1932
0
  1. Home
  2. A-Z Publications
  3. Annual Review of Condensed Matter Physics
  4. Volume 13, 2022
  5. Article

Annual Review of Condensed Matter Physics

Volume 13, 2022

The Hubbard Model: A Computational Perspective

Abstract

The Hubbard model is the simplest model of interacting fermions on a lattice and is of similar importance to correlated electron physics as the Ising model is to statistical mechanics or the fruit fly to biomedical science. Despite its simplicity, the model exhibits an incredible wealth of phases, phase transitions, and exotic correlation phenomena. Although analytical methods have provided a qualitative description of the model in certain limits, numerical tools have shown impressive progress in achieving quantitative accurate results over the past several years. This article gives an introduction to the model, motivates common questions, and illustrates the progress that has been achieved over recent years in revealing various aspects of the correlation physics of the model.

Keyword(s): model Hamiltonians, quantum many-body theory, strongly correlated electron systems
Loading

Article metrics loading...

/content/journals/10.1146/annurev-conmatphys-090921-033948
2022-03-10
2024-06-13
Loading full text...

Full text loading...

/deliver/fulltext/conmatphys/13/1/annurev-conmatphys-090921-033948.html?itemId=/content/journals/10.1146/annurev-conmatphys-090921-033948&mimeType=html&fmt=ahah

