1932

Abstract

The repulsive Hubbard model has been immensely useful in understanding strongly correlated electron systems and serves as the paradigmatic model of the field. Despite its simplicity, it exhibits a strikingly rich phenomenology reminiscent of that observed in quantum materials. Nevertheless, much of its phase diagram remains controversial. Here, we review a subset of what is known about the Hubbard model based on exact results or controlled approximate solutions in various limits, for which there is a suitable small parameter. Our primary focus is on the ground state properties of the system on various lattices in two spatial dimensions, although both lower and higher dimensions are discussed as well. Finally, we highlight some of the important outstanding open questions.

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2022-03-10
2024-06-17
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Literature Cited

  1. 1. 
    Hubbard J. 1963. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 276:1365238–57
    [Google Scholar]
  2. 2. 
    Qin M, Schäfer T, Andergassen S, Corboz P, Gull E. 2022. Annu. Rev. Condens. Matter Phys. 13:275302
    [Google Scholar]
  3. 3. 
    Anderson PW. 1987. Science 235:47931196–98
    [Google Scholar]
  4. 4. 
    Emery VJ. 1987. Phys. Rev. Lett. 58:262794–97
    [Google Scholar]
  5. 5. 
    Scalapino DJ. 2012. Rev. Mod. Phys. 84:41383–417
    [Google Scholar]
  6. 6. 
    Lieb EH, Loss M, McCann RJ. 1993. J. Math. Phys. 34:3891–98
    [Google Scholar]
  7. 7. 
    Yang CN. 1989. Phys. Rev. Lett. 63:192144–47
    [Google Scholar]
  8. 8. 
    Zhang S. 1991. Int. J. Mod. Phys. B 5:1153–67
    [Google Scholar]
  9. 9. 
    Yang CN, Zhang SC. 1990. Mod. Phys. Lett. B 04:11759–66
    [Google Scholar]
  10. 10. 
    Lieb EH. 1989. Phys. Rev. Lett. 62:101201–4
    [Google Scholar]
  11. 11. 
    Lieb E, Mattis D. 1962. J. Math. Phys. 3:4749–51
    [Google Scholar]
  12. 12. 
    Tasaki H. 1989. Phys. Rev. B 40:139192–93
    [Google Scholar]
  13. 13. 
    Tasaki H. 2021. Physics and Mathematics of Quantum Many-Body Systems Cham, Switz: Springer
    [Google Scholar]
  14. 14. 
    Nagaoka Y. 1966. Phys. Rev. 147:1392–405
    [Google Scholar]
  15. 15. 
    Thouless DJ. 1965. Proc. Phys. Soc. 86:5893–904
    [Google Scholar]
  16. 16. 
    Lieb E, Schultz T, Mattis D. 1961. Ann. Phys. 16:3407–66
    [Google Scholar]
  17. 17. 
    Yamanaka M, Oshikawa M, Affleck I. 1997. Phys. Rev. Lett. 79:61110–13
    [Google Scholar]
  18. 18. 
    Oshikawa M. 2000. Phys. Rev. Lett. 84:71535–38
    [Google Scholar]
  19. 19. 
    Hastings MB. 2004. Phys. Rev. B 69:10104431
    [Google Scholar]
  20. 20. 
    Laughlin RB. 1981. Phys. Rev. B 23:105632–33
    [Google Scholar]
  21. 21. 
    Parameswaran SA, Turner AM, Arovas DP, Vishwanath A. 2013. Nat. Phys. 9:5299–303
    [Google Scholar]
  22. 22. 
    Watanabe H, Po HC, Vishwanath A, Zaletel M. 2015. PNAS 112:4714551–56
    [Google Scholar]
  23. 23. 
    Schumann R. 2002. Ann. Phys. 11:149–87
    [Google Scholar]
  24. 24. 
    Scalapino DJ, Trugman SA. 1996. Philos. Mag. B 74:5607–10
    [Google Scholar]
  25. 25. 
    Yao H, Kivelson SA. 2010. Phys. Rev. Lett. 105:16166402
    [Google Scholar]
  26. 26. 
    White SR, Chakravarty S, Gelfand MP, Kivelson SA. 1992. Phys. Rev. B 45:95062–65
    [Google Scholar]
  27. 27. 
