Strong Electronic Correlations: Dynamical Mean-Field Theory and Beyond

  • Hartmut HafermannEmail author
  • Frank Lechermann
  • Alexei N. Rubtsov
  • Mikhail I. Katsnelson
  • Antoine Georges
  • Alexander I. Lichtenstein
Part of the Lecture Notes in Physics book series (LNP, volume 843)


This chapter aims at a pedagogical introduction to theoretical approaches of strongly correlated materials based on dynamical mean-field theory (DMFT) and its extensions. The goal of this theoretical construction is to retain the many-body aspects of local atomic physics within the extended solid. After introducing the main concept at the level of the Hubbard model, we briefly review the theoretical insights into the Mott metal-insulator transition that DMFT provides. We then describe realistic extensions of this approach which combine the accuracy of first-principle Density-Functional Theory with the treatment of local many-body effects within DMFT. We further provide an elementary discussion of the continuous-time Quantum Monte Carlo schemes for the numerical solution of the DMFT effective quantum impurity problem. Finally, the effects of short-range non-local correlations within cluster extensions of the DMFT scheme, as well as long-range fluctuations within the fully renormalized dual-fermion perturbation scheme are discussed extensively.


Green Function Local Density Approximation Mott Transition Lattice Fermion Density Matrix Renormalization Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We are greatly indebted to the input of our collaborators and colleagues Markus Aichhorn, Vladimir Anisimov, Matthias Balzer, Silke Biermann, Lewin Boehnke, Sergej Brener, Emanuel Gull, Václav Janiš, Christoph Jung, Martin Kecker, Gabriel Kotliar, Gang Li, Andrew Millis, Hartmut Monien, Alexander Poteryaev, Michael Potthoff, Leonid Pourovskii, Matouš Ringel and Philipp Werner.


  1. 1.
    Prange, R.E., Girvin, S.M.: The Quantum Hall Effect. Springer, New York (1997)Google Scholar
  2. 2 .
    Stewart, G.R.: Heavy-fermion systems. Rev. Mod. Phys. 56, 755 (1984)ADSCrossRefGoogle Scholar
  3. 3.
    Löhneysen, H.V., Rosch, A., Vojta, M., Wölfle, P.: Fermi-liquid instabilities at magnetic quantum phase transitions. Rev. Mod. Phys. 79, 1015 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Hewson, A.C.: The Kondo Problem to Heavy Fermions. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  5. 5.
    Anderson, P.W.: The Theory of Superconductivity in High- \(T_c \) Cuprates. Princeton University Press, Princeton (1997)Google Scholar
  6. 6.
    Scalapino, D.J.: The case for \(d_{x^2-y^2}\) pairing in the cuprate superconductors. Phys. Rep 250, 329 (1995)Google Scholar
  7. 7.
    Biermann, S., Poteryaev, A., Lichtenstein, A.I., Georges, A.: Dynamical singlets and correlation-assisted Peierls transition in \({\hbox{VO}}_2\). Phys. Rev. Lett. 94, 026404 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    Kamihara, Y., Watanabe, T., Hirano, M., Hosono, H.: Iron-Based Layered Superconductor La \({\rm O}_{1-x} {\hbox{F}}_x\)FeAs (x =0.05-0.12) with \(T_{c} = 26 {\hbox{K}}\). J. Am. Chem. Soc. 130, 3296 (2008)Google Scholar
  9. 9.
    Imada, M., Fujimori, A., Tokura, Y.: Metal-insulator transitions. Rev. Mod. Phys. 70, 1039 (1998)ADSCrossRefGoogle Scholar
  10. 10.
    Anisimov, V.I., Zaanen, J., Andersen, O.K.: Band theory and Mott insulators: Hubbard U instead of Stoner I. Phys. Rev. B 44, 943 (1991)ADSCrossRefGoogle Scholar
  11. 11.
    Aryasetiawan, F., Gunnarsson, O.: The GW method. Rep. Prog. Phys. 61, 237 (1998)ADSCrossRefGoogle Scholar
  12. 12.