Literature Cited

  1. 1. 
    Hubbard J 1963. Proc. R. Soc. Lond. Ser. A 276:1365238–57
     [Google Scholar]
  2. 2. 
    Kanamori J 1963. Prog. Theor. Phys. 30:3275–89
     [Google Scholar]
  3. 3. 
    Gutzwiller MC 1963. Phys. Rev. Lett. 10:5159–62
     [Google Scholar]
  4. 4. 
    Gull E, Millis AJ 2015. Nat. Phys. 11:10808–10
     [Google Scholar]
  5. 5. 
    LeBlanc JPF, Antipov AE, Becca F, Bulik IW, Chan GKL et al. 2015. Phys. Rev. X 5:4041041
     [Google Scholar]
  6. 6. 
    Schäfer T, Wentzell N, Šimkovic F, He YY, Hille C et al. 2021. Phys. Rev. X 11:1011058
     [Google Scholar]
  7. 7. 
    Chubukov AV 1993. Phys. Rev. B 48:21097–104
     [Google Scholar]
  8. 8. 
    Dagotto E 1994. Rev. Mod. Phys. 66:3763–840
     [Google Scholar]
  9. 9. 
    Bulut N 2002. Adv. Phys. 51:71587–667
     [Google Scholar]
  10. 10. 
    Maier T, Jarrell M, Pruschke T, Hettler MH 2005. Rev. Mod. Phys. 77:31027–80
     [Google Scholar]
  11. 11. 
    Tremblay AMS, Kyung B, Sénéchal D 2006. Low Temp. Phys. 32:4424–51
     [Google Scholar]
  12. 12. 
    Scalapino DJ. 2007. Handbook of High-Temperature Superconductivity JR Schrieffer 495–526 New York: Springer
     [Google Scholar]
  13. 13. 
    Scalapino DJ 2012. Rev. Mod. Phys. 84:41383–417
     [Google Scholar]
  14. 14. 
    Arovas DP, Berg E, Kivelson SA, Raghu S. 2022. Annu. Rev. Condens. Matter Phys. 13:239–74
     [Google Scholar]
  15. 15. 
    Daré AM, Albinet G 2000. Phys. Rev. B 61:74567–75
     [Google Scholar]
  16. 16. 
    Staudt R, Dzierzawa M, Muramatsu A. 2000. Eur. Phys. J. B 17:3411–15
     [Google Scholar]
  17. 17. 
    Rohringer G, Toschi A, Katanin A, Held K 2011. Phys. Rev. Lett. 107:25256402
     [Google Scholar]
  18. 18. 
    Kent PRC, Jarrell M, Maier TA, Pruschke T 2005. Phys. Rev. B 72:6060411
     [Google Scholar]
  19. 19. 
    Hirschmeier D, Hafermann H, Gull E, Lichtenstein AI, Antipov AE 2015. Phys. Rev. B 92:14144409
     [Google Scholar]
  20. 20. 
    Kozik E, Burovski E, Scarola VW, Troyer M 2013. Phys. Rev. B 87:20205102
     [Google Scholar]
  21. 21. 
    Rohringer G, Hafermann H, Toschi A, Katanin AA, Antipov AE et al. 2018. Rev. Mod. Phys. 90(2):025003. https://doi.org/10.1103/RevModPhys.90.025003
    [Crossref] [Google Scholar]
  22. 22. 
    Andersen O, Liechtenstein A, Jepsen O, Paulsen F 1995. J. Phys. Chem. Solids 56:121573–91
     [Google Scholar]
  23. 23. 
    Pavarini E, Dasgupta I, Saha-Dasgupta T, Jepsen O, Andersen OK 2001. Phys. Rev. Lett. 87:4047003
     [Google Scholar]
  24. 24. 
    Zheng H, Changlani HJ, Williams KT, Busemeyer B, Wagner LK 2018. Front. Phys. 6:43
     [Google Scholar]
  25. 25. 
    Hirayama M, Yamaji Y, Misawa T, Imada M 2018. Phys. Rev. B 98:13134501
     [Google Scholar]
  26. 26. 
    Hirayama M, Misawa T, Ohgoe T, Yamaji Y, Imada M 2019. Phys. Rev. B 99:24245155
     [Google Scholar]
  27. 27. 
    Damascelli A, Hussain Z, Shen ZX 2003. Rev. Mod. Phys. 75:2473–541
     [Google Scholar]
  28. 28. 
    Hybertsen MS, Schlüter M, Christensen NE 1989. Phys. Rev. B 39:139028–41
     [Google Scholar]
  29. 29. 
    Zhang FC, Rice TM 1988. Phys. Rev. B 37:73759–61
     [Google Scholar]
  30. 30. 
    Markiewicz RS, Sahrakorpi S, Lindroos M, Lin H, Bansil A 2005. Phys. Rev. B 72:5054519
     [Google Scholar]
  31. 31. 
    Kim C, White PJ, Shen ZX, Tohyama T, Shibata Y et al. 1998. Phys. Rev. Lett. 80:194245–48
     [Google Scholar]
  32. 32. 
    Coldea R, Hayden SM, Aeppli G, Perring TG, Frost CD et al. 2001. Phys. Rev. Lett. 86:235377–80
     [Google Scholar]
  33. 33. 
    Kitatani M, Si L, Janson O, Arita R, Zhong Z, Held K 2020. npj Quantum Mater. 5:159
     [Google Scholar]
  34. 34. 
    Powell BJ, McKenzie RH 2006. J. Phys. Condens. Matter 18:45R827–66
     [Google Scholar]
  35. 35. 
    Georgescu IM, Ashhab S, Nori F 2014. Rev. Mod. Phys. 86:1153–85
     [Google Scholar]
  36. 36. 
    Mazurenko A, Chiu CS, Ji G, Parsons MF, Kanász-Nagy M et al. 2017. Nature 545:7655462–66
     [Google Scholar]
  37. 37. 
    Byrnes T, Kim NY, Kusudo K, Yamamoto Y 2008. Phys. Rev. B 78:7075320
     [Google Scholar]
  38. 38. 
    Singha A, Gibertini M, Karmakar B, Yuan S, Polini M et al. 2011. Science 332:60341176–79
     [Google Scholar]
  39. 39. 
    Hensgens T, Fujita T, Janssen L, Li X, Van Diepen CJ et al. 2017. Nature 548:766570–73
     [Google Scholar]
  40. 40. 
    Schneider U, Hackermüller L, Ronzheimer JP, Will S, Braun S et al. 2012. Nat. Phys. 8:3213–18
     [Google Scholar]
  41. 41. 
    White IG, Hulet RG, Hazzard KRA 2019. Phys. Rev. A 100:3033612
     [Google Scholar]
  42. 42. 
    Somma R, Ortiz G, Gubernatis JE, Knill E, Laflamme R 2002. Phys. Rev. A 65:4042323
     [Google Scholar]
  43. 43. 
    Arute F, Arya K, Babbush R, Bacon D, Bardin JCet al 2020. arXiv:2010.07965
  44. 44. 
    Wecker D, Hastings MB, Wiebe N, Clark BK, Nayak C, Troyer M 2015. Phys. Rev. A 92:6062318
     [Google Scholar]
  45. 45. 