    Altman E, Auerbach A. 2002. Phys. Rev. B 65:10104508
    [Google Scholar]
  28. 28. 
    Tsai WF, Kivelson SA. 2006. Phys. Rev. B 73:21214510 Erratum. 2007. Phys. Rev. B 76:139902
    [Google Scholar]
  29. 29. 
    Yao H, Tsai WF, Kivelson SA. 2007. Phys. Rev. B 76:16161104
    [Google Scholar]
  30. 30. 
    Baruch S, Orgad D. 2010. Phys. Rev. B 82:13134537
    [Google Scholar]
  31. 31. 
    Wachtel G, Baruch S, Orgad D. 2017. Phys. Rev. B 96:6064527
    [Google Scholar]
  32. 32. 
    Ying T, Mondaini R, Sun XD, Paiva T, Fye RM, Scalettar RT. 2014. Phys. Rev. B 90:7075121
    [Google Scholar]
  33. 33. 
    Karakonstantakis G, Berg E, White SR, Kivelson SA. 2011. Phys. Rev. B 83:5054508
    [Google Scholar]
  34. 34. 
    Lieb EH, Wu FY. 1968. Phys. Rev. Lett. 20:251445–48
    [Google Scholar]
  35. 35. 
    Lin HH, Balents L, Fisher MPA. 1997. Phys. Rev. B 56:116569–93
    [Google Scholar]
  36. 36. 
    Emery VJ, Kivelson SA, Zachar O. 1997. Phys. Rev. B 56:106120–47
    [Google Scholar]
  37. 37. 
    Giamarchi T. 2003. Quantum Physics in One Dimension 121 Oxford, UK: Clarendon Press
    [Google Scholar]
  38. 38. 
    Kohn W, Luttinger JM. 1965. Phys. Rev. Lett. 15:12524–26
    [Google Scholar]
  39. 39. 
    Scalapino DJ, Loh E, Hirsch JE. 1986. Phys. Rev. B 34:118190–92
    [Google Scholar]
  40. 40. 
    Bealmonod M, Bourbonnais C, Emery V. 1986. Phys. Rev. B 34:117716–20
    [Google Scholar]
  41. 41. 
    Maiti S, Chubukov AV. 2013. AIP Conf. Proc. 1550:13–73
    [Google Scholar]
  42. 42. 
    Polchinski J. 1993. Theoretical Advanced Study Institute in Elementary-Particle Physics: Recent Directions in Particle Theory: From Superstrings and Blackholes to the Standard Model (TASI '92)235–74 Singapore: World Sci.
    [Google Scholar]
  43. 43. 
    Shankar R. 1994. Rev. Mod. Phys. 66:1129–92
    [Google Scholar]
  44. 44. 
    Kagan MY, Chubukov AV. 1988. Sov. J. Exp.Theor. Phys. Lett. 47:614–17
    [Google Scholar]
  45. 45. 
    Raghu S, Kivelson SA, Scalapino DJ. 2010. Phys. Rev. B 81:22224505
    [Google Scholar]
  46. 46. 
    Hlubina R. 1999. Phys. Rev. B 59:149600–5
    [Google Scholar]
  47. 47. 
    Deng Y, Kozik E, Prokof'ev NV, Svistunov BV 2015. Europhys. Lett. 110:557001
    [Google Scholar]
  48. 48. 
    Markiewicz RS. 1997. J. Phys. Chem. Solids 58:81179–310
    [Google Scholar]
  49. 49. 
    Dzyaloshinskii IE. 1987. J. Exp. Theor. Phys. 66:4848–54
    [Google Scholar]
  50. 50. 
    Schulz HJ. 1987. Europhys. Lett. 4:5609–15
    [Google Scholar]
  51. 51. 
    Abrikosov AA, Beneslavskii SD. 1971. J. Low Temp. Phys. 5:2141–54
    [Google Scholar]
  52. 52. 
    Abrikosov AA. 1974. J. Exp. Theor. Phys 66:1443–60
    [Google Scholar]
  53. 53. 
    Sun K, Yao H, Fradkin E, Kivelson SA. 2009. Phys. Rev. Lett. 103:4046811
    [Google Scholar]
  54. 54. 
