the last expression dominates the long time behavior of the Green s function G loc f (τ), so that the kernel (18) decays as 1/τ2, similarly to the case of the metallic island (9). There is therefore Coulomb blockade in this regime, as expected. 2 Kondo effect in quantum dots: first discussion At temperatures below the interlevel spacing δe in the dot, the discreteness of the spectrum becomes sizeable. Let us assume here that a single level sits close to the Fermi energy, so that we can forget all other levels. In this case, we can argue here that the factorization (11-12) reproduces the Kondo physics correctly. This property is well known if, instead of the phase variable θ, one uses a slave boson representation of the interaction (in the case of the E c = limit) [44]. This method has been extended recently by two of us to the case of finite E c, using the phase representation d pσ = f pσ eiθ [4], and shown to lead to a good approximation scheme for dealing with the Kondo effect. In quantum dots, the Kondo effect manifests itself by the buildup of a many-body resonance close to the Fermi level that allows the coherent transport of charge through the structure. The existence of the Friedel sum rule (to be discussed later on) guarantees that the conductance through the dot can recover the unitarity limit (i.e. G = 2e 2 /h) at low temperature. In conclusion, we can therefore expect that our selfconsistent approach interpolates between the coherent Kondo regime (associated with a large and increasing conductance with decreasing temperature) and the 23

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 6 Coulomb blockade (which leads to a suppression of the charge fluctuations on the dot and a decreasing conductance). In practice, we still need to solve the selfconsistent phase problem (11-12). This will be done in the next section. IV. INTERMEDIATE CONDUCTION REGIMES IN QUANTUM DOTS A. Solution of the phase action in the spherical limit The simplest, yet non trivial treatment of the bosonic action (17) is to take its spherical limit [4, 45, 46] (this is also equivalent to the large number of component limit for the sigma-model in the field theory litterature [47]). We introduce for this purpose the complex phase field X(τ) e iθ(τ), with its correlator G X (τ) X(τ)X (). This representation is perfectly equivalent to the original problem if the hard constraint X(τ) 2 = 1 is maintained exactly. The approximation that we will make in order to solve the (self-consistent) phase action (17) is to impose this constraint on average only. For this we introduce a Lagrange multiplier λ to enforce the equality G X (τ = ) = X(τ) 2 = 1. The main drawback (on a qualitative level) of this approximation is that, although it works fine close to the center of the charge plateaus, it eventually breaks down at the degeneracy points [4]. We will also enforce the constraint pσ f pσ f pσ = L on average. In the symmetric case (i.e. at the center of a plateau), this simply implies that δµ =. The bosonic action is now purely quadratic, leading to the set of self-consistent equations [54]: [ ] ν 2 1 G X (iν n ) = n + λ α(iν n ) 2E c (25) G X (τ = ) = 1 [ (26) G loc f (iω n ) = 1 1 N L p 1/(iω n ǫ p ) Σ f(iω n ) ] 1 (27) α(τ) = N (τ)g loc f (τ) (28) Σ f (τ) = (τ)g X (τ) (29) with the notation: (iω n ) = k N L t 2 iω n ε k (3) We have noted here ω n (resp. ν n ) a fermionic (resp. bosonic) Matsubara frequency. In addition to the relative simplicity of this scheme, we note that it allows to obtain a solution for real frequency quantities (doing the analytic continuation of these equations), as required for computing spectral and transport properties of quantum dots. A few technical remarks: The numerical solution of the system of coupled integral equations (25-3) is straightforward using Fast Fourier Transforms (we will also give analytical arguments later on). In all calculations that follow, we take N = 2 (single channel of conduction per spin). The Coulomb energy is chosen as the reference, E c = 1, and the bandwidth of the continuous density of states of the electrodes is Λ = 5, which is the largest energy scale of the problem. The precise form of the spectrum in the electrodes playing little role in the low energy limit, we have chosen a semi-elliptic density of states: (iω) = N L t 2 8 ( Λ 2 iω isgn(ω) ) ω 2 + (Λ/2) 2 (31) We define also the single-level width Γ (i + ) /N L = 4t 2 /Λ (see equation (4)), which characterizes the strength of the coupling of the electrons in the dot to the reservoirs. Other important parameters, that we will also investigate, are the separation between the electronic levels of the dot, δe, and the number of states in the dot, N L. We also define a typical bandwidth of the dot, W = δe(n L 1) (although the spectrum is made up of dicrete states). A technical aspect worth mentioning is the method of computation of the conductance through the dot. In the same spirit of the decoupling performed in equations (11-12), we will compute the Green s function of the physical electron as G loc d (τ) = Gloc f (τ)g X(τ). An analytical continuation (performed numerically) allows then to get the (interacting) density of states in the dot, ρ d (ω) = (1/π)Im G loc d (ω + i+ ). We will also use the interacting Landauer formula for the conductance: G = Ne2 h Γ ( dω n ) F(ω) πρ d (ω) (32) ω where n F (ω) = 1/(e βω + 1) is the Fermi function. This expression involves the local density of states (i.e. summed over all p = 1...N L ) because of the form of the local coupling to the reservoir, see equation (1). In the following, we explore situations that are intermediate to the single level Kondo effect and the Coulomb blockade regime, depending on the internal structure of the quantum dot. We insist on the fact that the results that we will obtain are not easily accessible to usual approaches of the hamiltonian (1), unless only a few levels in the dot are considered (for which case the Numerical Renormalization Group is quite successful [37, 48]). B. Results for finite interlevel spacing: Kondo effect and Coulomb blockade We now analyze the case of a few, well separated energy levels in the dot. We start by discussing the Coulomb blockade using the spherical limit (for temperature larger than the interlevel spacing), then we give analytical arguments in favor of the existence of a Kondo resonance at 24

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 7 lower temperature, and finally we show the full numerical solution of the self-consistent equations corresponding to the usual situation found in experiments. 1 Renormalized Coulomb energy As discussed in section III C 1, for T δe the quantum dot is in the Coulomb blockade regime, and the bosonic kernel (28) behaves at long times as α(τ) α t /τ 2, see eq.(9). Here α t is a measure of the dimensionless high temperature conductance. One can now focus on equation (26), that we write (at low temperature) as: dν 2π 1 ν 2 /(2E c ) + πα t ν + λ α(i) = 1 (33) This is easily solved to obtain the Lagrange parameter: λ α(i) = 2π 2 α 2 t E ce π2 α t E c (34) that we naturally interpret as the renormalized Coulomb energy Ec, since this quantity appears as a mass term in the phase propagator (25), which decays exponentially over time scales of order h/e c. This quick calculation allows us to understand the origin of the Coulomb blockade for quantum dots in our formalism. 2 Single level Kondo effect: analytical proof In quantum dots, the basic manifestations of the Kondo effect are twofold: existence of a small energy scale, the Kondo temperature T K, and restoration of the unitarity limit (at T < T K ), as discussed in section III C 2. To prove the first point from our integral equations (25-3), we examine qualitatively the solution of the selfconsistent equations at temperatures smaller than the interlevel spacing δe. There, the fermionic Green s function (27) is dominated by the ǫ p = pole, so that G loc f (τ) 1/(2N L) Sgn(τ), at long times. From equation (28), this produces in turn a rotor self-energy α(iν) NΓ log ν/t at low frequency (here T is an undetermined cutoff, in practice of order E c Γ). Upon lowering the temperature, this self-energy can reach the charging energy E c (assumed to be only weakly renormalized), indicating the suppression of Coulomb blockade and the beginning of the Kondo regime. This happens (very roughly) for NΓ log(t K /T ) E c, giving the estimate: ( T K (N L ) = T exp C E ) c NΓ (35) where C = π/2 (the simple argument here does not allow to compute this precise value). This estimates of T K (in agreement with [4]) explains that the key step to the experimental observation of the Kondo effect in quantum dots [13] lies in the realization of strongly coupled structures, having Γ comparable to E c in magnitude, such that the Kondo temperature remains accessible. We finish by explaining precisely the restoration of full coherence below the Kondo temperature. In order to do this, we set the temperature to zero and perform an exact low-energy solution of the system of equations (25-3) (this was done in Appendix-C of [4]). This analysis, valid while δe for an arbitrary number of levels, leads to the following value of the zero temperature, zero frequency density of states in our approximation: ρ d (ω = T = ) = 1 π/2 ( πγ N + 1 tan N π/2 ) N + 1 (36) The interpretation of this relation is that, whenever the temperature is lower than T K, the density of states is pinned at its non-interacting value, no matter how large the Coulomb energy E c is. This constatation reflects indirectly the presence of the Kondo resonance at low energy. The exact Friedel s sum rule however states that ρ exact d (ω = T = ) = 1/(πΓ), also independently of N. The difference between equation (36) and this exact result is a consequence of the decoupling approximation made above. It is nevertheless quite small in practice, since (36) goes to the exact value for N large, and is only 1% off for N = 2. The Friedel s sum rule provides also a simple explanation for the restoration of the unitary limit in the conductance below the Kondo temperature, since the Landauer formula (32) at zero temperature leads to: G(T = ) = Ne2 [ πγρd () ] (37) h 3 Kondo effect: numerical solution and crossover to the Coulomb blockade regime The numerical solution of the set of self-consistent equations (25-3) allows in principle to investigate all regimes of parameters. We will first concentrate on the gradual suppression of the Kondo effect with decreasing values of the single-level coupling Γ. In order to maintain the Coulomb blockade effect, we will keep in this section the total (multilevel) coupling fixed, Γ multi N L Γ =.5, and vary the number of levels N L to allow changes in Γ = Γ multi /N L. We will also fix the total bandwidth of the dot, W = 1, so that the level spacing (assumed to be uniform) is also decreasing, δe = W/(N L 1). This way of proceeding allows to interpolate from the few, well-sepatated levels situation relevant for small quantum dots in the Kondo regime to the case of larger dots with small level spacing and many levels N L which show only Coulomb blockade. The figure 1 demonstrates indeed how the low temperature local density of states evolves from N L = 1 (single level: regular Kondo effect) to N L = (continuum of levels: Coulomb blockade only). In particular the rapid suppression of the Kondo peak for diminishing values of 25