    Deng Y, Kozik E, Prokof'ev NV, Svistunov BV 2015. Europhys. Lett. 110:557001
     [Google Scholar]
  46. 46. 
    Šimkovic F, Liu XW, Deng Y, Kozik E 2016. Phys. Rev. B 94:8085106
     [Google Scholar]
  47. 47. 
    Mermin ND, Wagner H 1966. Phys. Rev. Lett. 17(22):1133–36. Erratum. 1966. Phys. Rev. Lett. 17:1307–8
     [Google Scholar]
  48. 48. 
    Maier TA, Jarrell M, Schulthess TC, Kent PRC, White JB 2005. Phys. Rev. Lett. 95:23237001
     [Google Scholar]
  49. 49. 
    Fradkin E, Kivelson SA, Tranquada JM 2015. Rev. Mod. Phys. 87:2457–82
     [Google Scholar]
  50. 50. 
    Hirsch JE 1985. Phys. Rev. B 31:74403–19
     [Google Scholar]
  51. 51. 
    Hirsch JE, Tang S 1989. Phys. Rev. Lett. 62:5591–94
     [Google Scholar]
  52. 52. 
    White SR, Scalapino DJ, Sugar RL, Loh EY, Gubernatis JE, Scalettar RT 1989. Phys. Rev. B 40:1506–16
     [Google Scholar]
  53. 53. 
    Varney CN, Lee CR, Bai ZJ, Chiesa S, Jarrell M, Scalettar RT 2009. Phys. Rev. B 80:7075116
     [Google Scholar]
  54. 54. 
    Dagotto E, Moreo A, Ortolani F, Poilblanc D, Riera J 1992. Phys. Rev. B 45:1810741–60
     [Google Scholar]
  55. 55. 
    Qin M, Shi H, Zhang S 2016. Phys. Rev. B 94:8085103
     [Google Scholar]
  56. 56. 
    Qin M, Shi H, Zhang S 2017. Phys. Rev. B 96:7075156
     [Google Scholar]
  57. 57. 
    Kim AJ, Simkovic F, Kozik E 2020. Phys. Rev. Lett. 124:11117602
     [Google Scholar]
  58. 58. 
    Prokof'ev NV, Svistunov BV 1998. Phys. Rev. Lett. 81:122514–17
     [Google Scholar]
  59. 59. 
    Lenihan C, Kim AJ, Šimkovic F IV, Kozik E 2021. Phys. Rev. Lett. 126:10105701
     [Google Scholar]
  60. 60. 
    Chakravarty S, Halperin BI, Nelson DR 1988. Phys. Rev. Lett. 60:111057–60
     [Google Scholar]
  61. 61. 
    Moreo A, Scalapino DJ 1991. Phys. Rev. Lett. 66:7946–48
     [Google Scholar]
  62. 62. 
    Schäfer T, Geles F, Rost D, Rohringer G, Arrigoni E et al. 2015. Phys. Rev. B 91:12125109
     [Google Scholar]
  63. 63. 
    Rohringer G, Toschi A 2016. Phys. Rev. B 94:12125144
     [Google Scholar]
  64. 64. 
    Jarrell M, Gubernatis J 1996. Phys. Rep. 269:3133–95
     [Google Scholar]
  65. 65. 
    Van Damme M, Vanhove R, Haegeman J, Verstraete F, Vanderstraeten L 2021. Phys. Rev. B 104:115142
     [Google Scholar]
  66. 66. 
    Furukawa N, Imada M 1992. J. Phys. Soc. Jpn. 61:93331–54
     [Google Scholar]
  67. 67. 
    Assaad FF, Imada M 1996. Phys. Rev. Lett. 76:173176–79
     [Google Scholar]
  68. 68. 
    Vitali E, Shi H, Qin M, Zhang S 2016. Phys. Rev. B 94:8085140
     [Google Scholar]
  69. 69. 
    Yang C, Feiguin AE 2016. Phys. Rev. B 93:8081107
     [Google Scholar]
  70. 70. 
    White SR 1992. Phys. Rev. Lett. 69:2863
     [Google Scholar]
  71. 71. 
    Bulut N, Scalapino DJ, White SR 1994. Phys. Rev. Lett. 72:5705–8
     [Google Scholar]
  72. 72. 
    Preuss R, Hanke W, von der Linden W 1995. Phys. Rev. Lett. 75:71344–47
     [Google Scholar]
  73. 73. 
    Preuss R, Hanke W, Gröber C, Evertz HG 1997. Phys. Rev. Lett. 79:61122–25
     [Google Scholar]
  74. 74. 
    Gröber C, Eder R, Hanke W 2000. Phys. Rev. B 62:74336–52
     [Google Scholar]
  75. 75. 
    Schollwöck U 2011. Ann. Phys. 326:96–192
     [Google Scholar]
  76. 76. 
    Bohrdt A, Demler E, Pollmann F, Knap M, Grusdt F 2020. Phys. Rev. B 102:3035139
     [Google Scholar]
  77. 77. 
    Georges A, Kotliar G, Krauth W, Rozenberg MJ 1996. Rev. Mod. Phys. 68:113–125
     [Google Scholar]
  78. 78. 
    Lin N, Gull E, Millis AJ 2010. Phys. Rev. B 82(4):045104. https://doi.org/10.1103/PhysRevB.82.045104
    [Crossref] [Google Scholar]
  79. 79. 
    Klett M, Wentzell N, Schäfer T, Simkovic F, Parcollet O et al. 2020. Phys. Rev. Res. 2:3033476
     [Google Scholar]
  80. 80. 
    Šimkovic F, LeBlanc JPF, Kim AJ, Deng Y, Prokof'ev NV et al. 2020. Phys. Rev. Lett. 124:1017003
     [Google Scholar]
  81. 81. 
    Mott NF 1949. Proc. Phys. Soc. Sect. A 62:7416–22
     [Google Scholar]
  82. 82. 
    Anderson PW 1959. Phys. Rev. 115:12–13
     [Google Scholar]
  83. 83. 
    Anderson P 1997. Adv. Phys. 46:13–11
     [Google Scholar]
  84. 84. 
    Slater JC 1951. Phys. Rev. 82:4538–41
     [Google Scholar]
  85. 85. 
    Gull E, Werner P, Wang X, Troyer M, Millis AJ 2008. Europhys. Lett. 84:337009
     [Google Scholar]
  86. 86. 
    Moukouri S, Jarrell M 2001. Phys. Rev. Lett. 87:16167010
     [Google Scholar]
  87. 87. 
    Manousakis E 1991. Rev. Mod. Phys. 63:11–62
     [Google Scholar]
  88. 88. 
    Sandvik AW, Evertz HG 2010. Phys. Rev. B 82:2024407
     [Google Scholar]
  89. 89. 
    Borejsza K, Dupuis N 2004. Phys. Rev. B 69:8085119
     [Google Scholar]
  90. 90. 
    Fradkin E 2013. Field Theories of Condensed Matter Physics Cambridge, UK: Cambridge Univ. Press, 2nd ed.
     [Google Scholar]
  91. 91. 
    Schäfer T, Toschi A 2021. J. Phys. Condens. Matter 33:21214001
     [Google Scholar]
  92. 92. 
    Scalettar RT, Scalapino DJ, Sugar RL, Toussaint D 1989. Phys. Rev. B 39:74711–14