    Vafek O, Yang K 2010. Phys. Rev. B 81:4041401
    [Google Scholar]
  55. 55. 
    Balents L, Fisher MPA. 1996. Phys. Rev. B 53:1812133–41
    [Google Scholar]
  56. 56. 
    Pujari S, Lang TC, Murthy G, Kaul RK. 2016. Phys. Rev. Lett. 117:8086404
    [Google Scholar]
  57. 57. 
    Chubukov AV. 2009. Phys. C Supercond. 469:9640–50
    [Google Scholar]
  58. 58. 
    Ulmke M. 1998. Eur. Phys. J. B Condens. Matter Complex Syst. 1:3301–4
    [Google Scholar]
  59. 59. 
    Mielke A. 1991. J. Phys. A Math. Gen. 24:2L73–77
    [Google Scholar]
  60. 60. 
    Mielke A. 1992. J. Phys. A Math. Gen. 25:164335–45
    [Google Scholar]
  61. 61. 
    Shastry BS, Krishnamurthy HR, Anderson PW. 1990. Phys. Rev. B 41:42375–79
    [Google Scholar]
  62. 62. 
    Emery VJ, Kivelson SA, Lin HQ. 1990. Phys. Rev. Lett. 64:4475–78
    [Google Scholar]
  63. 63. 
    Liu L, Yao H, Berg E, White SR, Kivelson SA. 2012. Phys. Rev. Lett. 108:12126406
    [Google Scholar]
  64. 64. 
    Abrikosov AA, Khalatnikov IM. 1958. Sov. Phys. J. Exp. Theor. Phys. 6:5888–92
    [Google Scholar]
  65. 65. 
    Engelbrecht JR, Randeria M, Zhang L. 1992. Phys. Rev. B 45:1710135–38
    [Google Scholar]
  66. 66. 
    Galitskii VM. 1958. Sov. Phys. J. Exp. Theor. Phys. 7:1104–12
    [Google Scholar]
  67. 67. 
    Chubukov AV. 1993. Phys. Rev. B 48:21097–104
    [Google Scholar]
  68. 68. 
    Affleck I, Marston JB. 1988. Phys. Rev. B 37:73774–77
    [Google Scholar]
  69. 69. 
    Arovas DP, Auerbach A. 1988. Phys. Rev. B 38:1316–32
    [Google Scholar]
  70. 70. 
    Marston JB, Affleck I. 1989. Phys. Rev. B 39:1611538–58
    [Google Scholar]
  71. 71. 
    Read N, Sachdev S. 1991. Phys. Rev. Lett. 66:131773–76
    [Google Scholar]
  72. 72. 
    Hewson AC. 1993. Cambridge Studies in Magnetism Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  73. 73. 
    Auerbach A. 1994. Interacting Electrons and Quantum Magnetism New York: Springer-Verlag
    [Google Scholar]
  74. 74. 
    Fradkin E. 2013. Field Theories of Condensed Matter Physics Cambridge, UK: Cambridge Univ. Press. , 2nd ed..
    [Google Scholar]
  75. 75. 
    Schulz HJ. 1995. J. Low Temp. Phys. 99:3615–24
    [Google Scholar]
  76. 76. 
    Kivelson SA, Emery VJ. 1994. Strongly Correlated Electronic Materials, Proceedings of The Los Alamos Symposium, 1993 KS Bedell, Z Wang, DE Meltzer, AV Balatsky, E Abrahams 619–56 Redding, MA: Addison-Wesley
    [Google Scholar]
  77. 77. 
    Zaanen J, Gunnarsson O. 1989. Phys. Rev. B 40:107391–94
    [Google Scholar]
  78. 78. 
    Schulz HJ. 1989. J. Phys. France 50:182833–49
    [Google Scholar]
  79. 79. 
    Machida K. 1989. Phys. C Supercond. 158:1192–96
    [Google Scholar]
  80. 80. 
    Schulz HJ. 1990. Phys. Rev. Lett. 64:121445–48
    [Google Scholar]
  81. 81. 
    White SR. 1992. Phys. Rev. Lett. 69:192863–66
    [Google Scholar]
  82. 82. 
    Schollwöck U. 2011. Ann. Phys. 326:196–192
    [Google Scholar]
  83. 83. 