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 8 Γ at increasing N L is in accordance with our previous discussion of the Kondo temperature, equation (35)..5.4.4.2 shifting of the minimum of conductance. The Coulomb blockade is present at higher temperature, as shown by the decrease of G(T) for T K < T < E c upon lowering T [16]. For the last curve with a continuum of states in the dot, this behavior persists up to zero temperature. The inset in log-scale on the same plot allows to grasp more clearly the saturation of conductance below the Kondo temperature..3 -.4 -.2.2.4 2 2.2.1-4 -3-2 -1 1 2 3 4 FIG. 1: Local density of states ρ d(ω) at E c = 1, β = 8, and fixed W = (N L 1)δE and Γ multi = N LΓ =.5. The four curves correspond to different values of the number of levels in the dot: N L = 1,3, 5, (from top to bottom, following the evolution of the central peak). The temperature dependence of the electronic spectrum is presented for the three level case N L = 3 in figure 2. When temperature is lowered, the zero frequency density of states starts diminishing (by Coulomb blockade of states with different charge). One then reaches a minimum, before ρ d () begins shooting up, towards the unitarity limit (Friedel s sum rule) at zero temperature..4.3.2.1-2 -1 1 2 FIG. 2: Temperature dependence of ρ d(ω) for the case N L = 3; temperatures correspond to β = 8 (sharp central peak), β = 25 (broad peak), β = 1 (deep minima) and β = 2 (shallow minima filled by thermal excitations). It is useful to compare this evolution of the density of states to the variations of the conductance G(T, N L ) with temperatures, figure 3. This figure illustrates the reduction of the Kondo temperature with Γ by the downward 1.5 1.5 1.5 1.5.1.1.1.1.1.2.3.4.5 FIG. 3: Conductance G(T, N L) in units of e 2 /h for the same parameters as in figure 1 in function of temperature; curves with N L = 1,3, 5, follow from top to bottom in the extreme left of the plot. Inset: G(T, N L = 1) in temperaturelogarithmic scale. C. Effects due to overlapping resonances in multilevel dots The analysis in the previous section describes the usual situation where a crossover from the Coulomb blockade regime to the Kondo effect takes place as the temperature is lowered below the interlevel spacing in the dot. A different behavior can be expected when there is a set of overlapping resonances at low energies within the dot, i.e. an ensemble of levels of individual width greater than their separation, Γ δe, that act together as a single effective level with enhanced coupling to the leads (the typical bandwidth of this set of level should also be smaller than the charging energy). The presence of broad resonances near the Fermi level can be relevant to some experimental situations [35, 36]. [DETAIL?] We give first some qualitative arguments in order to describe the new effects expected in this regime, and then present explicit calculations using the integral equations. 1 Inverse crossover from the Kondo effect to Coulomb blockade In practical situations encountered in quantum dots, the single-level Kondo temperature T K is usually much 26

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 9 smaller than the level spacing δe (which is typically comparable to E c ). Therefore Coulomb blockade occurs (if it does) inevitably before the Kondo effect sets in, as was shown at length in the previous section. There is however a simple mechanism that allows to obtain Kondo temperatures greater than both level spacing and renormalized Coulomb energy Ec. The idea is that when the individual level width Γ exceeds the interlevel spacing δe, many levels are involved in the formation of the Kondo resonance. This leads to a multi-level Kondo temperature [3, 31]: ( ) T multi K T exp E c C NN eff Γ (38) which can be greatly enhanced with respect to the single level estimate, equation (35), by the presence of many levels N eff > 1 acting together (increasing the number of channels N might also contributes to this effect). This way of enhancing the Kondo temperature allows to obtain a new regime where TK multi Ec,δE, so that the Kondo effect can now occur before the Coulomb blockade (when lowering the temperature), in an inverse manner as observed traditionally in quantum dots. The fact that the Coulomb energy can be strongly renormalized to smaller values adds credibility to this idea. One further notes that, at even lower temperatures, a Kondo peak associated to the formation of a resonance which involves only one electronic state in the dot will ultimately emerge. One therefore has a two-stage Kondo effect (if the single-level Kondo temperature is not vanishingly small). In order to be more precise, we will first give a concrete example with a limiting case that one can understand independently of any approximation scheme. Then, we will illustrate in detail this inverse crossover using our integral equations. 2 Limit of exactly degenerate levels We consider here the extreme limit in which N L levels in the dot are simultaneously put to zero: ǫ p = for all p (the total bandwidth W is therefore also zero). We can formulate the model after a redefinition of the fermionic operators (unitary transformation) { d pσ} { c pσ }, such that: c 1σ = 1 N L d pσ (39) NL p=1 In this case, the remaining fermionic degrees of freedom, c pσ for p > 1, simply decouple from the problem, leaving an effective single level Anderson model describing the fermion c 1σ. Actually, a capacitive coupling persists between this fermion and all the other ones, but this influence gets frozen at low temperature. As the conductance through the dot is obtained from the Green s function: G loc d (τ) 1 d N pσ ()d p σ(τ) = c 1σ ()c 1σ(τ) L pp (4) one gets a full Kondo effect and a complete restoration of unitary conductance at low temperature. Furthermore, the width of this effective level is simply N L Γ, as one checks by inserting c 1σ in equation (4). This leads then to an enhanced Kondo temperature, ( ) T deg. K = T exp E c C NN L Γ (41) as discussed in the introductory part of this section (because δe = in this limiting case, one has N eff = N L ). We can easily check that our self-consistent scheme preserves this interesting property of the model. Indeed, when all levels ǫ p are exactly degenerate (W = ), one gets from equation (27) the f-electrons Green s function G f (iω) = [iω Σ f (iω)] 1. We obtain therefore the Kondo effect of a single effective level [4]. The following paragraph will allow us to give more weight to this discussion, by studying the more realistic case of a quantum dot in the regime Γ > δe, corresponding to nearly degenerate levels. 3 Inverse crossover: illustration We now illustrate the inverse crossover discussed qualitatively in paragraph IV C1 by solving our integral equations in the regime Γ > δe, where multi-level effects play an important role. We will assume here that Γ E c so that one can neglect the single level Kondo effect at low temperature (see however the next paragraph). For this computation, we have fixed Γ =.4, and taken N L = 9 states in the dot, varying the interlevel spacing from δe = (exactly degenerate level case considered in the previous paragraph) to δe =.1 < Γ (possibility of multilevel effect) to δe =.8 > Γ (absence of multilevel effect). The low-temperature local density of states displayed in figure 4 shows the expected multi-level enhanced Kondo peak at δe = corresponding to formula (41). Upon increasing the level spacing to δe =.1, Coulomb blockade sets in at a scale Ec < TK multi, however coherence effects remain around TK multi (since we are in a regime with δe < Γ). This results in a surprising splitting of the Kondo resonance at low energy. The last curve is taken with δe =.8 > Γ, so that no multilevel Kondo effect is possible, and only Coulomb blockade is observed. Another interesting consequence of this phenomenon is that the temperature dependance of the conductance is reversed with respect to the usual signature of the Kondo effect in quantum dots, i.e. to figure 3. Indeed, upon lowering the temperature, one remarks an initial increase of the conductance (due to the multilevel 27

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 1.8.6.4.2-3 -2-1 1 2 3 FIG. 4: Density of states ρ d(ω) on the dot for E c = 1, Γ =.4, inverse temperature β = 4, N L = 9 states, and different values of the interlevel spacing: δe =,.1,.8. Kondo effect), then a sharp decrease of the conductance because of the Coulomb blockade. This is illustrated by the middle curve in figure 5. 2 1.5 1.5.5.1.15.2 the order of E c, so that one finds the final expression: p (T multi K ) 2 + ǫ p 2 (T multi K ) 2 + ( ǫ p + T ) 2 = (T K) 2 (44) where T K is the single level Kondo temperature (35). A similar result was obtained previously by a renormalization group argument[3]. The limiting cases studied before are obviously contained in the previous equation: TK multi reduces to T K for widely separated levels (δe large) and in the opposite limit of exactly degenerate levels (or if W TK multi ), TK multi = T (T K /T ) 1/NL, consistently with equation (41). In general, TK multi is enhanced with respect to the single level Kondo temperature T K. As an example, formula (44) shows that in the regime δe TK multi W, one obtains then TK multi T exp[ E c δe/(nγt K multi )], so that the number of effective levels taking part to the multilevel Kondo resonance is N eff = TK multi /δe, as discussed qualitatively at equation (38). [COMMENTAIRE?] 4 Two-stage Kondo effect Finally we consider again a multilevel case, δe Γ, but now with Γ < E c, so that the single-level Kondo resonance is accessible to the low temperature regime. Therefore, one witnesses a further increase of the conductance at low temperature, taking place after the inverse Kondoto-Coulomb crossover that we discussed previously. The occurence of such a two-stage Kondo effect is depicted for the conductances shown in figure 6. For this calculation, we have taken N L = 5 levels, Γ =.1 and various level spacings between zero and 2.5Γ. The two curves 2 1.5 FIG. 5: Conductance G(T) for the same parameters as in figure 4. One can also perform a general evaluation of the multilevel Kondo temperature, starting from the leading behavior of the f-green s function at high temperature: G loc f (iω) 1 N L Equation (28) then leads to [55]: α(iν) = NΓ 2 p 1 (42) iω ǫ p ν 2 + ǫ p 2 log ν 2 + ( ǫ p + T ) 2 (43) p The Kondo temperature is reached when this kernel is of 1.5.1.2.3 FIG. 6: Conductance G(T) in the case N L = 5, Γ =.1 and various level spacings: δe =,.25,.5,.1,.25. with < δe < Γ show indeed this two-stage Kondo effect: a first rise of the conductance at high temperature due to the multilevel resonance at T TK multi, then the Coulomb blockade at T Ec and then a further increase at T T K (smallest scale of the problem). The last two 28

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 11 curves, with δe =.1,.25, show the usual crossover that was illustrated on the figure 3 in section IVB3. A last point that was checked is that our results are weakly sensitive to the addition of more levels outside the energy window of width Γ. This calculation is shown in figure 7. 2 1.5 1.5.5.1.15 FIG. 7: Lower curve: conductance G(T) in the case N L = 5, Γ =.1 and δe =.15. Upper curve: similar model, but 1 additional levels (with larger spacing δe =.2) have been superimposed to the previous ones. 5 Summary of the different regimes of transport A sketch of the different regimes analyzed in this paper is given in figure 8, as a function of the level spacing, δe, and single level width, Γ. We assume that the number of electrons in the dot is odd, so that the ground state, in the absence of the coupling to the leads is degenerate. The transition between different regimes is a smooth crossover. The occurence of Kondo effect is signaled by a non monotonous temperature dependance of the conductance, and is observable at Γ δe for temperatures smaller than the single-level Kondo temperature T K given by equation (35). Depending on the relative values of level spacing δe and Coulomb energy E c, Coulomb blockade might be also present above the Kondo temperature, which is the usual experimental situation encountered in quantum dots. We emphasize that if Γ is much smaller than E c the Kondo temperature is extremely small and the Kondo effect inobservable in practice; this correspond to weakly coupled dots, and this situation shows only Coulomb blockade. In the case δe Γ we have defined a region MultiLevel which corresponds actually to various regimes discussed previously. This describes the inverse crossover (Kondo to Coulomb Blockade), as well as the two-stage Kondo effect shown in section IVC4. This region can also imply that the Kondo effect is observed in the usual manner, but with a Kondo temperature greater than the single-level estimate. Note that the Kondo effect involving many levels can also occur in dots with an even number of electrons, in a similar manner to the Kondo resonance which arises at a singlet-triplet crossing in an applied magnetic field[49]. Finally, the region Γ E c of the diagram (denoted Coherent ) is associated with large conductances that are weakly modulated with temperature or applied gate voltage. Γ/ Ec 1 MultiLevel Coherent Kondo + Coulomb Blockade 1 Kondo δε/ Ec FIG. 8: Sketch of the the different regimes discussed in the paper, as function of the charging energy, E C, level spacing within the dot, δe, and level widths, Γ. The notation Kondo corresponds to a single-level Kondo effect. The region Kondo + Coulomb-Blockade is the usual situation in quantum dots. The Multilevel region is also associated to the Kondo effect, but with important renormalization of the Kondo temperature, as discussed in the text. The Coherent regime implies a temperature insensitive conductance. V. CONCLUSIONS In this paper we have presented a method of calculation for strongly correlated mesoscopic systems in terms of a collective phase variable and the electronic degrees of freedom. The scheme is valid both for the study of the Coulomb blockade regime and of the Kondo effect. We have shown examples of the crossover between a Coulomb blockade regime at temperatures below the charging energy, and the formation of a Kondo resonance at temperatures lower than the separation between levels within the dot. In addition, we have described an inverse regime, where a Kondo like resonance is splitted at low energies by Coulomb blockade effects. This regime is associated to the existence of many conduction channels, or overlapping resonances within the dot, which contribute collectively to the Kondo effect. The coherence of this state is destroyed at low energies by Coulomb effects. At even lower energies, a narrow Kondo peak, asociated to a single level within the dot, will emerge. Because the self-consistent approach used here was succesfully applied to model of strongly correlated electrons 29