     [Google Scholar]
  93. 93. 
    Fuchs S, Gull E, Troyer M, Jarrell M, Pruschke T 2011. Phys. Rev. B 83:23235113
     [Google Scholar]
  94. 94. 
    Blankenbecler R, Scalapino DJ, Sugar RL 1981. Phys. Rev. D 24:82278–86
     [Google Scholar]
  95. 95. 
    Gull E, Millis AJ, Lichtenstein AI, Rubtsov AN, Troyer M, Werner P 2011. Rev. Mod. Phys. 83:2349–404
     [Google Scholar]
  96. 96. 
    Paiva T, Loh YL, Randeria M, Scalettar RT, Trivedi N 2011. Phys. Rev. Lett. 107:8086401
     [Google Scholar]
  97. 97. 
    Fuchs S, Gull E, Pollet L, Burovski E, Kozik E et al. 2011. Phys. Rev. Lett. 106:3030401
     [Google Scholar]
  98. 98. 
    Schäfer T, Katanin AA, Held K, Toschi A 2017. Phys. Rev. Lett. 119:4046402
     [Google Scholar]
  99. 99. 
    Jaksch D, Zoller P 2005. Ann. Phys. 315:152–79
     [Google Scholar]
  100. 100. 
    Esslinger T 2010. Annu. Rev. Condens. Matter Phys. 1:129–52
     [Google Scholar]
  101. 101. 
    Jördens R, Tarruell L, Greif D, Uehlinger T, Strohmaier N et al. 2010. Phys. Rev. Lett. 104:18180401
     [Google Scholar]
  102. 102. 
    Greif D, Uehlinger T, Jotzu G, Tarruell L, Esslinger T 2013. Science 340:61381307–10
     [Google Scholar]
  103. 103. 
    Hart RA, Duarte PM, Yang TL, Liu X, Paiva T et al. 2015. Nature 519:7542211–14
     [Google Scholar]
  104. 104. 
    Ibarra-García-Padilla E, Mukherjee R, Hulet RG, Hazzard KRA, Paiva T, Scalettar RT 2020. Phys. Rev. A 102:3033340
     [Google Scholar]
  105. 105. 
    Schulz HJ 1990. Phys. Rev. Lett. 64:121445–48
     [Google Scholar]
  106. 106. 
    Dombre T 1990. J. Phys. France 51:9847–56
     [Google Scholar]
  107. 107. 
    Frésard R, Dzierzawa M, Wölfle P 1991. Europhys. Lett. 15:3325–30
     [Google Scholar]
  108. 108. 
    Igoshev PA, Timirgazin MA, Katanin AA, Arzhnikov AK, Irkhin VY 2010. Phys. Rev. B 81:9094407
     [Google Scholar]
  109. 109. 
    Shraiman BI, Siggia ED 1989. Phys. Rev. Lett. 62:131564–67
     [Google Scholar]
  110. 110. 
    Chubukov AV, Frenkel DM 1992. Phys. Rev. B 46:1811884–901
     [Google Scholar]
  111. 111. 
    Chubukov AV, Musaelian KA 1995. Phys. Rev. B 51:1812605–17
     [Google Scholar]
  112. 112. 
    Kotov VN, Sushkov OP 2004. Phys. Rev. B 70:19195105
     [Google Scholar]
  113. 113. 
    Halboth CJ, Metzner W 2000. Phys. Rev. Lett. 85:245162–65
     [Google Scholar]
  114. 114. 
    Husemann C, Salmhofer M 2009. Phys. Rev. B 79:19195125
     [Google Scholar]
  115. 115. 
    Metzner W, Salmhofer M, Honerkamp C, Meden V, Schönhammer K 2012. Rev. Mod. Phys. 84:1299–352
     [Google Scholar]
  116. 116. 
    Yamase H, Eberlein A, Metzner W 2016. Phys. Rev. Lett. 116:9096402
     [Google Scholar]
  117. 117. 
    Scalapino D 1995. Phys. Rep. 250:6329–65
     [Google Scholar]
  118. 118. 
    Bickers NE, Scalapino DJ, White SR 1989. Phys. Rev. Lett. 62:8961–64
     [Google Scholar]
  119. 119. 
    Neumayr A, Metzner W 2003. Phys. Rev. B 67:3035112
     [Google Scholar]
  120. 120. 
    Kyung B, Landry JS, Tremblay AMS 2003. Phys. Rev. B 68:17174502
     [Google Scholar]
  121. 121. 
    Raghu S, Kivelson SA, Scalapino DJ 2010. Phys. Rev. B 81:22224505
     [Google Scholar]
  122. 122. 
    Šimkovic F IV, Deng Y, Kozik E 2021. Phys. Rev. B 104:L020507
     [Google Scholar]
  123. 123. 
    Zanchi D, Schulz HJ 2000. Phys. Rev. B 61:2013609–32
     [Google Scholar]
  124. 124. 
    Honerkamp C, Salmhofer M, Furukawa N, Rice TM 2001. Phys. Rev. B 63:3035109
     [Google Scholar]
  125. 125. 
    Honerkamp C, Salmhofer M 2001. Phys. Rev. Lett. 87:18187004
     [Google Scholar]
  126. 126. 
    Eberlein A, Metzner W 2014. Phys. Rev. B 89:3035126
     [Google Scholar]
  127. 127. 
    Reiss J, Rohe D, Metzner W 2007. Phys. Rev. B 75:7075110
     [Google Scholar]
  128. 128. 
    Wang J, Eberlein A, Metzner W 2014. Phys. Rev. B 89:12121116
     [Google Scholar]
  129. 129. 
    Irkhin VY, Katanin AA, Katsnelson MI 2001. Phys. Rev. B 64:16165107
     [Google Scholar]
  130. 130. 
    Hlubina R, Sorella S, Guinea F 1997. Phys. Rev. Lett. 78:71343–46
     [Google Scholar]
  131. 131. 
    Katanin AA, Kampf AP 2003. Phys. Rev. B 68:19195101
     [Google Scholar]
  132. 132. 
    Vilardi D, Bonetti PM, Metzner W 2020. Phys. Rev. B 102:24245128
     [Google Scholar]
  133. 133. 
    Taranto C, Andergassen S, Bauer J, Held K, Katanin A et al. 2014. Phys. Rev. Lett. 112:19196402
     [Google Scholar]
  134. 134. 
    Vilardi D, Taranto C, Metzner W 2019. Phys. Rev. B 99:10104501
     [Google Scholar]
  135. 135. 
    Vilk YM, Tremblay AMS 1997. J. Phys. I France 7:111309–68
     [Google Scholar]
  136. 136. 
    Kyung B, Hankevych V, Daré AM, Tremblay AMS 2004. Phys. Rev. Lett. 93:14147004
     [Google Scholar]
  137. 137. 
    Wu W, Scheurer MS, Chatterjee S, Sachdev S, Georges A, Ferrero M 2018. Phys. Rev. X 8:2021048
     [Google Scholar]
  138. 138. 
    van Loon EGCP, Hafermann H, Katsnelson MI 2018. Phys. Rev. B 97:8085125