    Kotliar, G., Savrasov, S.Y., Haule, K., Oudovenko, V.S., Parcollet, O., Marianetti, C.A.: Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 78, 865 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Kotliar, G., Vollhardt, D.: Strongly correlated materials: insights from dynamical mean-field theory. Phys. Today 57, 53 (2004)Google Scholar
  14. 14.
    Georges, A., Kotliar, G., Krauth, W., Rozenberg, M.J.: Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Mott, N.F.: Metal-Insulator Transitions. Taylor and Francis, London (1974)Google Scholar
  16. 16.
    Anisimov, V.I., Poteryaev, A.I., Korotin, M.A., Anokhin, A.O., Kotliar, G.: First-principles calculations of the electronic structure and spectra of strongly correlated systems: dynamical mean-field theory. J. Phys.: Condens. Matter 9, 7359 (1997)ADSCrossRefGoogle Scholar
  17. 17.
    Lichtenstein, A.I., Katsnelson, M.I.: Ab initio calculations of quasiparticle band structure in correlated systems: LDA++ approach. Phys. Rev. B 57, 6884 (1998)ADSCrossRefGoogle Scholar
  18. 18.
    Lichtenstein, A.I., Katsnelson, M.I., Kotliar, G.: Finite-temperature magnetism of transition metals: an ab initio dynamical mean-field theory. Phys. Rev. Lett. 87, 067205 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Lichtenstein, A.I., Katsnelson, M.I.: Antiferromagnetism and d-wave superconductivity in cuprates: a cluster dynamical mean-field theory. Phys. Rev. B 62, R9283 (2000)ADSCrossRefGoogle Scholar
  20. 20.
    Kotliar, G., Savrasov, S.Y., Pálsson, G., Biroli, G.: Cellular dynamical mean field approach to strongly correlated systems. Phys. Rev. Lett. 87, 186401 (2001)ADSCrossRefGoogle Scholar
  21. 21.
    Potthoff, M., Aichhorn, M., Dahnken, C.: Variational cluster approach to correlated electron systems in low dimensions. Phys. Rev. Lett. 91, 206402 (2003)ADSCrossRefGoogle Scholar
  22. 22.
    Maier, T., Jarrell, M., Pruschke, T., Hettler, M.H.: Quantum cluster theories. Rev. Mod. Phys. 77, 1027 (2005)ADSCrossRefGoogle Scholar
  23. 23.
    Irkhin, V.Y., Katanin, A.A., Katsnelson, M.I.: Robustness of the Van Hove scenario for high- \(T_{c} \) superconductors. Phys. Rev. Lett. 89, 076401 (2002)Google Scholar
  24. 24.
    Slezak, C., Jarrell, M., Maier, T., Deisz, J.: Multi-scale extensions to quantum cluster methods for strongly correlated electron systems. J. Phys.: Condens. Matter 21, 435604 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    Toschi, A., Katanin, A.A., Held, K.: Dynamical vertex approximation: a step beyond dynamical mean-field theory. Phys. Rev. B 75, 045118 (2007)ADSCrossRefGoogle Scholar
  26. 26.
    Kusunose, H.: Influence of spatial correlations in strongly correlated electron systems: extension to dynamical mean field approximation. J. Phys. Soc. Jpn 75, 054713 (2006)Google Scholar
  27. 27.
    Georges, A., Kotliar, G.: Hubbard model in infinite dimensions. Phys. Rev. B 45, 6479 (1992)ADSCrossRefGoogle Scholar
  28. 28.
    Metzner, W., Vollhardt, D.: Correlated lattice fermions in \(d=\infty \) dimensions. Phys. Rev. Lett. 62, 324 (1989)ADSCrossRefGoogle Scholar
  29. 29.
    Georges, A.: In: Avella A., and Mancini F. (eds.) Lectures on the Physics of Highly Correlated Electron Systems VIII, American Institute of Physics (2004) (cond-mat/0403123)Google Scholar
  30. 30.