ÖØ Ð Ó Ö Ò Ò ÓÙÐÓÑ ÐÓ Ò Ò Ð Ð ØÖÓÒ Ú 12 (Hubbard model) in a previous work [4], we can also envision possible applications of this work to the physics of granular materials or quantum dot arrays in the vicinity of the metal-insulator transition [5 53]. Disorder effects, that were neglected here, should also be included in future work along these lines. Other possible development is an exact (numerical) solution of the self-consistent action (11-12) instead of taking the spherical limit as was done in section IVA, which brings some limitations (as the restriction to valleys of conduction). Also importantly, we believe that the present work, both by the technical concepts and the physical ideas, connects in an original manner the fields of strongly correlated systems and mesoscopic physics. VI. ACKNOWLEDGEMENTS SF and AG are grateful to D. Esteve, H. Pothier and C. Urbina for a useful discussion. We appreciate the hospitality of the Kavli Institute of Theoretical Physics, where this work was initiated. The KITP is supported by NSF through grant PHY99-7949. P. San José and F. Guinea are thankful to MCyT (Spain) for financial support through grant MAT22-495-C2-1. SF acknowledges the support of the CNRS-PICS(162) program. REFERENCES [1] M. Kastner, Rev. Mod. Phys. 64, 849 (1992). [2] H. Grabert and M. Devoret, eds., Single Electron Tunneling (Plenum Press, 1992). [3] Y. Alhassid, Rev. Mod. Phys. 72, 895 (2). [4] I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep. 358, 39 (22). [5] D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb (Elsevier, 1991). [6] T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. 59, 19 (1987). [7] P. Lafarge, H. Pothier, E. R. Williams, D. Esteve, C. Urbina, and M. H. Devoret, Z. Phys. B 85, 327 (1991). [8] P. Joyez, V. Bouchiat, D. Esteve, C. Urbina, and M. H. Devoret, Phys. Rev. Lett. 79, 1349 (1997). [9] J. Kondo, Prog. Theor. Phys. 32, 37 (1964). [1] A. Hewson, The Kondo problem to heavy fermions (Cambridge, 1996). [11] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452 (1988). [12] T. K. Ng and P. Lee, Phys. Rev. Lett. 61, 1768 (1988). [13] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. A. Kastner, Nature 391, 156 (1998). [14] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science 281, 54 (1998). [15] L. Kouwenhoven and L. Glazman, Physics World 14, 33 (21). [16] L. Glazman and M. Pustilnik (23), cond-mat/32159,. [17] T.-L. Ho, Phys. Rev. Lett. 51, 26 (1983). [18] E. Ben-Jacob, E. Mottola, and G. Schön, Phys. Rev. Lett. 51, 264 (1983). [19] G. Göppert and H. Grabert, Phys. Rev. B 63, 12537 (21). [2] F. Guinea and G. Schön, Europhys. Lett. 1, 585 (1986). [21] S. V. Panyukov and A. D. Zaikin, Phys. Rev. Lett. 67, 3168 (1991). [22] G. Schön and A. D. Zaikin, Phys. Rep. 198, 237 (199). [23] G. Göppert, X. Wang, and H. Grabert, Phys. Rev. B 55, 1213 (1997). [24] P. Joyez, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. 8, 1956 (1998). [25] D. S. Golubev, J. König, H. Schoeller, and G. Schön, Phys. Rev. B 56, 15782 (1997). [26] C. P. Herrero, G. Schön, and A. D. Zaikin, Phys. Rev. B 59, 5728 (1999). [27] P. W. Anderson, Phys. Rev. 124, 41 (1961). [28] Y. Meir, N. S. Wingreen, and P. Lee, Phys. Rev. Lett. 66, 348 (1991). [29] L. P. Glazman and M. Pustilnik, Phys. Rev. Lett. 87, 21661 (21). [3] K. Yamada, K. Yosida, and K. Hanzawa, Prog. Theor. Phys. 71, 45 (1984). [31] T. Inoshita, A. Shimizu, Y. Kuramoto, and H. Sakaki, Phys. Rev. B 48, 14725 (1993). [32] T. Pohjola, J. König, M. M. Salomaa, J. Schmid, H. Schoeller, and G. Schön, Europhys. Lett. 4, 189 (1997). [33] J. J. Palacios, L. Liu, and D. Yoshioka, Phys. Rev. B 55, 15735 (1999). [34] A. L. Yeyati, F. Flores, and A. Martín-Rodero, Phys. Rev. Lett. 83, 6 (1999). [35] P. G. Silvestrov and Y. Imry, Phys. Rev. Lett. 85, 2565 (2). [36] P. G. Silvestrov and Y. Imry, Phys. Rev. B 65, 3539 (21). [37] W. Hofstetter and H. Schoeller, Phys. Rev. Lett. 88, 1683 (22). [38] F. Guinea, Phys. Rev. B 67, 19514 (23). [39] C. Fühner, U. F. Keyser, R. J. Haug, D. Reuter, and A. D. Wieck, Phys. Rev. B 66, 16135 (22). [4] S. Florens and A. Georges, Phys. Rev. B 66, 165111 (22). [41] T. Pruschke and N. Grewe, Z. Phys. B 74, 439 (1989). [42] O. Parcollet and A. Georges, Phys. Rev. Lett. 79, 4665 (1997). [43] D. Chouvaev, L. S. Kuzmin, D. S. Golubev, and A. D. Zaikin, Phys. Rev. B 59, 1599 (1999). 21

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ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PHYSICAL REVIEW B 69, 54426 24 Non-Fermi-liquid behavior from two-dimensional antiferromagnetic fluctuations: A renormalization-group and large-n analysis Sergey Pankov, 1 Serge Florens, 2 Antoine Georges, 2 Gabriel Kotliar, 1 and Subir Sachdev 3 1 Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 8854, USA 2 Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond, 75231 Paris Cedex 5, France 3 Department of Physics, Yale University, P.O. Box 2812, New Haven, Connecticut 652-812, USA Received 6 May 23; published 27 February 24 We analyze the Hertz-Moriya-Millis theory of an antiferromagnetic quantum critical point, in the marginal case of two dimensions (d 2,z 2). Up to next-to-leading order in the number of components N of the field, we find that logarithmic corrections do not lead to an enhancement of the Landau damping. This is in agreement with a renormalization-group analysis, for arbitrary N. Hence, the logarithmic effects are unable to account for the behavior reportedly observed in inelastic neutron scattering experiments on CeCu 6 x Au x.we also examine the extended dynamical mean-field treatment local approximation of this theory, and find that only subdominant corrections to the Landau damping are obtained within this approximation, in contrast to recent claims. DOI: 1.113/PhysRevB.69.54426 PACS number s : 75.2.Hr, 71.27. a, 71.1.Hf I. INTRODUCTION Materials in the vicinity of a quantum critical point QCP continue to be the subject of intensive investigations. 1 Particular attention has been paid to the heavy fermion system CeCu 6 x Au x which undergoes a phase transition from a paramagnetic, heavy-fermion metal to an antiferromagnetic metal as a function of chemical composition and pressure. 2,3 Another remarkable case is the stoichiometric compound YbRh 2 Si 2 which is apparently located very close on the magnetic side of an antiferromagnetic QCP. 4 Hydrostatic pressure stabilizes the magnetic phase in that case. As the critical point is approached, these systems exhibit an enhancement of the specific heat coefficient C/T, magnetic susceptibility, resistivity and thermoelectric power, which is not easily described within the conventional theory of a three-dimensional quantum critical antiferromagnet. Neutron scattering experiments 3,5 7 on CeCu 6 x Au x have revealed that the spin fluctuation spectrum of this compound has a two-dimensional character in a wide range of temperatures. Indeed, maxima of the neutron scattering intensity are observed along rodlike structures in reciprocal space along which spin fluctuations are independent of one component of the wave vector. This motivated Rosch et al. 5 to propose that the observed anomalies are the results of the coupling of three-dimensional electrons to quasi-two-dimensional critical spin fluctuations. In such a situation, conventional Hertz- Moriya-Millis theory 8 11 of quantum critical behavior leads to C/T ln(1/t), T, consistent with experimental findings. A similar observation applies to the thermoelectric power. 12 The microscopic reason for the two-dimensional nature of spin fluctuations is not clear however for CeCu 6 x Au x, while magnetic frustration might provide a natural explanation 4 in the case of YbRh 2 Si 2 for which inelastic neutron scattering data are not yet available. The approach of Rosch et al. 5 requires fine-tuning on a microscopic level since a three-dimensional dispersion of the electrons will generically lead to three-dimensional critical fluctuations at the QCP but it is internally consistent on a theoretical level. Indeed, the vertex corrections to the electron-spin fluctuation interaction is finite 12 in the case of a coupling to a three-dimensional fermion spectrum. Note that the situation is different for two-dimensional fermions, which leads to a singular vertex. 13 Since the vertex is nonsingular, the physics of long-wavelength magnetic fluctuations is described by a 4 field theory with Ohmic damping. The quartic term is marginally irrelevant for d 2, which is the upper critical dimension of this theory. Hence, the flow of the quartic term will lead to logarithmic deviations from mean-field critical behavior. These effects have not been investigated previously for the frequency-dependent response function in this context. Inelastic neutron scattering 3,6,7 on CeCu 6 x Au x at the QCP (x c.1) have revealed an anomalous enhancement of the Landau damping. Whether logarithmic effects at the upper critical dimension could account for such an enhancement in the range of experimentally accessible frequencies is an outstanding open question which we address in this paper. We use the number of components N of the field as a control parameter, and perform a calculation of the dynamical susceptibility to order 1/N. We find that, to this order, the sign of the corrections to Landau damping cannot explain the experimental data. An alternative theoretical viewpoint on the physics at the QCP is that the spin-fluctuation self-energy at the QCP is purely local i.e., momentum independent and has an anomalous power-law dependence on frequency and temperature. This was first proposed 3,6 as a phenomenological fit 1 to the inelastic neutron scattering data in the form Im q Q f ( /T) with.7.8. Si et al. 14 have further developed this point of view in the framework of the extended dynamical mean-field EDMFT theory of the Kondo lattice. 15 17 It has been argued in this context that the separation of electronic degrees of freedom and long-wavelength magnetic fluctuations is not legitimate, 14,18 because the Kondo coherence scale below which electron degrees of freedom can be eliminated might vanish at the QCP. Re- 163-1829/24/69 5 /54426 12 /$22.5 69 54426-1 24 The American Physical Society 231