    Ostlund S, Rommer S. 1995. Phys. Rev. Lett. 75:193537–40
    [Google Scholar]
  84. 84. 
    Dolfi M, Bauer B, Keller S, Troyer M. 2015. Phys. Rev. B 92:19195139
    [Google Scholar]
  85. 85. 
    Verstraete F, Wolf M, Pérez-García D, Cirac JI. 2006. Int. J. Mod. Phys. B 20:30n315142–53
    [Google Scholar]
  86. 86. 
    White SR, Feiguin AE. 2004. Phys. Rev. Lett. 93:7076401
    [Google Scholar]
  87. 87. 
    Yang C, Feiguin AE. 2019. Phys. Rev. B 99:23235117
    [Google Scholar]
  88. 88. 
    Haldane FDM. 1983. Phys. Lett. A 93:9464–68
    [Google Scholar]
  89. 89. 
    Chakravarty S. 1996. Phys. Rev. Lett. 77:214446–49
    [Google Scholar]
  90. 90. 
    Jiang HC, Yao H, Balents L. 2012. Phys. Rev. B 86:2024424
    [Google Scholar]
  91. 91. 
    Figueirido F, Karlhede A, Kivelson S, Sondhi S, Rocek M, Rokhsar DS. 1990. Phys. Rev. B 41:74619–32
    [Google Scholar]
  92. 92. 
    Gong SS, Zhu W, Sheng DN, Motrunich OI, Fisher MPA. 2014. Phys. Rev. Lett. 113:2027201
    [Google Scholar]
  93. 93. 
    Noack RM, White SR, Scalapino DJ. 1995. Europhys. Lett. 30:3163–68
    [Google Scholar]
  94. 94. 
    Troyer M, Tsunetsugu H, Rice TM. 1996. Phys. Rev. B 53:1251–67
    [Google Scholar]
  95. 95. 
    Poilblanc D, Chiappa O, Riera J, White SR, Scalapino DJ. 2000. Phys. Rev. B 62:22R14633–36
    [Google Scholar]
  96. 96. 
    Gannot Y, Jiang YF, Kivelson SA. 2020. Phys. Rev. B 102:11115136
    [Google Scholar]
  97. 97. 
    Luther A, Emery VJ. 1974. Phys. Rev. Lett. 33:10589–92
    [Google Scholar]
  98. 98. 
    White SR, Affleck I, Scalapino DJ. 2002. Phys. Rev. B 65:16165122
    [Google Scholar]
  99. 99. 
    Schulz HJ. 1980. Phys. Rev. B 22:115274–77
    [Google Scholar]
  100. 100. 
    White SR, Scalapino DJ. 1997. Phys. Rev. B 55:2214701–4
    [Google Scholar]
  101. 101. 
    White SR, Scalapino DJ. 1999. Phys. Rev. B 60:2R753–56
    [Google Scholar]
  102. 102. 
    Huang EW, Mendl CB, Jiang HC, Moritz B, Devereaux TP. 2018. NPJ Quantum Mater. 3:22
    [Google Scholar]
  103. 103. 
    Jiang HC, Devereaux TP. 2019. Science 365:64601424–28
    [Google Scholar]
  104. 104. 
    Jiang HC, Weng ZY, Kivelson SA. 2018. Phys. Rev. B 98:14140505
    [Google Scholar]
  105. 105. 
    Chung CM, Qin M, Zhang S, Schollwöck U, White SR. 2020. Phys. Rev. B 102:4041106
    [Google Scholar]
  106. 106. 
    Dodaro JF, Jiang HC, Kivelson SA. 2017. Phys. Rev. B 95:15155116
    [Google Scholar]
  107. 107. 
    Zheng BX, Chung CM, Corboz P, Ehlers G, Qin MP et al. 2017. Science 358:63671155–60
    [Google Scholar]
  108. 108. 
    Jiang YF, Zaanen J, Devereaux TP, Jiang HC. 2020. Phys. Rev. Res. 2:3033073
    [Google Scholar]
  109. 109. 
    Hager G, Wellein G, Jeckelmann E, Fehske H. 2005. Phys. Rev. B 71:7075108
    [Google Scholar]
  110. 110. 
    White SR, Scalapino DJ. 2003. Phys. Rev. Lett. 91:13136403
    [Google Scholar]
  111. 111. 