    Bulla, R., Costi, T.A., Pruschke, T.: Numerical renormalization group method for quantum impurity systems. Rev. Mod. Phys. 80, 395 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Kotliar, G.: Driving the electron over the edge. Science 302, 67 (2003)Google Scholar
  32. 32.
    Lechermann, F., Georges, A., Poteryaev, A., Biermann, S., Posternak, M., Yamasaki, A., Andersen, O.K.: Dynamical mean-field theory using Wannier functions: a flexible route to electronic structure calculations of strongly correlated materials. Phys. Rev. B 74, 125120 (2006)ADSCrossRefGoogle Scholar
  33. 33.
    Amadon, B., Lechermann, F., Georges, A., Jollet, F., Wehling, T.O., Lichtenstein, A.I.: Plane-wave based electronic structure calculations for correlated materials using dynamical mean-field theory and projected local orbitals. Phys. Rev. B 77, 205112 (2008)ADSCrossRefGoogle Scholar
  34. 34.
    Miyake, T., Aryasetiawan, F.: Screened Coulomb interaction in the maximally localized Wannier basis. Phys. Rev. B 77, 085122 (2008)Google Scholar
  35. 35.
    Anisimov, V.I., Aryasetiawan, F., Lichtenstein, A.I.: First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA + U method. J. Phys.: Condens. Matter 9, 767 (1997)ADSCrossRefGoogle Scholar
  36. 36.
    Pruschke, T., Bulla, R.: Hund’s coupling and the metal-insulator transition in the two-band Hubbard model. Eur. Phys. J. B 44, 217 (2005)ADSCrossRefGoogle Scholar
  37. 37.
    Pourovskii, L.V., Delaney, K.T., Vande Walle, C.G., Spaldin, N.A, Georges, A.: Role of atomic multiplets in the electronic structure of rare-earth semiconductors and semimetals. Phys. Rev. Lett. 102, 096401 (2009)ADSCrossRefGoogle Scholar
  38. 38.
    Mo, S.-K., Denlinger, J.D., Kim, H.-D., Park, J.-H., Allen, J.W., Sekiyama, A., Yamasaki, A., Kadono, K., Suga, S., Saitoh, Y., Muro, T., Metcalf, P., Keller, G., Held, K., Eyert, V., Anisimov, V.I., Vollhardt, D.: Prominent quasiparticle peak in the photoemission spectrum of the metallic phase of \(\text{V}_2\text{O}_3\). Phys. Rev. Lett. 90, 186403 (2003)ADSCrossRefGoogle Scholar
  39. 39.
    Panaccione, G., Altarelli, M., Fondacaro, A., Georges, A., Huotari, S., Lacovig, P., Lichtenstein, A., Metcalf, P., Monaco, G., Offi, F., Paolasini, L., Poteryaev, A., Tjernberg, O., Sacchi, M.: Coherent peaks and minimal probing depth in photoemission spectroscopy of Mott-Hubbard systems. Phys. Rev. Lett. 97, 116401 (2006)ADSCrossRefGoogle Scholar
  40. 40.
    Haule, K., Shim, J.H., Kotliar, G.: Correlated electronic structure of \({\hbox{LaO}}_{1-x} \) \({\hbox{F}}_ {x} {\hbox{FeAs}}\). Phys. Rev. Lett. 100, 226402 (2008)Google Scholar
  41. 41.
    Haule, K., Kotliar, G.: Coherence–incoherence crossover in the normal state of iron oxypnictides and importance of Hund’s rule coupling. New J. Phys. 11, 025021 (2009)ADSCrossRefGoogle Scholar
  42. 42.
    Anisimov, V.I., Korotin, D.M., Korotin, M.A., Kozhevnikov, A.V., Kunes, J., Shorikov, A.O., Skornyakov, S.L., Streltsov, S.V.: Coulomb repulsion and correlation strength in LaFeAsO from density functional and dynamical mean-field theories. J. Phys.: Condens. Matter 21, 075602 (2009)ADSCrossRefGoogle Scholar
  43. 43.
    Shorikov A.O., Korotin M.A., Streltsov S.V., Skornyakov S.L., Korotin D.M., Anisimov V.I.: Coulomb correlation effects in LaFeAsO: An LDA + DMFT(QMC) study, JETP 108:121 (2009)Google Scholar
  44. 44.