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PANKOV, FLORENS, GEORGES, KOTLIAR, AND SACHDEV PHYSICAL REVIEW B 69, 54426 24 cently however, Grempel and Si showed 19 that starting above the Kondo temperature the EDMFT of the Kondo lattice can be mapped, using bosonization tricks, onto the EDMFT of the 4 theory for magnetic modes see also the earlier work by Sengupta and one of us 17. They further suggested that, in two dimensions, this theory would lead to an anomalous Landau damping, with the power-law form mentioned above ( 1). We study this effective theory in the EDMFT approximation, using both a strict 1/N expansion and a numerical solution using the large-n expansion as an impurity solver. We find that, contrary to the statements of Ref. 19, no anomalous power-law is generated. In fact, the EDMFT approximation to the 4 field theory in the marginal case does not correctly reproduce the logarithmic corrections to the damping rate found in our direct calculation for d 2, and leads only to subdominant O( 2 ) corrections at low energy. This extends to the marginal case the previous work of Motome and two of us 2 on the EDMFT approximation applied to critical behavior, in which it was indeed found that the approximation is better in higher dimensions, and becomes unreliable below the upper critical dimensions. Hence, the general conclusion of our paper is that logarithmic effects at the upper critical dimensions in the conventional theory of an antiferromagnetic QCP cannot explain the behavior observed experimentally for CeCu 6 x Au x, at least at dominant order in the 1/N expansion. This paper is organized as follows. In Sec. II, we perform a general renormalization-group analysis of the 4 theory with damping. In Sec. III, we take a large-n perspective, and present an explicit solution up to next to leading order in the 1/N expansion. We also describe the local EDMFT approximation to this theory, up to order 1/N. In Sec. IV we discuss in more details the EDMFT approach, and solve numerically the EDMFT equations using a large-n impurity solver. We conclude with a discussion of the implications of our work for the understanding of non-fermi-liquid behavior in heavy-fermion materials close to an antiferromagnetic QCP. II. RENORMALIZATION GROUP ANALYSIS A. Model and general renormalization-group framework In this paper, we consider the following field theory 8,1,11 for an N-component field a (a 1,...,N): S 1 D 1 2 q,i 2 a q,i q, a u d 4! 2 2 d xd a x, a 1 in which the free propagator reads, for imaginary frequencies D 1 (q,i ) r q 2. This theory describes the vicinity of an antiferromagnetic quantum critical point, corresponding to the bare value z 2 of the dynamical critical exponent. In this context, the dissipative term is induced by the coupling to electronic degrees of freedom Landau damping. For a different physical realization of this model see Ref. 21. The q-vector is the difference from the ordering wave vector, and r is a parameter which measures the distance to the critical point. We are particularly interested in this paper in the two-dimensional case, but we shall also briefly consider lower dimensions 1 d 2 Sec. II C. In this section, we shall use the following normalized coupling constant: 2 g S d u with S d d/2 4. d/2 When considering the bare theory, an ultraviolet cutoff on momenta is introduced. We consider the bare irreducible two-point function 1, which is the inverse of the correlation function dynamical susceptibility (q,i ) (q,i ) (q, i ) ]. The renormalization group RG specifies 22 how changes upon a rescaling of the cutoff. The corresponding RG equation is obtained by imposing that the renormalized two-point function R Z is independent of the original cutoff with Z the field or wave function renormalization. Focusing first, for simplicity, on the field-theory 1 in the critical massless case, satisfies the following equation all Green s functions in this and the next subsection are at T ): q,i ;g, 2 Z 1 q, Z i ;g 2, 1. In this expression, Z and the running coupling constant g are defined from the usual RG functions 2 3 d d g g g 1 g, 4 d d ln Z g Z 1 1. 5 Let us emphasize a key aspect of Eq. 3, namely, that the damping term does not require an independent renormalization so that the frequency dependence in the right-hand side RHS of Eq. 3 involves only the wave-function renormalization Z. 23,24 This can be proven to hold to all orders, using a field theoretic RG. 25 For a use of this method in a different context see Ref. 26. We have calculated the RG function at one-loop order and the function at two-loop order. The diagrams contributing to the two-point function up to this order are depicted in Fig. 1. We obtain with d 2) g g b 2 g 2 O g 3, g c 2 g 2 O g 3, where the coefficients b 2 and c 2 read, in the N-component theory 54426-2 232

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ NON-FERMI-LIQUID BEHAVIOR FROM TWO-... PHYSICAL REVIEW B 69, 54426 24 FIG. 1. Graphs contributing to the two-point function at twoloop order. b 2 N 8 6, c 2 N 2 12 2. 6 144 Details of calculation of the coefficient c 2 are given in Appendix A. B. The marginal case dä2: logarithmic corrections to Landau damping We focus here on the marginal case d 2 at the quantumcritical point, and integrate the above RG equations in order to obtain the corrections to the Landau damping at T. Integrating the RG equations 4, 5 using Eq. 6 yields, to this order g g 1 b 2 g ln, Z exp c 2 g 2 ln 1 b 2 g ln. As expected in the marginal case, g flows logarithmically to zero as, while Z tends to some nonuniversal constant Z*. To analyze the frequency dependence of we use the general RG equation 3, setting q and choosing * such that Z * /( * ) 2 1. This leads to q,i ;g, g * in which we use the notation (g) (q,i i;g, 1). We then expand the RHS of Eq. 8 in powers of g *, with g * 1/(b 2 ln *) 2/ b 2 ln(z* / 2 ). This expansion actually starts at second order in g, since the two-point function is subtracted to insure that one sits at the critical point in other words, the tadpoles I 1 do not contribute to the frequency dependence. Noting that I 2 (q,i ;g, ) I 2 (q,i,g, ) cg 2, coefficient c computed in Appendix A, we finally obtain a correction to the frequency dependence of the two-point function at the T QCP of the form 7 8 q,i 1 6 2 ln 2 11 3 N 2 1. 12 N 8 2 ln 2 Z*/ 2 9 Hence, only subdominant corrections to the Landau damping are generated in the marginal case d 2. Furthermore, the positive coefficient in the above expression can be interpreted as an effective exponent 2/z 1. This is in agreement with the 1/N expansion, which we consider later in this paper. C. Ä2Àd expansion of critical exponents and corrections to scaling We consider here the case d 2, in which the quartic term is relevant, and the effective coupling g tends to a nontrivial fixed point. The case d 1 is relevant, for example, when considering 11 an Ising chain in a transverse field (N 1) or a chain of quantum rotors (N 1) coupled to a dissipative environment. We make use of the general RG equation 3 by choosing * such that q/ * 1. For, we now have g g* and Z with the critical exponent (g*)]. Hence, we obtain at T ) for the dynamical susceptibility 1 at low-frequency and small momentum q, q 2 /cq 2. 1 In this expression, is a universal scaling function associated with the fixed point. From this expression, the dynamical critical exponent is identified as z 2. 11 An expression which holds to all orders, and is the result of the existence of a unique RG function, as explained above. This scaling form can actually be generalized to finite temperature in the quantum critical regime in the form q, 1 T T,c 1q T 1/z 12 with a universal scaling function and c 1 a nonuniversal constant. In particular, at zero-momentum, T (q, ) isan entirely universal scaling function of /T. At small /T, we expect an analytic dependence on /T, with (, )/T C 1 ic 2 /T. In contrast, at large /T we have (, )/T ic 3 /T, and the subdominant terms are not analytic, but are related to those in Eq. 19 below. Here C 1 3 are all nontrivial universal constants. The value of C 3 is determined in the expansion by the computations in Appendix A C 3 1 c( /b 2 ) 2, and this universality is clearly related to the universal logarithmic correction in Eq. 9. The values of C 1,2 require a computation of the damping at T, and accurate determination of these is likely to require a self-consistent treatment of the loop corrections, as discussed in Ref 27. In particular, C 2 C 3 and hence 54426-3 233

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PANKOV, FLORENS, GEORGES, KOTLIAR, AND SACHDEV PHYSICAL REVIEW B 69, 54426 24 Im q, Im q, lim lim lim lim. 13 T T So the Landau damping co-efficient is sensitive to the order of limits of and T ; we expect that similar considerations will also apply to the logarithmic corrections in d 2 noted in Eq. 9. From the RG expressions in Sec. II A, the expansion of the critical exponent 2 z reads, to order 2 N 2 12 2 4 N 8 2 2 O 3. 14 We have also obtained the correlation length exponent at order : 1 2 N 2 4 N 8 O 2. 15 Finally, we comment briefly on the next-to-leading corrections at T to the Landau damping at the QCP. These are obtained from corrections to scaling, i.e., depend on the manner in which g flows to the fixed point: with g g* g g* g* O 2. 16 17 The RG equation then yields the following form for the twopoint function, including corrections to scaling q, q 2 q z q 2 2 q z. 18 Taking the limit q, this yields the following form of the Landau damping term q,i C (2 )/z C 2 (2 )/z. 19 In the present case, the exact identity z 2 implies that the dominant term is simply, while a correction of the form 1 /(2 ) is found. The logarithmic correction found above can be seen as the limiting behavior of this correction in the marginal case d 2 ( ). III. LARGE-N ANALYSIS OF THE 4 THEORY OF AN ANTIFERROMAGNETIC QUANTUM CRITICAL POINT IN TWO DIMENSIONS zä2 In this section, we treat the field theory 1 within the large-n expansion. For this purpose, we scale the coupling constant in Eq. 1 by 1/N and set u 6u N. 2 We denote by and 1 the contributions to the self-energy of order 1/N and 1/N, respectively, ( 1 D 1 ). The skeleton diagrams contributing to and 1 are depicted in Fig. 2. The corresponding expressions read where FIG. 2. Self-energy graphs in the 1/N expansion. q,i ut n 1 q,i 2 u N T n q,i T n d2 k 2 2 D k,i n, 21 d2 k D k q,i n i n, 2 2 1 u k,i n 22 d2 p 2 2 D p,i n D p q,i n i n. 23 In this expression, D denotes the full propagator to order 1/N i.e., including the self-energy ). Hence, Eq. 21 should be viewed as a self-consistent equation for.we first consider this equation at q, defining a gap (T) r (q,i ) related to the correlation length by 2. vanishes ( diverges at the T QCP. At the saddle point (N ) level (T) (T ) (T). We show in Appendix B that, at this order, the self-consistent equation for the gap reads see also Ref. 28 2 2 1 u 2 ln ln q2 q2ln q2 q2, q2 24 where q 2 is a cutoff for q 2, q2 q 2 /(2 T) and /(2 T). In the limit 1 and q2 Eq. 24 reduces to 2 2 1 u 2 ln ln. 25 We dropped the index in q 2 as it is the only scale we use. In the limit u ln 1 the approximate solution of Eq. 25 is 28 ln ln /2 T T ln /2 T. 26 Hence, vanishes faster than T as T. The contributions of the marginally irrelevant coupling are smaller than temperature or energy itself. As we shall now see, this also applies to the T contributions to the Landau damping at the critical point, which is the main quantity of interest in this paper. We perform a calculation of the damping rate at the T QCP up to order 1/N. At the saddle-point level (1/N ), the propagator at the QCP reads D(q,i ) ( q 2 ) 1. 54426-4 234