     [Google Scholar]
  139. 139. 
    Eckhardt CJ, Honerkamp C, Held K, Kauch A 2020. Phys. Rev. B 101:15155104
     [Google Scholar]
  140. 140. 
    Hille C, Rohe D, Honerkamp C, Andergassen S 2020. Phys. Rev. Res. 2:3033068
     [Google Scholar]
  141. 141. 
    Wu W, Ferrero M, Georges A, Kozik E 2017. Phys. Rev. B 96:4041105
     [Google Scholar]
  142. 142. 
    Sénéchal D, Tremblay AMS 2004. Phys. Rev. Lett. 92:12126401
     [Google Scholar]
  143. 143. 
    Vojta M 2009. Adv. Phys. 58:6699–820
     [Google Scholar]
  144. 144. 
    Kloss T, Montiel X, Carvalho VS, Freire H, Pépin C 2016. Rep. Prog. Phys. 79:8084507
     [Google Scholar]
  145. 145. 
    Agterberg DF, Davis JS, Edkins SD, Fradkin E, Van Harlingen DJ et al. 2020. Annu. Rev. Condens. Matter Phys. 11:231–70
     [Google Scholar]
  146. 146. 
    Tranquada JM, Sternlieb BJ, Axe JD, Nakamura Y, Uchida S 1995. Nature 375:6532561–63
     [Google Scholar]
  147. 147. 
    Zaanen J, Gunnarsson O 1989. Phys. Rev. B 40:107391–94
     [Google Scholar]
  148. 148. 
    Poilblanc D, Rice TM 1989. Phys. Rev. B 39:139749–52
     [Google Scholar]
  149. 149. 
    Machida K 1989. Phys. C: Superconduct. 158:1–2192–96
     [Google Scholar]
  150. 150. 
    Schulz H 1989. J. Phys. 50:182833–49
     [Google Scholar]
  151. 151. 
    Kato M, Machida K, Nakanishi H, Fujita M 1990. J. Phys. Soc. Jpn. 59:1047–58
     [Google Scholar]
  152. 152. 
    White SR, Scalapino DJ 1998. Phys. Rev. Lett. 80:61272–75
     [Google Scholar]
  153. 153. 
    White SR, Scalapino DJ 1998. Phys. Rev. Lett. 81:153227–30
     [Google Scholar]
  154. 154. 
    White SR, Scalapino DJ 2003. Phys. Rev. Lett. 91:13136403
     [Google Scholar]
  155. 155. 
    White SR, Scalapino DJ 1999. Phys. Rev. B 60:2R753–56
     [Google Scholar]
  156. 156. 
    Himeda A, Kato T, Ogata M 2002. Phys. Rev. Lett. 88:11117001
     [Google Scholar]
  157. 157. 
    White SR, Scalapino DJ 2009. Phys. Rev. B 79:22220504
     [Google Scholar]
  158. 158. 
    Corboz P, White SR, Vidal G, Troyer M 2011. Phys. Rev. B 84:4041108
     [Google Scholar]
  159. 159. 
    Corboz P, Rice TM, Troyer M 2014. Phys. Rev. Lett. 113:4046402
     [Google Scholar]
  160. 160. 
    Zheng BX, Chan GKL 2016. Phys. Rev. B 93:3035126
     [Google Scholar]
  161. 161. 
    Berg E, Fradkin E, Kim EA, Kivelson SA, Oganesyan V et al. 2007. Phys. Rev. Lett. 99:12127003
     [Google Scholar]
  162. 162. 
    Li Q, Hücker M, Gu GD, Tsvelik AM, Tranquada JM 2007. Phys. Rev. Lett. 99:6067001
     [Google Scholar]
  163. 163. 
    Raczkowski M, Capello M, Poilblanc D, Frésard R, Oles AM 2007. Phys. Rev. B 76:14140505
     [Google Scholar]
  164. 164. 
    Yokoyama H, Shiba H 1988. J. Phys. Soc. Jpn. 57:72482–93
     [Google Scholar]
  165. 165. 
    Gros C 1988. Phys. Rev. B 38:1931–34
     [Google Scholar]
  166. 166. 
    Giamarchi T, Lhuillier C 1991. Phys. Rev. B 43:1612943–51
     [Google Scholar]
  167. 167. 
    Sorella S, Martins GB, Becca F, Gazza C, Capriotti L et al. 2002. Phys. Rev. Lett. 88:11117002
     [Google Scholar]
  168. 168. 
    Eichenberger D, Baeriswyl D 2007. Phys. Rev. B 76:18180504
     [Google Scholar]
  169. 169. 
    Misawa T, Imada M 2014. Phys. Rev. B 90:11115137
     [Google Scholar]
  170. 170. 
    Tocchio LF, Becca F, Sorella S 2016. Phys. Rev. B 94:19195126
     [Google Scholar]
  171. 171. 
    Lichtenstein AI, Katsnelson MI 2000. Phys. Rev. B 62:14R9283–86
     [Google Scholar]
  172. 172. 
    Capone M, Kotliar G 2006. Phys. Rev. B 74:5054513
     [Google Scholar]
  173. 173. 
    Becca F, Capriotti L, Sorella S 2001. Phys. Rev. Lett. 87:16167005
     [Google Scholar]
  174. 174. 
    Hu WJ, Becca F, Sorella S 2012. Phys. Rev. B 85:8081110
     [Google Scholar]
  175. 175. 
    Jordan J, Orús R, Vidal G, Verstraete F, Cirac JI 2008. Phys. Rev. Lett. 101:25250602
     [Google Scholar]
  176. 176. 
    Zhang S, Carlson J, Gubernatis JE 1997. Phys. Rev. B 55:127464–77
     [Google Scholar]
  177. 177. 
    Chang CC, Zhang S 2010. Phys. Rev. Lett. 104:11116402
     [Google Scholar]
  178. 178. 
    Peters R, Kawakami N 2014. Phys. Rev. B 89:15155134
     [Google Scholar]
  179. 179. 
    Knizia G, Chan GKL 2012. Phys. Rev. Lett. 109:18186404
     [Google Scholar]
  180. 180. 
    Zheng BX, Chung CM, Corboz P, Ehlers G, Qin MP et al. 2017. Science 358:63671155–60
     [Google Scholar]
  181. 181. 
    Ido K, Ohgoe T, Imada M 2018. Phys. Rev. B 97:4045138
     [Google Scholar]
  182. 182. 
    Tocchio LF, Montorsi A, Becca F 2019. SciPost Phys. 7:2021
     [Google Scholar]
  183. 183. 
    Huang EW, Mendl CB, Jiang HC, Moritz B, Devereaux TP 2018. npj Quantum Mater. 3:122
     [Google Scholar]
  184. 184. 
    Ehlers G, White SR, Noack RM 2017. Phys. Rev. B 95:125125
     [Google Scholar]
  185. 185. 
    Jiang HC, Devereaux TP 2019. Science 365:64601424–28
     [Google Scholar]
  186. 186. 
    Jiang YF, Zaanen J, Devereaux TP, Jiang HC 2020. Phys. Rev. Res. 2:3033073