    Anisimov, V.I., Korotin, D.M., Streltsov, S.V., Kozhevnikov, A.V., Kuneš, J., Shorikov, A.O., Korotin, M.A.: Coulomb parameter U and correlation strength in LaFeAsO. JETP Lett. 88, 729 (2008)ADSCrossRefGoogle Scholar
  45. 45.
    Aichhorn, M., Pourovskii, L., Vildosola, V., Ferrero, M., Parcollet, O., Miyake, T., Georges, A., Biermann, S.: Dynamical mean-field theory within an augmented plane-wave framework: assessing electronic correlations in the iron pnictide LaFeAsO. Phys. Rev. B 80, 085101 (2009)ADSCrossRefGoogle Scholar
  46. 46.
    Nakamura, K., Arita, R., Imada, M.: Ab initio Derivation of Low-Energy Model for Iron-Based Superconductors LaFeAsO and LaFePO. J. Phys. Soc. Jpn. 77, 093711 (2008)ADSCrossRefGoogle Scholar
  47. 47.
    Miyake, T., Pourovskii, L., Vildosola, V., Biermann, S., Georges, A.: d- and f-orbital correlations in the REFeAsO compounds. J. Phys. Soc. Jpn. 77, (Supp. c) 99 (2008)Google Scholar
  48. 48.
    Miyake, T., Nakamura, K., Arita, R., Imada, M.: Comparison of ab initio low-energy models for LaFePO, LaFeAsO, \(\text{BaFe}_ 2 \text{As}_ 2\), LiFeAs, FeSe, and FeTe, electron correlation and covalency. J. Phys. Soc. Jpn. 79, 044705 (2010)Google Scholar
  49. 49.
    Craco, L., Laad, M.S., and Leoni, S.: \(\alpha\)-FeSe as an orbital-selective incoherent metal: An LDA + DMFT study, arXiv:0910.3828, unpublished (2009).Google Scholar
  50. 50.
    Aichhorn, M., Biermann, S., Miyake, T., Georges, A., Imada, M.: Theoretical evidence for strong correlations and incoherent metallic state in FeSe. Phys. Rev. B 82, 064504 (2010)ADSCrossRefGoogle Scholar
  51. 51.
    Werner, P., Gull, E., Troyer, M., Millis, A.J.: Spin freezing transition and non-Fermi-liquid self-energy in a three-orbital model. Phys. Rev. Lett. 101, 166405 (2008)Google Scholar
  52. 52.
    Scalapino, D.J., Sugar, R.L.: Method for performing Monte Carlo calculations for systems with fermions. Phys. Rev. Lett. 46, 519 (1981)Google Scholar
  53. 53.
    Blankenbecler, R., Scalapino, D.J., Sugar, R.L.: Monte Carlo calculations of coupled boson-fermion systems I. Phys. Rev. D 24, 2278 (1981)Google Scholar
  54. 54.
    Hirsch, J.E.: Two-dimensional Hubbard model: numerical simulation study. Phys. Rev. B 31, 4403 (1985)ADSCrossRefGoogle Scholar
  55. 55.
    Hirsch, J.E., Fye, R.M.: Monte Carlo method for magnetic impurities in metals. Phys. Rev. Lett. 56, 2521 (1986)ADSCrossRefGoogle Scholar
  56. 56.
    Rubtsov, A.N., Savkin, V.V., Lichtenstein, A.I.: Continuous-time quantum Monte Carlo method for fermions. Phys. Rev. B 72, 035122 (2005)ADSCrossRefGoogle Scholar
  57. 57.
    Rubtsov, A.N., Lichtenstein, A.I.: Continuous-time quantum Monte Carlo method for fermions: beyond auxiliary field framework. JETP Lett. 80, 61 (2004)ADSCrossRefGoogle Scholar
  58. 58.