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ NON-FERMI-LIQUID BEHAVIOR FROM TWO-... The polarization bubble (q,i ) defined by Eq. 23 can then be calculated exactly. Details are given in Appendix C. Here we present the result 8 2 q,i 2 4 q2 q 2 ln 2ln q2 PHYSICAL REVIEW B 69, 54426 24 2 ln s q ln q2 2 s 2 q 2 2 s Li 2 2 q 2 Li 2 q 2 2 q2 2 q Li 2 q 2 Li 2 q 2 Li s 2 s q2 Li s 2 s q2, 27 where s ( q 2 ) 2 4q 2 and Li 2 (x) x dy ln(1 y)/y. The above expression has the following asymptotics at /, q 2 / : 4 2 ln q2, q 2, 2 2 4 2ln, q 2, 2 q 2, 2 2 4 2ln, q 2, q 2 2. 28 FIG. 3. Self-energy at the d 2,z 2 QCP, up to order 1/N, asa function of imaginary frequency. The calculation is for u/(2 ) 3 5, and is measured in units of the cutoff. FIG. 4. Effective exponent ( ), as defined in the text, as a function of imaginary frequency. The parameters are as in Fig. 3. We use the explicit expression 27 into Eq. 22 and perform the frequency and momentum integrals numerically. The resulting self-energy 1 (q,i ) is plotted in Fig. 3. It is seen that the 1/N corrections to the damping rate are less singular than at low frequency. Alternatively, the corrections can be put in the form of an effective, scale-dependent exponent ( ), defined by ( ) ln 1 (i ) / with 1 (i ) 1 (q,i ) 1 (q,i ) the q dynamical susceptibility at the QCP. This effective exponent is plotted on Fig. 4: it is seen to be only weakly dependent on frequency in the range where it is displayed. Hence, the logarithmic corrections in the marginal case can be mimicked by a power law, but correspond to an effective exponent 1, in contrast to the experimental observation 1 for the two compounds mentioned above. Incidentally, we would like to comment on the data analysis made in Ref. 6 in order to support /T scaling of the inelastic neutron scattering data in CeCu 5.9 Au.1. To this aim, we perform a similar analysis on our analytical result, by plotting T (,T) versus /T Fig. 5. In this plot, the frequency-dependent susceptibility at finite temperature and q is approximated by (,T) 1 i (T) 1 (q, ) 1 (q, ) this approximation does not include the temperature dependence of the Bose functions in the calculations which would be required to obtain the correct damping coefficient, as is required to obtain Eq. 13. Such a scaling plot is attempted for a varying range of, and the optimal value of the exponent is chosen such as to yield the best collapse, 6 as measured by the standard deviation plotted in Fig. 6. It is seen that an excellent collapse is obtained with 1.9. We emphasize that /T scaling, in a strict asymptotic sense, does apply here, but with a trivial Gaussian exponent and scaling function T (,T) i/ ( /T). The logarithmic corrections characteristic of the marginal case apparently mimic nontrivial scaling properties over quite an extended range of /T. This should serve as a warning for the interpretation of the experimental results. We end this section by considering the local approxima- 54426-5 235

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PANKOV, FLORENS, GEORGES, KOTLIAR, AND SACHDEV D PHYSICAL REVIEW B 69, 54426 24 1 2 4 ln 2/z e i 2/z d 1 dx ln x 2/z cos x 2 2 1 4 z. 3 The 1/N correction to the spin-fluctuation self-energy is 1 2u2 N D 3 1 N 2u 2 4 3 z 3 3 31 FIG. 5. Scaling plot for T Im (,T,q ) versus /T. The temperatures displayed are T.3.9. The choice 1.9 provides a good collapse of the data over this frequency range. Different symbols correspond to different values of T. tion to the damped 4 theory in the marginal case, up to order 1/N. In a previous paper, Motome and two of us 2 have investigated the EDMFT approximation to the critical behavior of various models, and shown that this local approximation is quite satisfactory above the upper critical dimension i.e., here for d z 4), when the quartic coupling is irrelevant. In this approach, all skeleton graphs are taken to be local. For arbitrary z and d 2, the local propagator at the T QCP reads D i qdq 2 1 2 1 z q 2 4 ln 2/z. 29 2/z Performing the Fourier transform, this yields the asymptotic behavior at long imaginary time FIG. 6. Standard deviation ln(t ), measuring the quality of the /T collapse Ref. 6, versus. which corresponds to the following low-frequency behavior with 1 a cutoff of order 1 : 1 i 1 2u 2 N 3 4 3 z 3 2 1 ln ln 2. 32 For real frequencies Eq. 31 corresponds to corrections to the Landau damping of the form Im 1 ( i ) 2 sgn( ). This is much smaller by a factor of order, up to logarithms than the corrections obtained from the direct calculation in d 2 detailed above Fig. 3. Hence, we conclude that the local EDMFT approximation is not very reliable at the upper critical dimension marginal case. As we shall see in the following section, a numerical solution of the EDMFT equations for the Kondo lattice using an approximate impurity solver based on the large-n expansion leads to a similar conclusion: only subdominant corrections to the Landau damping rate are generated instead of the anomalous power-law-like enhancement observed experimentally. IV. LOCALLY CRITICAL POINT: SELF-CONSISTENT LARGE-N SOLUTION OF THE EDMFT EQUATIONS A. EDMFT of the Kondo lattice and mapping onto a local spinfluctuation model It has been recently argued 14,18 that the understanding of non-fermi liquid behavior in heavy-fermion compounds close to a QCP requires a formalism in which electronic degrees of freedom and spin fluctuations can be treated on the same footing, at least as a starting point. This is a nontrivial task, which is however made easier in the context of the extended dynamical mean-field theory EDMFT Refs. 15 17 of the Kondo lattice model, which we consider here. Let us consider the Kondo lattice Hamiltonian with an explicit exchange coupling between localized spins, chosen to be Ising-like: H k k c k c k J K S i c i c i I ij S z i S z j. i i j 33 54426-6 236

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ NON-FERMI-LIQUID BEHAVIOR FROM TWO-... PHYSICAL REVIEW B 69, 54426 24 In this expression, J K is the antiferromagnetic Kondo coupling, and stands for the Pauli matrices. EDMFT maps this model onto a local impurity model with effective action S d d c G 1 c S z 1 S z d JK S c c. 34 Both G and are effective Weiss fields which must be determined in such a way that the following self-consistency conditions hold: G i n c i n c i n S d i n c i n, 35 i n S z i n S z i n S I d M i n. 36 In this expression, c (i n ) and M(i n ) are, respectively, the fermionic and local spin self-energies calculated within the impurity problem 34, i.e., G 1 i n G 1 i n c i n, 1 i n 1 i n M i n. 37 38 The densities of states and I associated, respectively, with the conduction band and to the spin interactions read ( ) k ( k ), I ( ) q ( I q ). The selfconsistency conditions 35, 36 simply express that the impurity model Green s functions and self-energies should coincide with their lattice counterpart assuming momentum independence of the self-energies. They can be recast into a more compact form, relating directly the local correlation functions to the two Weiss fields appearing in the effective action 34 : G i n d i n d i n G 1 i n G 1 i n, I 1 i n 1 i n. 39 4 The key question at this stage is whether the electronic degrees of freedom can be integrated out, particularly near the QCP. The local EDMFT framework allows one to handle this issue in a more controlled manner. Indeed, electronic degrees of freedom can be integrated out 17 from the impurity model 34 using an expansion in the spin-flip part of the Kondo coupling in the manner of Anderson-Yuval-Hammann. 29,3 This can be conveniently performed, for example, using bosonization methods. 19,31,32 This leads to the following action for the spin degrees of freedom: S d d S z 1 K S z d S x. 41 In this expression, K( ) 1/( ) 2 is a long-range interaction induced by the electronic modes. It corresponds to the bare Landau damping K(i n ) n at low frequency. The coupling in front of the spin-flip term is proportional to the Kondo coupling J K with a short-time cutoff. Hence, the problem has been mapped onto an quantum Ising model in a transverse field with long range interactions. 17,19,32 B. Numerical solution based on the large-n approximation This mapping has been used in order to perform analytical 17 and numerical 32 studies of the behavior at the QCP in mean-field models with random exchange couplings I ij. It was demonstrated that this case is formally similar 33,37 to the Hertz-Moriya-Millis theory of an antiferromagnetic QCP in d 3. Recently, Grempel and Si have analyzed the EDMFT theory of the QCP in the two-dimensional case using this mapping. 19 Their claim is that anomalous Landau damping (, 1) and /T scaling can be demonstrated in this case. Here, we present a solution of this problem in which the local impurity problem is solved using a large-n method. In contrast to the claims of Ref. 19, we find that only subdominant corrections to the Landau damping are generated. Following Ref. 17, we deal with the local action 41 for the transverse field Ising model by extending it to an N-component rotor model n : S 2 1 d n 2 n N 2 g 1 2 d d n 1 K n. 42 In this expression, g plays a role similar to that of the transverse field in the original Ising model disordering term. We define a self-energy correction for the local spin susceptibility ( ) S z ()S z ( ) : 1 i n 2 n /g K i n 1 i n i n. 43 Let us first consider the problem at the saddle-point level (N ), for a given Weiss function. The self-energy is then frequency independent as noted by Grempel and Si 19 and is given by the saddle-point value of the Lagrange multiplier such that i n 44 54426-7 237