     [Google Scholar]
  187. 187. 
    Sorella S. 2021. arXiv:2101.07045 [cond-mat]
  188. 188. 
    Darmawan AS, Nomura Y, Yamaji Y, Imada M 2018. Phys. Rev. B 98:20205132
     [Google Scholar]
  189. 189. 
    Simons Collab. on the Many-Electron Problem, Qin M, Chung CM, Shi H, Vitali E et al. 2020. Phys. Rev. X 10:3031016
     [Google Scholar]
  190. 190. 
    Emery VJ, Kivelson SA, Lin HQ 1990. Phys. Rev. Lett. 64:4475–78
     [Google Scholar]
  191. 191. 
    White SR, Scalapino DJ 2000. Phys. Rev. B 61:96320–26
     [Google Scholar]
  192. 192. 
    Ohgoe T, Hirayama M, Misawa T, Ido K, Yamaji Y, Imada M 2020. Phys. Rev. B 101:4045124
     [Google Scholar]
  193. 193. 
    Ponsioen B, Chung SS, Corboz P 2019. Phys. Rev. B 100:19195141
     [Google Scholar]
  194. 194. 
    Simons Collab. on the Many-Electron Problem, Chung CM, Qin M, Zhang S, Schollwöck U, White SR 2020. Phys. Rev. B 102:4041106
     [Google Scholar]
  195. 195. 
    Dodaro JF, Jiang HC, Kivelson SA 2017. Phys. Rev. B 95:15155116
     [Google Scholar]
  196. 196. 
    Jiang HC, Weng ZY, Kivelson SA 2018. Phys. Rev. B 98:14140505
     [Google Scholar]
  197. 197. 
    Yanagisawa T, Koike S, Yamaji K 2001. Phys. Rev. B 64:18184509
     [Google Scholar]
  198. 198. 
    Yanagisawa T, Miyazaki M, Yamaji K 2009. J. Phys. Soc. Jpn. 78:1013706
     [Google Scholar]
  199. 199. 
    Weber C, Giamarchi T, Varma CM 2014. Phys. Rev. Lett. 112:11117001
     [Google Scholar]
  200. 200. 
    Zegrodnik M, Biborski A, Fidrysiak M, Spałek J 2019. Phys. Rev. B 99:10104511
     [Google Scholar]
  201. 201. 
    Biborski A, Zegrodnik M, Spałek J 2020. Phys. Rev. B 101:21214504
     [Google Scholar]
  202. 202. 
    Zegrodnik M, Biborski A, Fidrysiak M, Spałek J 2021. J. Phys. Condens. Matter 33:415601
     [Google Scholar]
  203. 203. 
    Cui ZH, Sun C, Ray U, Zheng BX, Sun Q, Chan GKL 2020. Phys. Rev. Res. 2:4043259
     [Google Scholar]
  204. 204. 
    Weber C, Yee C, Haule K, Kotliar G 2012. Europhys. Lett. 100:337001
     [Google Scholar]
  205. 205. 
    Mai P, Balduzzi G, Johnston S, Maier TA 2021. Phys. Rev. B 103:14144514
     [Google Scholar]
  206. 206. 
    White SR, Scalapino DJ 2015. Phys. Rev. B 92:20205112
     [Google Scholar]
  207. 207. 
    Huang EW, Mendl CB, Liu S, Johnston S, Jiang HC et al. 2017. Science 358:63671161–64
     [Google Scholar]
  208. 208. 
    Chiciak A, Vitali E, Zhang S 2020. Phys. Rev. B 102:21214512
     [Google Scholar]
  209. 209. 
    Alloul H, Ohno T, Mendels P 1989. Phys. Rev. Lett. 63:161700–3
     [Google Scholar]
  210. 210. 
    Hüfner S, Hossain MA, Damascelli A, Sawatzky GA 2008. Rep. Prog. Phys. 71:6062501
     [Google Scholar]
  211. 211. 
    Kanigel A, Norman MR, Randeria M, Chatterjee U, Souma S et al. 2006. Nat. Phys. 2:7447–51
     [Google Scholar]
  212. 212. 
    Shen KM, Ronning F, Lu DH, Baumberger F, Ingle NJC et al. 2005. Science 307:5711901–4
     [Google Scholar]
  213. 213. 
    Kastner MA, Birgeneau RJ, Shirane G, Endoh Y 1998. Rev. Mod. Phys. 70:3897–928
     [Google Scholar]
  214. 214. 
    Maier T, Jarrell M, Pruschke T, Keller J 2000. Eur. Phys. J. B Condens. Matter Complex Syst. 13:4613–24
     [Google Scholar]
  215. 215. 
    Huscroft C, Jarrell M, Maier T, Moukouri S, Tahvildarzadeh AN 2001. Phys. Rev. Lett. 86:1139–42
     [Google Scholar]
  216. 216. 
    Macridin A, Jarrell M, Maier T, Kent PRC, D'Azevedo E 2006. Phys. Rev. Lett. 97:3036401
     [Google Scholar]
  217. 217. 
    Gunnarsson O, Schäfer T, LeBlanc JPF, Gull E, Merino J et al. 2015. Phys. Rev. Lett. 114(23):236402. https://doi.org/10.1103/PhysRevLett.114.236402
    [Crossref] [Google Scholar]
  218. 218. 
    Parcollet O, Biroli G, Kotliar G 2004. Phys. Rev. Lett. 92:22226402
     [Google Scholar]
  219. 219. 
    Civelli M, Capone M, Kancharla SS, Parcollet O, Kotliar G 2005. Phys. Rev. Lett. 95:10106402
     [Google Scholar]
  220. 220. 
    Kyung B, Kancharla SS, Sénéchal D, Tremblay AMS, Civelli M, Kotliar G 2006. Phys. Rev. B 73:16165114
     [Google Scholar]
  221. 221. 
    Zhang YZ, Imada M 2007. Phys. Rev. B 76:4045108
     [Google Scholar]
  222. 222. 
    Liebsch A, Tong NH 2009. Phys. Rev. B 80:16165126
     [Google Scholar]
  223. 223. 
    Gull E, Ferrero M, Parcollet O, Georges A, Millis AJ 2010. Phys. Rev. B 82:15155101
     [Google Scholar]
  224. 224. 
    Gull E, Parcollet O, Werner P, Millis AJ 2009. Phys. Rev. B 80:24245102
     [Google Scholar]
  225. 225. 
    Ferrero M, Cornaglia PS, De Leo L, Parcollet O, Kotliar G, Georges A 2009. Phys. Rev. B 80:6064501
     [Google Scholar]
  226. 226. 
    Ferrero M, Cornaglia PS, De Leo L, Parcollet O, Kotliar G, Georges A 2009. Europhys. Lett. 85:557009
     [Google Scholar]
  227. 227. 
    Sordi G, Sémon P, Haule K, Tremblay AMS 2012. Phys. Rev. Lett. 108:21216401
     [Google Scholar]
  228. 228. 
    Terletska H, Vuččević J, Tanasković D, Dobrosavljević V 2011. Phys. Rev. Lett. 107:2026401