    Werner, P., Comanac, A., de’Medici, L., Troyer, M., Millis, A.J.: Continuous-time solver for quantum impurity models. Phys. Rev. Lett. 97, 076405 (2006)ADSCrossRefGoogle Scholar
  59. 59.
    Werner, P., Millis, A.J.: Hybridization expansion impurity solver: general formulation and application to Kondo lattice and two-orbital models. Phys. Rev. B 74, 155107 (2006)ADSCrossRefGoogle Scholar
  60. 60.
    Haule, K.: Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base. Phys. Rev. B 75, 155113 (2007)ADSCrossRefGoogle Scholar
  61. 61.
    Gull, E., Millis, A.J., Lichtenstein, A.I., Rubtsov, A.N., Troyer, M., and Werner, P.: Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349 (2011).Google Scholar
  62. 62.
    Gull, E., Werner, P., Millis, A., Troyer, M.: Performance analysis of continuous-time solvers for quantum impurity models. Phys. Rev. B 76, 235123 (2007)ADSCrossRefGoogle Scholar
  63. 63.
    Negele, J.W., Orland, H.: Quantum Many-Particle Systems. Westview Press, Boulder (1998)Google Scholar
  64. 64.
    Prokof’ev, N.V., Svistunov, B.V., Tupitsyn, I.S.: Exact quantum Monte Carlo process for the statistics of discrete systems. JETP Lett. 64, 911 (1996)ADSCrossRefGoogle Scholar
  65. 65.
    Yoo, J., Chandrasekharan, S., Kaul, R.K., Ullmo, D., Baranger, H.U.: On the sign problem in the Hirsch & Fye algorithm for impurity problems. J. Phys. A: Math. Gen. 38, 10307 (2005)zbMATHADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    Gull E., Continuous-Time Quantum Monte Carlo Algorithms for Fermions. Ph.D. thesis, ETH Zurich (2008)Google Scholar
  67. 67.
    Läuchli, A.M., Werner, P.: Krylov implementation of the hybridization expansion impurity solver and application to 5-orbital models. Phys. Rev. B 80, 235117 (2009)ADSCrossRefGoogle Scholar
  68. 68.
    Poteryaev, A.I., Lichtenstein, A.I., Kotliar, G.: Nonlocal Coulomb interactions and metal-insulator transition in Ti2O3: a cluster LDA+ DMFT approach. Phys. Rev. Lett. 93, 086401 (2004)ADSCrossRefGoogle Scholar
  69. 69.
    Saha-Dasgupta, T., Lichtenstein, A., Hoinkis, M., Glawion, S., Sing, M., Claessen, R., Valenti, R.: Cluster dynamical mean-field calculations for TiOCl. New J. Phys. 9, 380 (2007)ADSCrossRefGoogle Scholar
  70. 70.
    Fuhrmann, A., Okamoto, S., Monien, H., Millis, A.J.: Fictive-impurity approach to dynamical mean-field theory: a strong-coupling investigation. Phys. Rev. B 75, 205118 (2007)ADSCrossRefGoogle Scholar
  71. 71.
    Okamoto, S., Millis, A.J., Monien, H., Fuhrmann, A.: Fictive impurity models: an alternative formulation of the cluster dynamical mean-field method. Phys. Rev. B 68, 195121 (2003)ADSCrossRefGoogle Scholar
  72. 72.
    Biroli, G., Kotliar, G.: Cluster methods for strongly correlated electron systems. Phys. Rev. B 65, 155112 (2002)ADSCrossRefGoogle Scholar
  73. 73.
    Potthoff, M.: Self-energy-functional approach to systems of correlated electrons. Eur. Phys. J. B 32, 429 (2003)ADSCrossRefGoogle Scholar
  74. 74.
    Schiller, A., Ingersent, K.: Systematic \(1/d\) corrections to the infinite-dimensional limit of correlated lattice electron models. Phys. Rev. Lett. 75, 113 (1995)ADSCrossRefGoogle Scholar
  75. 75.
    Sadovskii, M.V., Nekrasov, I.A., Kuchinskii, E.Z., Pruschke, T., Anisimov, V.I.: Pseudogaps in strongly correlated metals: a generalized dynamical mean-field theory approach. Phys. Rev. B 72, 155105 (2005)ADSCrossRefGoogle Scholar
  76. 76.