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PANKOV, FLORENS, GEORGES, KOTLIAR, AND SACHDEV PHYSICAL REVIEW B 69, 54426 24 1. 45 In this expression, is the local susceptibility at the saddlepoint (N ) level 1 i n 2 n /g K i n 1 i n. 46 Both and depend on the Weiss function. We approximate the self-energy by its first two terms in the 1/N expansion i n i n 1 i n 1 i, 1 2 N, with i n 1 i n and 2. 47 48 49 The EDMFT equations are thus solved by iterating numerically the following procedure. For a given (i n ), the saddle-point quantities and are computed by solving Eqs. 45, 46. Then, the self-energy is computed from Eqs. 47, 48, 49. This is inserted into the EDMFT self-consistency condition in order to obtain the local susceptibility i n d I 2 n /g K i n i n. 5 From this, an updated value of the Weiss function is obtained as 1 (i n ) n 2 /g K(i n ) (i n ) 1 (i n ), and the procedure is iterated until convergence is reached. The result of this numerical calculation for the local spin self-energy is displayed in Fig. 7. The behavior of the selfenergy at low imaginary frequency is well fitted by 1 (i ) 2 ln inset when one tunes the parameters to sit at criticality Fig. 8. This corresponds to subdominant corrections to the Landau damping Im ( i ) 2 sgn( ). These findings are in agreement with the expansion of the EDMFT equations for the 4 model at order 1/N discussed at the end of the previous section, and provide an independent check of this result. The method followed in the present section has been to use the large-n method as an approximate impurity solver for the local problem. It can be easily checked that if, instead, the equations used here are all expanded up to order 1/N, the equations of the local approximation presented in the previous section are recovered. V. CONCLUSION We analyzed two of the approaches to a critical theory of the antiferromagnetic metal to paramagnetic metal in the FIG. 7. Self-energy (i ) close to the critical point at inverse temperature 5 and g g c.223), as a function of imaginary frequency. For this calculation, we have set N 2, and used a flat density of states I ( ) of half-width equal to unity. Inset: plot of 2 (i ), showing the logarithmic behavior of this quantity at small frequencies. CeCu 6 x Au x and YbRh 2 Si 2 materials from the perspective of the antiferromagnetic spin fluctuations. In the context of the model of Rosch et al. 5 we analyzed the corrections to mean field theory due to the marginally irrelevant fourth order coupling. We find that while there are logarithmic corrections to mean field theory which vanish as 1/ln 2 the coefficient of this corrections is such that it can only be mimicked by a decreased Landau damping, in clear disagreement with experimental observations. In our view, this rules out this model, in spite of the fact that it can account nicely for many of the electronic properties of this systems. We also analyze the local version of this model using the large-n expansion. Again in this case, we find that it is not possible to account for the anomalous frequency depen- FIG. 8. Plot of the mass (i) as a function of temperature, demonstrating the existence of a continuous transition at T within our approximate solution of the EDMFT equations. From bottom to top: g.1 ordered state, g.223 critical at T ), g.3 disordered case. 54426-8 238

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ NON-FERMI-LIQUID BEHAVIOR FROM TWO-... PHYSICAL REVIEW B 69, 54426 24 dence of the damping coefficient. This is in disagreement with Ref. 19 but in agreement with a recent EDMFT study of the Anderson lattice model. 34 The anomalous damping has been recently observed in NMR measurements in both 35,36 YbRh 2 Si 2 and CeCu 6 x Au x. While it has different temperature dependencies in each system, it is clearly enhanced above the Fermiliquid temperature dependence. Possible resolution of this puzzle may require a better treatment of the effects of disorder or better treatments of models which retain the fermionic nature of the spin fluctuations as proposed by Coleman et al. 18 and Si et al. 14 ACKNOWLEDGMENTS We thank D. Grempel and Q. Si for useful discussions and correspondence. The authors acknowledge the hospitality of the Kavli Institute for Theoretical Physics UCSB, Santa Barbara, where part of this work was performed and supported under Grant No. NSF-PHY99-7949 and NSF Grant No. DMR-96462. We also acknowledge the support of an international collaborative grant from CNRS PICS 162. APPENDIX A: COMPUTATION OF RG FLOW The computations leading to the RG flow 6 are standard. Here we only provide some further details of the computation of the number c 2, and of the constant c required to obtain the universal logarithmic correction in Eq. 9. These computations involve the term in the propagator in an essential and novel manner. Repeating the standard field-theoretic derivation for the two-loop graph in Fig. 1 we have I 2 q,i u 2 N 2 d 1 18 2 d 2 2 dd q 1 2 d dd q 2 2 d 1 q 1 2 1 q 2 2 2 1 q q 1 q 2 2 1 2 g 2 q 2d 2 c 2 2 O. I 2 is to be evaluated for d 1, where it is convergent, and the result analytically continued to near d 2, where it is expected to have the small expansion shown on the RHS. The coefficient of the pole fixes the value of the constant c 2. Note that we are evaluating I 2 in an external momentum q, and in d 2 the pole on the RHS corresponds to the appearance of q 2 ln q term. In contrast, evaluation of I 2 (,i ) presented below shows that no pole appears, and hence there is no ln term in d 2. This absence of such a pole is, of course, the reason for the absence of an independent renormalization constant for the damping term noted below Eq. 5, and for the dependence of I 2 noted above Eq. 9. It is also responsible for the exponent identity z 2 for d 2. We now turn to the evaluation of I 2 in Eq. A1 in an external momentum q. First, we evaluate the integral over q 2 by the standard Feynman parameter method A1 I 2 q, u 2 N 2 2 d/2 18 4 d/2 d 1 2 d 2 dd q 1 1 1 dx 2 2 d q 2 1 1 1 q q 1 2 x 1 x 2 1 x 1 2 x. 2 d/2 A2 Similarly, performing the integral over q 1 with a Feynman parameter y we obtain I 2 q, u 2 N 2 3 d 18 4 d d 1 2 d 1 2 dx x 1 x /2 2 1 1 dyy /2 q 2 x 1 x y 1 y 1 2 x 1 x 1 y 1 y 1 x 2 xy. 1 A3 Now we perform the integral over 1 and 2 by using the useful formula 54426-9 239

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PANKOV, FLORENS, GEORGES, KOTLIAR, AND SACHDEV PHYSICAL REVIEW B 69, 54426 24 d 1 d 2 1 A B 1 C 2 D 1 2 4A2 B C D 2 B C C D B D. A4 The formula A4 is derived by explicitly performing the integrals over 1,2 over different regions in the 1,2 plane delineated by changes in signs of 1, 2, and 1 2. Note that Eq. A4 is to be evaluated at 1. Consequently, we obtain from Eq. A4 the factor ( 2) ( 1 ) 1/ O( ), which gives us the requisite pole in appearing on the RHS of Eq. A1. So we may safely set d 2 in the remaining terms in Eq. A3. In this manner, combining Eqs. A1, A3, and A4 we obtain c 2 N 2 1 36 dx 1 1 y x 1 x 1 y y 1 x xy dy x 1 y y 1 x 1 y y N 2 12 2. A5 144 Let us now present a few details of the evaluation of I 2 (,i ). In this case, Eq. A3 is replaced by I 2,i u 2 N 2 3 d 18 4 d d 1 2 d 1 2 dx x 1 x /2 2 1 1 dyy /2 1 2 x 1 x 1 y 1 y 1 x 2 xy. 1 A6 Now, instead of Eq. A4, we need the following integral: d 1 d 2 1 A 1 B 2 C 1 2 4 2 ABC 2 A 1 A 2 B 2 A 2 C 2 B 1 C1 B 2 A 2 B 2 C 2 C 2 A 2 C 2 B. 2 A7 As noted below Eq. A4, we need to evaluate Eq. A7 at 1 and pick out a possible pole in. Indeed, a possible pole does appear to be present in the ( 2) prefactor in Eq. A7. However, careful evaluation shows that the residue of such a pole vanishes, and Eq. A7 in fact has a smooth 1 limit: ln A A7 4ABC lim 1 A 2 B 2 A 2 C 2 ln B B 2 A 2 B 2 C 2 ln C C 2 A 2 C 2 B. 2 A8 The limit of the remaining terms in Eq. A6 is straightforward, and this establishes the claimed absence of a ln term in I 2 (,i ) ind 2. Evaluating the constant c in I 2 (,i ) I 2 (,) cg 2 we obtain c N 2 36 P 1 dx 1 dy x 2 1 y 1 2x y 2 x 2 1 y 2 ln x 1 y y N 2 6 2 ln 2 11 3. A9 1728 We performed integrations in Eq. A9 by changing variables to x x, y x(1 y)/y and integrating first over x and then over y. T ut n qdq 2 D q,i APPENDIX B: GAP EQUATION In this appendix we derive the gap equation Eq. 24. We need to evaluate 1 4 ut n n ln. B1 n q 2 54426-1 24

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ NON-FERMI-LIQUID BEHAVIOR FROM TWO-... PHYSICAL REVIEW B 69, 54426 24 We consider a q 2 cutoff q 2, which from now on we denote as, and a frequency cutoff 2 nt which is to be taken to infinity. For this choice of the sharp cutoff one can show S 1 T n ln n n q 2 ln 2 ln, B2 where /(2 T), /(2 T), and /(2 T). Taking the limit we have S 1 T ln 2 ln 2 ln. B3 The finite T self-energy is (T) (1/4 )uts 1 (T), and the zero temperature (T ) is T 1 4 uts 1 T T u ln 2 2 1. B4 Noticing that (T) (T ) (T) we write Eq. 24. APPENDIX C: POLARIZATION BUBBLE Here we show in detail how we compute (q,i ) ind 2 with z 2. We use a finite cutoff q 2 for q 2 in a momentum integration, and we use as a frequency cutoff. We use the Feynman parametrization to find q,i d2 p 2 2 d 2 D p,i D p q,i i 4 1 1 dx d 2 l x,q,,i 1 l x,q,,i 1, C1 where l(x,q,,i ) x(1 x)q 2 x (1 x) and D 1 q 2 at the QCP. Integrating out frequency we get q,i 1 8 2P 1 4x dx 1 2x ln 1 x q 2 x ln 1 2q 2 q2 s x q 2 x 1 2 q2 s, C2 where s ( q 2 ) 2 4q 2. It is easy to show 1 4x P dx 1 2x ln ax b 2 2a b ln a b a b 2lnb Li 2 a Li a 2b 2 a 2b a C3 for a b and b. From Eqs. C2 and C3, after simple algebra, Eq. 27 follows. 1 For a recent review, see G.R. Stewart, Rev. Mod. Phys. 73, 797 21. 2 H.v. Löhneysen, T. Pietrus, G. Portisch, H.G. Schlager, A. Schroder, M. Sieck, and T. Trappmann, Phys. Rev. Lett. 72, 3262 1994. 3 A. Schroder, G. Aeppli, R. Coldea, M. Adams, O. Stockert, H.v. Löhneysen, E. Bucher, R. Ramazashvili, and P. Coleman, Nature London 47, 351 2. 4 O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F.M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and F. Steglich, Phys. Rev. Lett. 85, 626 2. 5 A. Rosch, A. Schröder, O. Stockert, and H.v. Lohneysen, Phys. Rev. Lett. 79, 159 1997. 6 A. Schröder, G. Aeppli, E. Bucher, R. Ramazashvili, and P. Coleman, Phys. Rev. Lett. 8, 5623 1998. 7 O. Stockert, H.v. Löhneysen, A. Rosch, N. Pyka, and M. Loewenhaupt, Phys. Rev. Lett. 8, 5627 1998. 8 J. Hertz, Phys. Rev. B 14, 1165 1976. 9 T. Moriya and J. Kawabata, J. Phys. Soc. Jpn. 34, 639 1973 ; 35, 669 1973. 1 A.J. Millis, Phys. Rev. B 48, 7183 1993. 11 For a review, see: S. Sachdev, Quantum Phase Transitions Cambridge University Press, Cambridge, 1999. 12 I. Paul and G. Kotliar, Phys. Rev. B 64, 184414 21. 13 A. Abanov and A. Chubukov, Phys. Rev. Lett. 84, 568 2. 14 Q. Si, S. Rabello, K. Ingersent, and L. Smith, Nature London 413, 84 21. 15 Q. Si and J.L. Smith, Phys. Rev. Lett. 77, 3391 1996 ; Phys. Rev. B 61, 5184 2. 16 H. Kajueter and G. Kotliar unpublished ; H. Kajueter, Ph.D. thesis, Rutgers University, New Brunswick, NJ, 1996. 17 A.M. Sengupta and A. Georges, Phys. Rev. B 52, 1295 1995. 18 P. Coleman and C. Pépin, Acta Phys. Pol. B 34, 691 23. 19 D.R. Grempel and Q. Si, Phys. Rev. Lett. 91, 2641 23. 2 S. Pankov, G. Kotliar, and Y. Motome, Phys. Rev. B 66, 45117 22. 21 B. Spivak, A. Zyuzin, and M. Hruska, Phys. Rev. B 64, 13252 21. 22 See, e.g., J. Zinn-Justin, Quantum Field and Critical Phenomena Clarendon Press, Oxford, 1989, in particular Chap. 22. 23 J. Sak, Phys. Rev. B 8, 281 1973. 24 A. Gamba, M. Grilli, and C. Castellani, Nucl. Phys. B 556, 463 1999. 25 A. Aharony, in Phase Transitions and Critical Phenomena, edited 54426-11 241