     [Google Scholar]
  229. 229. 
    Wu W, Scheurer MS, Ferrero M, Georges A 2020. Phys. Rev. Res. 2:3033067
     [Google Scholar]
  230. 230. 
    Dong X, Gull E 2020. Phys. Rev. B 101:19195115
     [Google Scholar]
  231. 231. 
    White SR, Scalapino DJ, Sugar RL, Bickers NE, Scalettar RT 1989. Phys. Rev. B 39:1839–42
     [Google Scholar]
  232. 232. 
    Chen X, LeBlanc JPF, Gull E 2015. Phys. Rev. Lett. 115:11116402
     [Google Scholar]
  233. 233. 
    Maier TA, Jarrell M, Macridin A, Slezak C 2004. Phys. Rev. Lett. 92:2027005
     [Google Scholar]
  234. 234. 
    Haule K, Kotliar G 2007. Phys. Rev. B 76:10104509
     [Google Scholar]
  235. 235. 
    Civelli M 2009. Phys. Rev. B 79:19195113
     [Google Scholar]
  236. 236. 
    Gull E, Parcollet O, Millis AJ 2013. Phys. Rev. Lett. 110:21216405
     [Google Scholar]
  237. 237. 
    Gull E, Millis AJ 2012. Phys. Rev. B 86:24241106
     [Google Scholar]
  238. 238. 
    Monthoux P, Balatsky AV, Pines D 1991. Phys. Rev. Lett. 67:243448–51
     [Google Scholar]
  239. 239. 
    Maier TA, Jarrell MS, Scalapino DJ 2006. Phys. Rev. Lett. 96:4047005
     [Google Scholar]
  240. 240. 
    Maier TA, Jarrell M, Scalapino DJ 2006. Phys. Rev. B 74:9094513
     [Google Scholar]
  241. 241. 
    Maier TA, Jarrell M, Scalapino DJ 2007. Phys. Rev. B 75:13134519
     [Google Scholar]
  242. 242. 
    Maier TA, Macridin A, Jarrell M, Scalapino DJ 2007. Phys. Rev. B 76:14144516
     [Google Scholar]
  243. 243. 
    Gull E, Millis AJ 2014. Phys. Rev. B 90:4041110
     [Google Scholar]
  244. 244. 
    Kitatani M, Schäfer T, Aoki H, Held K 2019. Phys. Rev. B 99:4041115
     [Google Scholar]
  245. 245. 
    Perepelitsky E, Galatas A, Mravlje J, Žitko R, Khatami E et al. 2016. Phys. Rev. B 94:23235115
     [Google Scholar]
  246. 246. 
    Pruschke T, Jarrell M, Freericks J 1995. Adv. Phys. 44:2187–210
     [Google Scholar]
  247. 247. 
    Deng X, Mravlje J, Žitko R, Ferrero M, Kotliar G, Georges A 2013. Phys. Rev. Lett. 110:8086401
     [Google Scholar]
  248. 248. 
    Cha P, Patel AA, Gull E, Kim EA 2020. Phys. Rev. Res. 2:3033434
     [Google Scholar]
  249. 249. 
    Huang EW, Sheppard R, Moritz B, Devereaux TP 2019. Science 366:6468987–90
     [Google Scholar]
  250. 250. 
    Bulla R, Costi TA, Pruschke T 2008. Rev. Mod. Phys. 80:2395–450
     [Google Scholar]
  251. 251. 
    Vučičević J, Kokalj J, Žitko R, Wentzell N, Tanasković D, Mravlje J 2019. Phys. Rev. Lett. 123:3036601
     [Google Scholar]
  252. 252. 
    Taheridehkordi A, Curnoe SH, LeBlanc JPF 2020. Phys. Rev. B 102:4045115
     [Google Scholar]
  253. 253. 
    Vučičević J, Ferrero M 2020. Phys. Rev. B 101:7075113
     [Google Scholar]
  254. 254. 
    Fei J, Yeh CN, Gull E 2021. Phys. Rev. Lett. 126:5056402
     [Google Scholar]
  255. 255. 
    Stanescu TD, Kotliar G 2006. Phys. Rev. B 74:12125110
     [Google Scholar]
  256. 256. 
    Ulmke M, Scalettar RT, Nazarenko A, Dagotto E 1996. Phys. Rev. B 54:2316523–32
     [Google Scholar]
  257. 257. 
    Trivedi N, Randeria M 1995. Phys. Rev. Lett. 75:2312–15
     [Google Scholar]
  258. 258. 
    Rohringer G, Valli A, Toschi A 2012. Phys. Rev. B 86:12125114
     [Google Scholar]
  259. 259. 
    Basov DN, Timusk T 2005. Rev. Mod. Phys. 77:2721–79
     [Google Scholar]
  260. 260. 
    Ferrero M, Parcollet O, Georges A, Kotliar G, Basov DN 2010. Phys. Rev. B 82:5054502
     [Google Scholar]
  261. 261. 
    Lin N, Gull E, Millis AJ 2009. Phys. Rev. B 80:16161105
     [Google Scholar]
  262. 262. 
    Jarrell M, Freericks JK, Pruschke T 1995. Phys. Rev. B 51:1711704–11
     [Google Scholar]
  263. 263. 
    Haule K, Kotliar G 2007. Europhys. Lett. 77:227007
     [Google Scholar]
  264. 264. 
    Le Tacon M, Sacuto A, Georges A, Kotliar G, Gallais Y et al. 2006. Nat. Phys. 2:8537–43
     [Google Scholar]
  265. 265. 
    Devereaux TP, Hackl R 2007. Rev. Mod. Phys. 79:1175–233
     [Google Scholar]
  266. 266. 
    Lin N, Gull E, Millis AJ 2012. Phys. Rev. Lett. 109:10106401
     [Google Scholar]
  267. 267. 
    Sakai S, Blanc S, Civelli M, Gallais Y, Cazayous M et al. 2013. Phys. Rev. Lett. 111:10107001
     [Google Scholar]
  268. 268. 
    Gull E, Millis AJ 2013. Phys. Rev. B 88:7075127
     [Google Scholar]
  269. 269. 
    LeBlanc JPF, Li S, Chen X, Levy R, Antipov AE et al. 2019. Phys. Rev. B 100:7075123
     [Google Scholar]
  270. 270. 
    Li S, Gull E 2020. Phys. Rev. Res. 2:1013295
     [Google Scholar]
  271. 271. 
    Chen X, Leblanc JPF, Gull E 2017. Nat. Commun. 8:14986
     [Google Scholar]
  272. 272. 
    Sorella S, Tosatti E 1992. Europhys. Lett. 19:8699–704
     [Google Scholar]
  273. 273. 
    Sorella S, Otsuka Y, Yunoki S 2012. Sci. Rep. 2:1992
     [Google Scholar]
  274. 274. 
    Assaad FF, Herbut IF 2013. Phys. Rev. X 3:3031010
     [Google Scholar]
  275. 275. 
    Herbut IF 2006. Phys. Rev. Lett. 97:14146401
     [Google Scholar]
  276. 276. 