    Jiang HC, Kivelson SA. 2021. Phys. Rev. Lett. 127:097002
    [Google Scholar]
  112. 112. 
    Gong S, Zhu W, Sheng DN. 2021. Phys. Rev. Lett. 127:097003
    [Google Scholar]
  113. 113. 
    Jiang S, Scalapino DJ, White SR. 2021. arXiv:2104.10149
  114. 114. 
    Jiang HC, Kivelson SA. 2021. arXiv:2105.07048
  115. 115. 
    Gong SS, Sheng DN, Motrunich OI, Fisher MPA. 2013. Phys. Rev. B 88:16165138
    [Google Scholar]
  116. 116. 
    Gong SS, Zhu W, Sheng DN. 2015. Phys. Rev. B 92:19195110
    [Google Scholar]
  117. 117. 
    Jiang S, Mesaros A, Ran Y. 2014. Phys. Rev. X 4:3 031040
    [Google Scholar]
  118. 118. 
    Yang X, Zheng H, Qin M. 2021. Phys. Rev. B 103:15155110
    [Google Scholar]
  119. 119. 
    Qin M. 2021. arXiv e-prints arXiv:2104.14160
    [Google Scholar]
  120. 120. 
    Jiang HC. 2021. NPJ Quantum Mater. 6:71
    [Google Scholar]
  121. 121. 
    Szasz A, Motruk J, Zaletel MP, Moore JE. 2020. Phys. Rev. X 10:2021042
    [Google Scholar]
  122. 122. 
    Zhu Z, Sheng DN, Vishwanath A. 2020. arXiv:2007.11963
  123. 123. 
    Gong SS, Zhu W, Sheng DN. 2014. Sci. Rep. 4:6317
    [Google Scholar]
  124. 124. 
    Gong SS, Zheng W, Lee M, Lu YM, Sheng DN. 2019. Phys. Rev. B 100:24241111
    [Google Scholar]
  125. 125. 
    Hu S, Zhu W, Eggert S, He YC. 2019. Phys. Rev. Lett. 123:20207203
    [Google Scholar]
  126. 126. 
    Kalmeyer V, Laughlin RB. 1987. Phys. Rev. Lett. 59:182095–98
    [Google Scholar]
  127. 127. 
    Moessner R, Sondhi SL. 2001. Phys. Rev. Lett. 86:91881–84
    [Google Scholar]
  128. 128. 
    Rokhsar DS, Kivelson SA. 1988. Phys. Rev. Lett. 61:202376–79
    [Google Scholar]
  129. 129. 
    Laughlin RB. 1988. Science 242:525–33
    [Google Scholar]
  130. 130. 
    Rokhsar DS. 1993. Phys. Rev. Lett. 70:4493–96
    [Google Scholar]
  131. 131. 
    Balents L, Fisher MPA, Nayak C. 1998. Int. J. Mod. Phys. B 12:101033–68
    [Google Scholar]
  132. 132. 
    Jiang YF, Jiang HC. 2020. Phys. Rev. Lett. 125:15157002
    [Google Scholar]
  133. 133. 
    Venderley J, Kim EA. 2019. Phys. Rev. B 100:6060506
    [Google Scholar]
  134. 134. 
    Zeng C, Elser V. 1990. Phys. Rev. B 42:138436–44
    [Google Scholar]
  135. 135. 
    Chalker JT, Eastmond JFG. 1992. Phys. Rev. B 46:2114201–4
    [Google Scholar]
  136. 136. 
    Lecheminant P, Bernu B, Lhuillier C, Pierre L, Sindzingre P. 1997. Phys. Rev. B 56:52521–29
    [Google Scholar]
  137. 137. 
    Läuchli AM, Sudan J, Sørensen ES. 2011. Phys. Rev. B 83:21212401
    [Google Scholar]
  138. 138. 
    Nakano H, Sakai T. 2011. J. Phys. Soc. Jpn. 80:5053704
    [Google Scholar]
  139. 139. 
    Läuchli AM, Sudan J, Moessner R. 2019. Phys. Rev. B 100:15155142
    [Google Scholar]
  140. 140. 
    Changlani HJ, Kochkov D, Kumar K, Clark BK, Fradkin E. 2018. Phys. Rev. Lett. 120:11117202
    [Google Scholar]
  141. 141. 