ÖØ Ð ÆÓÒ¹ ÖÑ Ð ÕÙ Ú ÓÖ ÖÓÑØÛÓ¹ Ñ Ò ÓÒ Ð ÒØ ÖÖÓÑ Ò Ø ÙØÙ Ø ÓÒ PANKOV, FLORENS, GEORGES, KOTLIAR, AND SACHDEV PHYSICAL REVIEW B 69, 54426 24 by C. Domb and M. S. Green Academic, London, 1976, Vol. 6. 26 D. Boyanovsky and J.L. Cardy, Phys. Rev. B 26, 154 1982. 27 S. Sachdev, Phys. Rev. B 55, 142 1997. 28 S. Sachdev, A.V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14 874 1995. 29 P.W. Anderson, G. Yuval, and D.R. Hamann, Phys. Rev. B 1, 4464 197. 3 S. Chakravarty and J. Rudnick, Phys. Rev. Lett. 75, 51 1995. 31 G. Kotliar and Q. Si, Phys. Rev. B 53, 12373 1996. 32 D.R. Grempel and M.J. Rozenberg, Phys. Rev. B 6, 472 1999. 33 It is not yet clear at this stage whether the EDMFT approach to the antiferromagnetic Kondo lattice in the nonrandom case follows the conventional Hertz-Millis behavior, or undergoes a first-order transition from the Fermi liquid to the ordered phase. A large-n study for an SU(N)-symmetric exchange coupling in contrast to the Ising case studied here provides indications in favor of the latter possibility Ref. 35. 34 P. Sun and G. Kotliar, Phys. Rev. Lett. 91, 3729 23. 35 K. Ishida, K. Okamoto, Y. Kawasaki, Y. Kitaoka, O. Trovarelli, C. Geibel, and F. Steglich, Phys. Rev. Lett. 89, 1722 22. 36 R.E. Walstedt, H. Kojima, N. Butch, and N. Bernhoeft, Phys. Rev. Lett. 9, 6761 23. 37 S. Burdin, M. Grilli, and D. Grempel, Phys. Rev. B 67, 12114 23. 54426-12 242

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ÖØ Ð º ÝÒ Ñ ÐÅ Ò¹ Ð Ì ÓÖÝÓ Ê ÓÒ Ø Ò ¹Î Ð Ò ¹ ÓÒ ÒØ ÖÖÓÑ Ò Ø VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 21 Dynamical Mean-Field Theory of Resonating-Valence-Bond Antiferromagnets Antoine Georges, Rahul Siddharthan, and Serge Florens Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 5, France and Laboratoire de Physique des Solides, Université Paris-Sud, Bât. 51, 9145 Orsay, France (Received 25 June 21; published 13 December 21) We propose a theory of the spin dynamics of frustrated quantum antiferromagnets, which is based on an effective action for a plaquette embedded in a self-consistent bath. This approach, supplemented by a lowenergy projection, is applied to the Kagomé antiferromagnet. We find that a spin-liquid regime extends to very low energy, in which local correlation functions have a slow decay in time, well described by powerlaw behavior and v T scaling of the response function: x v ~v 2a F v T. DOI: 1.113/PhysRevLett.87.27723 PACS numbers: 75.1.Jm, 75.1.Nr, 75.4.Gb The possibility of frustrated quantum antiferromagnets (QAF) having a resonating-valence-bond (RVB) ground state, that is, a superposition of states where all spins are paired into singlets, was suggested many years ago [1]; such a ground state has neither long-range spin order (e.g., Néel order) nor spin-peierls order (an ordered arrangement of these singlet pairs). Recently, convincing evidence has been given that some frustrated two-dimensional QAF indeed have RVB physics [2,3]. In particular, numerical studies [2] of the spin-half Heisenberg QAF on the Kagomé lattice reveal that the model has no long-range order and displays a very small gap to magnetic (triplet) excitations (estimated to be of order J 2). Moreover, this gap is filled with an exponential number of singlet excitations, suggesting a possible continuum in the thermodynamic limit, and a large low-temperature entropy, in agreement with the RVB picture. We also note, though this is seldom emphasized, that the results of [2] suggest a correspondingly large number of triplet states immediately above the gap. This may be expected, since in any valence-bond component of an RVB state, one can replace any singlet pair by a triplet pair and continue to have an eigenstate of S. Resonance between such states would lower the excitation energy to much below J. These considerations suggest that, in a temperature range above the triplet gap, the spin correlations have a rapid decay in space, but a slow decay in time due to the large density of triplet excited states. Equivalently, the dynamical susceptibility x q, v would not display a narrow peak around a specific ordering wave vector, but x q, v would have weight at low frequency, characteristic of a spin liquid. Indeed, recent inelastic neutron scattering studies [4] on the S 3 2 Kagomé slab compound SrCr 9p Ga 1229p O 19 (SCGO) reveal such behavior above the freezing temperature T $ 4 K. The only relevant energy scale in this spin-liquid regime is apparently set by the temperature itself, with xloc P q x q, v obeying scaling behavior xloc v 2a F v T with a.4, corresponding to a slow decay in time of the local dynamical correlations S x, S x, t 1 t 12a 1 t.6. We observe that the uniform q susceptibility in this regime is also well fitted by x~t 2.4. In this Letter, we introduce a novel theoretical approach to the spin dynamics in the spin-liquid regime of frustrated QAF, which focuses on short-range spin correlations only. This approach draws inspiration from the dynamical mean-field theory (DMFT) of itinerant fermion models [5], and some of its extensions [5 1]. Our method is fairly general but is applied here to the concrete case of the spin-half Kagomé QAF. The determination of on-site and nearest-neighbor dynamical spin correlations is mapped onto the solution of a model of quantum spins on a triangular plaquette coupled to a self-consistent bath. (A singlesite approach is not adequate, since the essential physics of singlet formation involves at least nearest-neighbor sites.) By using a projection onto the low-energy sector of this model, we demonstrate that, in a temperature range extending down to T ø J, a slow (power-law) decay of temporal correlations is found. We view the Kagomé lattice as a triangular superlattice of up-pointing triangular plaquettes (Fig. 1). Sites are labeled by an index a 1, 2, 3 within a plaquette and by a triangular superlattice index I numbering the plaquette. We denote by x ab q, t2t the Fourier transform of the dynamical spin correlation function 1 3 S a,i t? S b,i t with respect to the plaquette coordinates I, I ( q is a vector of the supercell Brillouin zone). Our approach relies on approximating the correlation function by x q, in n 21 ab J ab q 1 M ab in n. (1) Here, n n 2np b is a bosonic Matsubara frequency, and all quantities are matrices in the internal plaquette indices a, b. J ab q is the supercell Fourier transform of the exchange couplings. M ab is a measure of how much the correlation function differs from that of a Gaussian model, for which x J 21, and hence plays the role of a spin self-energy matrix for the QAF. Obviously, the key assumption made in (1) is that the q dependence of this self-energy can be neglected. This approximation, which is 27723-1 31-97 1 87(27) 27723(4)$15. 21 The American Physical Society 27723-1 245

ÖØ Ð º ÝÒ Ñ ÐÅ Ò¹ Ð Ì ÓÖÝÓ Ê ÓÒ Ø Ò ¹Î Ð Ò ¹ ÓÒ ÒØ ÖÖÓÑ Ò Ø VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 21 2 2 3 1 3 2 3 1 1 through time-dependent external fields with a Gaussian correlator D ab. The latter will be determined by a selfconsistency requirement, which stipulates that the correlation functions on a plaquette calculated from the above action [x ab t2t 1 3 S a t? S b t S involve the same self-energy as the entire lattice. This reads xab 21 in n JD ab 1 D ab in n 1 M ab in n, where JD ab J 1 2d ab is the matrix of nearest-neighbor couplings on the cluster. Inserting this relation for the self-energy into (1) and imposing that P q x ab q x ab lead to the final form of the self-consistency requirement x ab in n X q J q 2 JD 1x 21 in n 2 D in n 21 ab. (3) FIG. 1. The Kagomé lattice, viewed as an up-pointing cluster (bold) embedded in a network of similar clusters, which is approximated in the DMFT approach by a self-consistent bath. likely to be reasonable when spatial correlations are short ranged, is analogous to the assumption of a momentumindependent self-energy made in the DMFT of correlated fermion models. Here, however, the DMFT concept is extended on two accounts: the local ansatz is made on the spin correlation function rather than on a single-fermion quantity, and a plaquette rather than a single site is considered. Extensions of DMFT to clusters [5 7] and to (spin or charge) response functions [8 1] have been previously considered separately in different contexts. Combining them is a unique aspect of our approach, which is necessary to capture the dynamical aspects of inter-site singlet formation at the heart of spin-liquid behavior. In order to calculate the dynamical self-energy matrix M ab in n, an effective action involving only the spins of a single triangular plaquette is introduced, which reads S S B 1 1 2 J X afib 1 1 2 Z b Z b dt S a? S b dt dt X D ab t2t S a t? S b t, ab (2) in which S B denote Berry phase terms. D ab t2t is a retarded interaction, generated by integrating out all spins outside the plaquette. Higher order interactions are also induced in this process, which have been neglected in (2). Equivalently, one can view the rest of the lattice as an external bath which couples to the spins in the cluster x Z s 3 M m 1 J 2 2 3 Je M s M m 1 J 2 2J 2 2 2 3 M r e de, s 2 M m Je We note that the matrix J q ab 2 JD ab involves only the exchange constants linking together different (uppointing) triangular plaquettes. In the limit where these interplaquette couplings vanish while the internal ones are kept fixed (decoupled triangles), our approach becomes exact since (3) implies D ab and (2) reduces to the action associated with the Heisenberg model on a single triangle. We also note that the above DMFT equations can be derived from a Baym-Kadanoff formalism in which a functional of the correlation function and self-energy matrix is introduced in the form V x,m 1 2 Tr ln M 1 J 2 1 2 Tr xm 1F x. (4) The stationarity conditions dv dx dv d M lead to the above equations when the exact functional F is approximated by its value for a single triangular plaquette. The free energy of the model can thus be calculated in the DMFT approach by inserting the self-consistent values of x and M into (4). Alternatively, a functional of the correlation function only can be used, along the lines of Ref. [9]. In a phase with unbroken translational invariance, the matrix x ab in n in fact reduces to two elements: the local susceptibility x loc x aa and the nearest-neighbor one x nn x ab a fi b (and similarly for D ab ). It is actually convenient to use the following linear combinations (proportional to the eigenvalues of the x ab and D ab matrices): x s 3 x loc 1 2x nn, x m 3 x loc 2x nn, D s 1 3 D loc 1 2D nn, and D m 1 3 D loc 2 D nn. Introducing the corresponding self-energies M s in n 3 x s in n 2 3D s in n, M m in n 3 x m in n 2 3D m in n, straightforward but tedious algebra allows us to recast (3) in the form of two scalar equations: x Z m 3 M s M m 2 J 2 1 1 3 J 2M m 2 MJ e M m 2 J M s M m 1 J 2 2J 2 2 2 3 M r e de. s 2 M m Je Here, e stands for cosq 1 1 cosq 2 1 cosq 3, where the q i are components of q along the three directions of the triangular 27723-2 27723-2 (5) 246