    Otsuka Y, Yunoki S, Sorella S 2016. Phys. Rev. X 6:1011029
     [Google Scholar]
  277. 277. 
    Nandkishore R, Levitov LS, Chubukov AV 2012. Nat. Phys. 8:2158–63
     [Google Scholar]
  278. 278. 
    Wang WS, Xiang YY, Wang QH, Wang F, Yang F, Lee DH 2012. Phys. Rev. B 85:3035414
     [Google Scholar]
  279. 279. 
    Gu ZC, Jiang HC, Sheng DN, Yao H, Balents L, Wen XG 2013. Phys. Rev. B 88:15155112
     [Google Scholar]
  280. 280. 
    Kanoda K, Kato R 2011. Annu. Rev. Condens. Matter Phys. 2:167–88
     [Google Scholar]
  281. 281. 
    Capriotti L, Trumper AE, Sorella S 1999. Phys. Rev. Lett. 82:193899–902
     [Google Scholar]
  282. 282. 
    White SR, Chernyshev AL 2007. Phys. Rev. Lett. 99:12127004
     [Google Scholar]
  283. 283. 
    Szasz A, Motruk J, Zaletel MP, Moore JE 2020. Phys. Rev. X 10:2021042
     [Google Scholar]
  284. 284. 
    Tocchio LF, Montorsi A, Becca F 2020. Phys. Rev. B 102:11115150
     [Google Scholar]
  285. 285. 
    Wietek A, Rossi R, Šimkovic F IV, Klett M, Hansmann P et al. 2021. Phys. Rev. X 11:041013
     [Google Scholar]
  286. 286. 
    Liao HJ, Xie ZY, Chen J, Liu ZY, Xie HD et al. 2017. Phys. Rev. Lett. 118:13137202
     [Google Scholar]
  287. 287. 
    Ohashi T, Kawakami N, Tsunetsugu H 2006. Phys. Rev. Lett. 97:6066401
     [Google Scholar]
  288. 288. 
    Jiang HC, Devereaux T, Kivelson SA 2017. Phys. Rev. Lett. 119:6067002
     [Google Scholar]
  289. 289. 
    Kaufmann J, Steiner K, Scalettar RT, Held K, Janson O 2021. Phys. Rev. B 104:165127
     [Google Scholar]
  290. 290. 
    Scalettar RT, Loh EY, Gubernatis JE, Moreo A, White SR et al. 1989. Phys. Rev. Lett. 62:121407–10
     [Google Scholar]
  291. 291. 
    Paiva T, dos Santos RR, Scalettar RT, Denteneer PJH 2004. Phys. Rev. B 69:18184501
     [Google Scholar]
  292. 292. 
    Hirsch JE, Scalapino DJ 1986. Phys. Rev. Lett. 56:252732–35
     [Google Scholar]
  293. 293. 
    Shi H, Rosenberg P, Chiesa S, Zhang S 2016. Phys. Rev. Lett. 117:4040401
     [Google Scholar]
  294. 294. 
    Emery VJ 1987. Phys. Rev. Lett. 58:262794–97
     [Google Scholar]
  295. 295. 
    Maier TA, Scalapino DJ 2011. Phys. Rev. B 84:18180513
     [Google Scholar]
  296. 296. 
    Imada M, Fujimori A, Tokura Y 1998. Rev. Mod. Phys. 70:41039–263
     [Google Scholar]
  297. 297. 
    Georges A, de’ Medici L, Mravlje J 2013. Annu. Rev. Condens. Matter Phys. 4:137–78
     [Google Scholar]
  298. 298. 
    Kotliar G, Savrasov SY, Haule K, Oudovenko VS, Parcollet O, Marianetti CA 2006. Rev. Mod. Phys. 78:3865–951
     [Google Scholar]
  299. 299. 
    Terletska H, Chen T, Gull E 2017. Phys. Rev. B 95:11115149
     [Google Scholar]
  300. 300. 
    Terletska H, Chen T, Paki J, Gull E 2018. Phys. Rev. B 97:11115117
     [Google Scholar]
  301. 301. 
    Jiang M, Hähner UR, Schulthess TC, Maier TA 2018. Phys. Rev. B 97:18184507
     [Google Scholar]
  302. 302. 
    Paki J, Terletska H, Iskakov S, Gull E 2019. Phys. Rev. B 99:24245146
     [Google Scholar]
  303. 303. 
    Stepanov EA, Peters L, Krivenko IS, Lichtenstein AI, Katsnelson MI, Rubtsov AN 2018. npj Quantum Mater. 3:54
     [Google Scholar]
  304. 304. 
    Oka T, Kitamura S 2019. Annu. Rev. Condens. Matter Phys. 10:387–408
     [Google Scholar]
  305. 305. 
    Aoki H, Tsuji N, Eckstein M, Kollar M, Oka T, Werner P 2014. Rev. Mod. Phys. 86:2779–837
     [Google Scholar]
  306. 306. 
    Messer M, Sandholzer K, Görg F, Minguzzi J, Desbuquois R, Esslinger T 2018. Phys. Rev. Lett. 121:23233603
     [Google Scholar]
  307. 307. 
    Sandholzer K, Murakami Y, Görg F, Minguzzi J, Messer M et al. 2019. Phys. Rev. Lett. 123:19193602
     [Google Scholar]
  308. 308. 
    Wietek A, He YY, White SR, Georges A, Stoudenmire EM 2021. Phys. Rev. X 11:031007
     [Google Scholar]
  309. 309. 
    Czarnik P, Dziarmaga J, Corboz P 2019. Phys. Rev. B 99:3035115
     [Google Scholar]
  310. 310. 
    Chen B-B, Chen C, Chen Z, Cui J, Zhai Y et al. 2021. Phys. Rev. B 103:4L041107
     [Google Scholar]
  311. 311. 
    Vanderstraeten L, Haegeman J, Verstraete F 2019. Phys. Rev. B 99:16165121
     [Google Scholar]
  312. 312. 
    Ponsioen B, Corboz P 2020. Phys. Rev. B 101:19195109
     [Google Scholar]
  313. 313. 
    Ferrari F, Becca F 2019. Phys. Rev. X 9:3031026
     [Google Scholar]
  314. 314. 
    Charlebois M, Imada M 2020. Phys. Rev. X 10:4041023
     [Google Scholar]
  315. 315. 
    Kugler FB, Lee SSB, von Delft J 2021. Phys. Rev. X 11:041006
     [Google Scholar]
/content/journals/10.1146/annurev-conmatphys-090921-033948
Loading
The Hubbard Model: A Computational Perspective
Annual Review of Condensed Matter Physics 13, 275 (2022); https://doi.org/10.1146/annurev-conmatphys-090921-033948
/content/journals/10.1146/annurev-conmatphys-090921-033948
/content/journals/10.1146/annurev-conmatphys-090921-033948
Loading

Data & Media loading...

  • Article Type: Review Article

Most Cited Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error