    Yan S, Huse DA, White SR. 2011. Science 332:60341173–76
    [Google Scholar]
  142. 142. 
    Jiang HC, Weng ZY, Sheng DN. 2008. Phys. Rev. Lett. 101:11117203
    [Google Scholar]
  143. 143. 
    Jiang HC, Wang Z, Balents L. 2012. Nat. Phys. 8:12902–5
    [Google Scholar]
  144. 144. 
    Depenbrock S, McCulloch IP, Schollwöck U. 2012. Phys. Rev. Lett. 109:6067201
    [Google Scholar]
  145. 145. 
    He YC, Zaletel MP, Oshikawa M, Pollmann F. 2017. Phys. Rev. X 7:3031020
    [Google Scholar]
  146. 146. 
    Jiang HC, Devereaux T, Kivelson SA. 2017. Phys. Rev. Lett. 119:6067002
    [Google Scholar]
  147. 147. 
    Peng C, Jiang YF, Sheng DN, Jiang HC. 2021. Adv. Quantum Technol. 4:32000126
    [Google Scholar]
  148. 148. 
    Blankenbecler R, Scalapino DJ, Sugar RL. 1981. Phys. Rev. D 24:82278–86
    [Google Scholar]
  149. 149. 
    Hirsch JE. 1983. Phys. Rev. B 28:74059–61
    [Google Scholar]
  150. 150. 
    White SR, Scalapino DJ, Sugar RL, Loh EY, Gubernatis JE, Scalettar RT. 1989. Phys. Rev. B 40:1506–16
    [Google Scholar]
  151. 151. 
    Assaad FF 2002. Quantum Simulations of Complex Many-Body Systems: From Theory to Algorithms 10 NIC Ser. J Grotendorst, D Marx, A Muyramatsu 99–156 Jülich, Ger: Neumann Inst. Comput.
    [Google Scholar]
  152. 152. 
    White SR, Scalapino DJ, Sugar RL, Bickers NE, Scalettar RT. 1989. Phys. Rev. B 39:1839–42
    [Google Scholar]
  153. 153. 
    Sorella S, Tosatti E. 1992. Europhys. Lett. 19:8699–702
    [Google Scholar]
  154. 154. 
    Paiva T, Scalettar RT, Zheng W, Singh RRP, Oitmaa J. 2005. Phys. Rev. B 72:8085123
    [Google Scholar]
  155. 155. 
    Meng ZY, Lang TC, Wessel S, Assaad FF, Muramatsu A. 2010. Nature 464:7290847–51
    [Google Scholar]
  156. 156. 
    Sorella S, Otsuka Y, Yunoki S. 2012. Sci. Rep. 2:11–5
    [Google Scholar]
  157. 157. 
    Assaad FF, Herbut IF. 2013. Phys. Rev. X 3:3031010
    [Google Scholar]
  158. 158. 
    Bulut N, Scalapino DJ, Scalettar RT. 1992. Phys. Rev. B 45:105577–84
    [Google Scholar]
  159. 159. 
    Scalettar RT, Cannon JW, Scalapino DJ, Sugar RL. 1994. Phys. Rev. B 50:1813419–27
    [Google Scholar]
  160. 160. 
    Bouadim K, Batrouni GG, Hébert F, Scalettar RT. 2008. Phys. Rev. B 77:14144527
    [Google Scholar]
  161. 161. 
    Lang TC, Meng ZY, Scherer MM, Uebelacker S, Assaad FF et al. 2012. Phys. Rev. Lett. 109:12126402
    [Google Scholar]
  162. 162. 
    Huang EW, Mendl CB, Liu S, Johnston S, Jiang HC et al. 2017. Science 358:63671161–64
    [Google Scholar]
  163. 163. 
    Huang EW, Sheppard R, Moritz B, Devereaux TP. 2019. Science 366:6468987–90
    [Google Scholar]
  164. 164. 
    Liu T, Jost D, Moritz B, Huang EW, Hackl R, Devereaux TP. 2021. Phys. Rev. B 103:134502
    [Google Scholar]
  165. 165. 
    Zhu X, Ying T, Guo H, Feng S. 2019. Chin. Phys. B 28:7077401
    [Google Scholar]
  166. 166. 