    Pairault, S., Sénéchal, D., Tremblay, A.-M.S.: Strong-coupling expansion for the Hubbard model. Phys. Rev. Lett. 80, 5389 (1998)ADSCrossRefGoogle Scholar
  77. 77.
    Pairault, S., Sénéchal, D., Tremblay, A.-M.: Strong-coupling perturbation theory of the Hubbard model. Eur. Phys. J. B 16, 85 (2000)ADSCrossRefGoogle Scholar
  78. 78.
    Sarker, S.K.: A new functional integral formalism for strongly correlated Fermi systems. J. Phys. C: Solid State Phys. 21, L667 (1988)ADSCrossRefGoogle Scholar
  79. 79.
    Stanescu, T.D., Kotliar, G.: Strong coupling theory for interacting lattice models. Phys. Rev. B 70, 205112 (2004)ADSCrossRefGoogle Scholar
  80. 80.
    Rubtsov, A.N.: Quality of the mean-field approximation: a low-order generalization yielding realistic critical indices for three-dimensional Ising-class systems. Phys. Rev. B 66, 052107 (2002)ADSCrossRefGoogle Scholar
  81. 81.
    Rubtsov A.N., Small parameter for lattice models with strong interaction, arXiv:cond-mat/0601333, unpublished (2006)Google Scholar
  82. 82.
    Rubtsov, A.N., Katsnelson, M.I., Lichtenstein, A.I.: Dual fermion approach to nonlocal correlations in the Hubbard model. Phys. Rev. B 77, 033101 (2008)Google Scholar
  83. 83.
    Hafermann, H.: Numerical Approaches to Spatial Correlations in Strongly Interacting Fermion Systems. Cuvillier Verlag, Göttingen (2010)Google Scholar
  84. 84.
    Schäfer, J., Schrupp, D., Rotenberg, E., Rossnagel, K., Koh, H., Blaha, P., Claessen, R.: Electronic quasiparticle renormalization on the spin wave energy scale. Phys. Rev. Lett. 92, 097205 (2004)ADSCrossRefGoogle Scholar
  85. 85.
    Eschrig, M., Norman, M.R: Neutron Resonance: Modeling photoemission and tunneling data in the superconducting state of \(\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+\delta} \). Phys. Rev. Lett. 85, 3261 (2000)Google Scholar
  86. 86.
    Schachinger, E., Tu, J.J., Carbotte, J.P.: Angle-resolved photoemission spectroscopy and optical renormalizations: phonons or spin fluctuations. Phys. Rev. B 67, 214508 (2003)ADSCrossRefGoogle Scholar
  87. 87.
    Claessen, R., Sing, M., Schwingenschlögl, U., Blaha, P., Dressel, M., Jacobsen, C.S.: Spectroscopic signatures of spin-charge separation in the quasi-one-dimensional organic conductor TTF-TCNQ. Phys. Rev. Lett. 88, 096402 (2002)ADSCrossRefGoogle Scholar
  88. 88.
    Rubtsov, A.N., Katsnelson, M.I., Lichtenstein, A.I., Georges, A.: Dual fermion approach to the two-dimensional Hubbard model: antiferromagnetic fluctuations and Fermi arcs. Phys. Rev. B 79, 045133 (2009)ADSCrossRefGoogle Scholar
  89. 89.
    Baym, G., Kadanoff, L.P.: Conservation laws and correlation functions. Phys. Rev. 124, 287 (1961)zbMATHADSMathSciNetCrossRefGoogle Scholar
  90. 90.
    Abrikosov, A.A., Gor’kov, L.P., Dzyaloshinskii, I.E.: Methods of Quantum Field Theory in Statistical Physics. Pergamon Press, New York (1965)Google Scholar
  91. 91.
    Irkhin, V.Y., Katsnelson, M.I.: Current carriers in a quantum two-dimensional antiferromagnet. J. Phys.: Condens. Matter 3, 6439 (1991)ADSCrossRefGoogle Scholar
  92. 92.