ÖØ Ð º ÝÒ Ñ ÐÅ Ò¹ Ð Ì ÓÖÝÓ Ê ÓÒ Ø Ò ¹Î Ð Ò ¹ ÓÒ ÒØ ÖÖÓÑ Ò Ø VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 21 lattice q 1 1 q 2 1 q 3, and r e is the corresponding density of states. The q summation in (3) simplifies into an e integration in (5). Equations (2) and (5) entirely define the plaquette- DMFT approach introduced here, and one could embark at this stage into a numerical determination of the two key dynamical quantities x loc t and x nn t using, e.g., a generalization of the quantum Monte Carlo algorithm recently introduced in [11] in the context of quantum spin glasses. Instead, we shall gain further insight into the problem by making use of a projection onto the lowenergy sector of the Hilbert space [12]. Interesting insights into the low-energy excited states of the singlet sector have been recently obtained by Mila and Mambrini starting from the limit of decoupled up-pointing triangles [13,14]. Each such triangle has a fourfold degenerate ground state with spin S 1 2 and a fourfold degenerate excited state with spin S 3 2. Hence the lattice of decoupled triangles has an exponentially large number of ground states, equal to 4 Ns 3. The physical picture of [13,14] is S low S B 1 Z b that the degenerate ground state broadens into a band as the intertriangle coupling is turned on. In the following, we shall project the DMFT equations onto the lowenergy Hilbert space corresponding to the S 1 2 sector of each triangle. This amounts to first diagonalizing the single-plaquette Hamiltonian (neglecting D ab ) and then rewriting the retarded term in the action (2) in this lowenergy sector. To this end, it is convenient to follow [12,13] and choose as a basis of the fourfold degenerate ground state the eigenvectors of the two following operators: the spin-1 2 operator corresponding to the total spin S in a plaquette, and a doublet of Pauli matrices T T x, T y with the meaning of a plaquette chirality operator. In terms of those, the low-energy projection of the three spin operators in a plaquette reads (with v expi2p 3 and for a 1, 2, 3) S low a 1 3 S 1 2 2v 12a T 2 2 2v a21 T 1. (6) The low-energy projection of the action (2) is then easily obtained as 1 dt dt 2 D s t 2t 1 D m t2t T t? T t S t? S t, (7) where D s and D m are as defined earlier. It turns out that x s and x m have simple interpretations as the total spin and mixed correlation functions: x s t 2 1 2 G s t G s 2t, x m t 2G s t G s 2t G t t G t 2t. (1) x s t2t 1 3 S t? S t, x m t2t 1 3 S t? S t T t? T t. To solve the local quantum problem defined by (7), we use an approximate technique that has proven successful in recent studies of quantum spin-glass models [1] and of impurities in quantum antiferromagnets [15]. We introduce two doublets of spin-1 2 fermions, s ", s # and t ", t #, subject to the constraints, s" 1 s " 1 s# 1 s # t" 1 t " 1 t# 1 t # 1, in order to represent the operators S and T as S 1 s" 1 s #, S 2 s# 1 s ", S z s" 1 s " 2 s# 1 s # 2; T 1 t" 1 t #, T 2 t# 1 t ". The interacting fermion problem corresponding to (7) is then solved in the self-consistent Hartree approximation. Also, the local constraint is approximated by its average (the associated Lagrange multiplier is zero due to particle-hole symmetry). Introducing the Green s functions G s and G t for the two fermion fields and defining self-energies by Gs 21 iv n iv n 2S s iv n, Gt 21 iv n iv n 2S t iv n [with v n 2n 1 1 p b a fermionic Matsubara frequency], yields the imaginary-time equations: S s t 2 3 8 D s t G s t 1 3D m t G s t G t t G t 2t, (9) S t t 3D m t G t t G s t G s 2t. Then the local problem [for a given bath D s, D m ] reduces to two coupled nonlinear integral equations. The correlation functions are given by (8) In practice, we use the following algorithm: we start with an initial guess for the bath D s,m, and obtain the Green s functions by iteration of Eq. (9). We can then calculate the susceptibilities from (1), as well as the self-energies M s,m. Inserting the latter into (5) yields new x s,m and, in turn, the new baths Ds,m new in n 2M s,m in n 3 1 1 x s,m in n. We now discuss our findings when solving these coupled equations numerically. The first key observation is that a paramagnetic solution can be stabilized down to the lowest temperature we could reach, with no sign of long-range ordering intervening. Long-range order is associated with a diverging eigenvalue of x ab q for some q and hence, within our scheme, to a vanishing denominator in (5), which we never observe [16]. In Fig. 2, we display our results for the temperature dependence of the local (i.e., R on-site, or q-integrated) susceptibility x loc T b x loc t dt. At high temperatures, x loc ~ 1 T obeys Curie s law. Below T.5J, the effective Curie constant decreases with decreasing T, indicating gradual quenching of the local moment. However, x loc itself continues to increase as the temperature is lowered. In the inset of Fig. 2, we display the effective exponent defined by a T 2 lnx loc lnt. A transient regime, corresponding to a slowly decreasing a, is apparent and extends over more than two decades. At the lowest temperatures, a appears to saturate to a value a.5. Correspondingly, the correlation function x loc t obeys a power-law dependence on time in the low-temperature 27723-3 27723-3 247

ÖØ Ð º ÝÒ Ñ ÐÅ Ò¹ Ð Ì ÓÖÝÓ Ê ÓÒ Ø Ò ¹Î Ð Ò ¹ ÓÒ ÒØ ÖÖÓÑ Ò Ø VOLUME 87, NUMBER 27 P H Y S I C A L R E V I E W L E T T E R S 31 DECEMBER 21 χτ.4.3.2.1 α.5.6.7.8.9 1-6 -5-4 -3-2 -1 log(t).5 1 T FIG. 2. Main plot: x loc obeys Curie law (~1 T) at high temperature, but diverges more slowly than 1 T at lower temperature. Inset: evolution of the effective exponent a from high to low temperature when fitting x loc T to 1 T a.(t is in units of J in all figures.) regime. Figure 3 shows x loc t x loc b 2 as a function of t b for several (large) values of b 1 T; the graphs collapse on each other for different b, and are well fitted by the expression x loc t x loc b 2 sinpt b 2 12a, with 1 2a.5. This scaling function is the one appearing in the study of quantum spin glasses [1] and quantum impurity models [15,17] and is dictated in the latter case by conformal invariance. It corresponds to the following scaling form of the dynamical susceptibility: xloc v ~v 2a F a v T with F a x x a jg 12a 2 1 i x 2p j2 sinh x 2. This also implies diverging behavior of the relaxation rate 1 T 1 ~ Txloc v v 1 T a. These findings can be interpreted as the formation of a spin-liquid regime with a large density of triplet excited states, consistent with Ref. [2]. Our approach fails to predict a triplet gap, however, because our triangular plaquette has a spin-1 2 ground state, and the classical bath to which it is coupled cannot fully screen this local moment. Hence the local susceptibility continues to increase at low temperatures in our approach, down to an nonphysically low-energy scale (of order J 1). In fact the triplet gap for the S 1 2 Kagomé QAF is known [2] to be quite small (of order J 2), so our description of the spin dynamics should be valid down to a rather lowtemperature scale. The power-law behavior of the spin correlations and the low-temperature increase of the susceptibility agree well with experimental findings on SCGO above its freezing temperature at T 4 K [4]. Since this is a S 3 2 system, it is conceivable that higher spin extends the range of applicability of our approach even farther. A promising application of our approach is the pyrochlore lattice: the natural plaquette is a tetrahedron, with a twofold degenerate singlet ground state, so that the very low-energy singlet sector should be accessible within our approach. In future work we plan to address these issues, and also study the low-temperature thermodynamics χ(τ)/χ(β/2) 5 4 3 2 1 Scaling func beta=11 beta=16 beta=21.2.4.6.8 1 τ/β FIG. 3. For different (low) temperatures, x loc t x loc b 2 plotted as a function of t b collapses on a single curve, well described by sin pt b 2.5 (solid curve). by solving the self-consistent local problem (2) using exact numerical techniques. We are grateful to O. Parcollet for his very useful help with numerical routines, and to C. Lhuillier, G. Misguich, R. Moessner, and S. Sachdev for discussions. We also thank C. Mondelli for sharing her data with us. [1] P. W. Anderson, Mater. Res. Bull. 8, 153 (1973). [2] C. Waldtmann, H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L. Pierre, Eur. Phys. J. B 2, 51 (1998); P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and P. Sindzingre, Phys. Rev. B 56, 2521 (1997). [3] R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (21); R. Moessner, S. L. Sondhi, and E. Fradkin, condmat/13396. [4] C. Mondelli, H. Mutka, C. Payen, B. Frick, and K. H. Andersen, Physica (Amsterdam) 284B, 1371 (2); C. Mondelli, Ph.D. thesis, Université Joseph Fourier, Grenoble, 2 (unpublished). [5] A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). [6] A. Schiller and K. Ingersent, Phys. Rev. Lett. 75, 113 (1995). [7] M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke, and H. R. Krishnamurthy, Phys. Rev. B 58, R7475 (1998). [8] J. Lleweilun Smith and Q. Si, Phys. Rev. B 61, 5184 (2). [9] R. Chitra and G. Kotliar, Phys. Rev. B 63, 11511 (21). [1] A. Georges, O. Parcollet, and S. Sachdev, Phys. Rev. Lett. 85, 84 (2); Phys. Rev. B 63, 13446 (21). [11] D. R. Grempel and M. J. Rozenberg, Phys. Rev. Lett. 8, 389 (1998). [12] V. Subrahmanyam, Phys. Rev. B 52, 1133 (1995). [13] F. Mila, Phys. Rev. Lett. 81, 2356 (1998). [14] M. Mambrini and F. Mila, Eur. Phys. J. B 17, 651 (2). [15] M. Vojta, C. Buragohain, and S. Sachdev, Phys. Rev. B 61, 15 152 (2). [16] In fact, a useful approximation to Eq. (5) is to replace the Hilbert transform of r e by its high-frequency behavior. [17] O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta, Phys. Rev. B 58, 3794 (1998). 27723-4 27723-4 248

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