    Varma CM. 1999. Phys. Rev. Lett. 83:173538–41
    [Google Scholar]
  167. 167. 
    Chakravarty S, Laughlin RB, Morr DK, Nayak C. 2001. Phys. Rev. B 63:9094503
    [Google Scholar]
  168. 168. 
    Himeda A, Kato T, Ogata M. 2002. Phys. Rev. Lett. 88:11117001
    [Google Scholar]
  169. 169. 
    Berg E, Fradkin E, Kim EA, Kivelson SA, Oganesyan V et al. 2007. Phys. Rev. Lett. 99:12127003
    [Google Scholar]
  170. 170. 
    Agterberg DF, Davis JS, Edkins SD, Fradkin E, Van Harlingen DJ et al. 2020. Annu. Rev. Condens. Matter Phys. 11:231–70
    [Google Scholar]
  171. 171. 
    Hsu CH, Wang Z, Chakravarty S. 2012. Phys. Rev. B 86:21214510
    [Google Scholar]
  172. 172. 
    Scalapino DJ, White SR, Affleck I. 2001. Phys. Rev. B 64:10100506(R)
    [Google Scholar]
  173. 173. 
    Nishimoto S, Jeckelmann E, Scalapino DJ. 2009. Phys. Rev. B 79:20205115
    [Google Scholar]
  174. 174. 
    Kung YF, Chen CC, Moritz B, Johnston S, Thomale R, Devereaux TP. 2014. Phys. Rev. B 90:22224507
    [Google Scholar]
  175. 175. 
    Fulde P, Ferrell RA. 1964. Phys. Rev. 135:3AA550–63
    [Google Scholar]
  176. 176. 
    Larkin AI, Ovchinnikov YN. 1964. Zh. Eksp. Teor. Fiz. 47:1136–46
    [Google Scholar]
  177. 177. 
    White SR, Scalapino DJ. 2009. Phys. Rev. B 79:22220504
    [Google Scholar]
  178. 178. 
    Devereaux TP. 2021. Numerical investigations of models of the cuprates (Harvard CMSC Colloquium). https://www.youtube.com/watch?v=mAYnLdi3vs4 Presents results of a large body of the most recent and unpublished DQMC results on the square lattice Hubbard model
  179. 179. 
    Berg E, Fradkin E, Kivelson SA. 2010. Phys. Rev. Lett. 105:14146403
    [Google Scholar]
  180. 180. 
    Jaefari A, Fradkin E. 2012. Phys. Rev. B 85:3035104
    [Google Scholar]
  181. 181. 
    Corboz P, Rice TM, Troyer M. 2014. Phys. Rev. Lett. 113:4046402
    [Google Scholar]
  182. 182. 
    Xu XY, Law KT, Lee PA. 2019. Phys. Rev. Lett. 122:16167001
    [Google Scholar]
  183. 183. 
    Eisenberg E, Berkovits R, Huse DA, Altshuler BL. 2002. Phys. Rev. B 65:13134437
    [Google Scholar]
  184. 184. 
    Sorella S. 2021. arXiv:2101.07045
  185. 185. 
    Nandkishore R, Levitov LS, Chubukov AV. 2012. Nat. Phys. 8:2158–63
    [Google Scholar]
  186. 186. 
    Nandkishore R, Thomale R, Chubukov AV. 2014. Phys. Rev. B 89:14144501
    [Google Scholar]
  187. 187. 
    Georges A, Kotliar G, Krauth W, Rozenberg MJ. 1996. Rev. Mod. Phys. 68:113–125
    [Google Scholar]
  188. 188. 
    LeBlanc JPF, Antipov AE, Becca F, Bulik IW, Chan GKL et al. 2015. Phys. Rev. X 5:4041041
    [Google Scholar]
  189. 189. 
    Kent PRC, Kotliar G. 2018. Science 361:6400348–54
    [Google Scholar]
  190. 190. 
    Paul A, Birol T. 2019. Annu. Rev. Mater. Res. 49:31–52
    [Google Scholar]
  191. 191. 
    Maier T, Jarrell M, Pruschke T, Hettler M. 2005. Rev. Mod. Phys. 77:31027–80
    [Google Scholar]
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