    Park, H., Haule, K., Kotliar, G.: Cluster dynamical mean field theory of the mott transition. Phys. Rev. Lett. 101, 186403 (2008)ADSCrossRefGoogle Scholar
  93. 93.
    Macridin, A., Jarrell, M., Maier, T., Kent, P.R.C., D’Azevedo, E.: Pseudogap and antiferromagnetic correlations in the Hubbard Model. Phys. Rev. Lett. 97, 036401 (2006)ADSCrossRefGoogle Scholar
  94. 94.
    Ferrero, M., Cornaglia, P.S., Leo, L.D., Parcollet, O., Kotliar, G., Georges, A.: Valence bond dynamical mean-field theory of doped Mott insulators with nodal/antinodal differentiation. Eur. phys. Lett. 85, 57009 (2009)ADSCrossRefGoogle Scholar
  95. 95.
    Brener, S., Hafermann, H., Rubtsov, A.N., Katsnelson, M.I., Lichtenstein, A.I.: Dual fermion approach to susceptibility of correlated lattice fermions. Phys. Rev. B 77, 195105 (2008)ADSCrossRefGoogle Scholar
  96. 96.
    Li, G., Lee, H., Monien, H.: Determination of the lattice susceptibility within the dual fermion method. Phys. Rev. B 78, 195105 (2008)ADSCrossRefGoogle Scholar
  97. 97.
    Lee, H., Li, G., Monien, H.: Hubbard model on the triangular lattice using dynamical cluster approximation and dual fermion methods. Phys. Rev. B 78, 205117 (2008)ADSCrossRefGoogle Scholar
  98. 98.
    Hafermann, H., Kecker, M., Brener, S., Rubtsov, A.N., Katsnelson, M.I., Lichtenstein, A.I.: Dual fermion approach to high-temperature superconductivity. J. Supercond. Nov. Magn. 22, 45 (2009)CrossRefGoogle Scholar
  99. 99.
    Hafermann, H., Brener, S., Rubtsov, A.N., Katsnelson, M.I., Lichtenstein, A.I.: Cluster dual fermion approach to nonlocal correlations. JETP Lett. 86, 677 (2007)ADSCrossRefGoogle Scholar
  100. 100.
    Hafermann, H., Li, G., Rubtsov, A.N., Katsnelson, M.I., Lichtenstein, A.I., Monien, H.: Efficient perturbation theory for quantum lattice models. Phys. Rev. Lett. 102, 206401 (2009)ADSCrossRefGoogle Scholar
  101. 101.
    Hafermann, H., Jung, C., Brener, S., Katsnelson, M.I., Rubtsov, A.N., Lichtenstein, A.I.: Superperturbation solver for quantum impurity models. Europhys. Lett. 85, 27007 (2009)ADSCrossRefGoogle Scholar
  102. 102.
    Schollwöck, U.: The density-matrix renormalization group. Rev. Mod. Phys. 77, 259 (2005)ADSCrossRefGoogle Scholar
  103. 103.
    Balzer, M., Hanke, W., Potthoff, M.: Mott transition in one dimension: benchmarking dynamical cluster approaches. Phys. Rev. B 77, 045133 (2008)ADSCrossRefGoogle Scholar
  104. 104.
    Mishchenko, A.S., Prokof’ev, N.V., Sakamoto, A., Svistunov, B.V.: Diagrammatic quantum Monte Carlo study of the Fröhlich polaron. Phys. Rev. B 62, 6317 (2000)ADSCrossRefGoogle Scholar
  105. 105.
    Migdal, A.B.: Theory of Finite Fermi Systems and Applications to Atomic Nuclei. Interscience Publishers, New York (1967)Google Scholar
  106. 106.
    Nozières, P.: Theory of Interacting Fermi Systems. Benjamin Day, New York (1964)zbMATHGoogle Scholar
  107. 107.
    Auerbach, A. (eds): Interacting Electrons and Quantum Magnetism. Springer, New York (1998)Google Scholar
  108. 108.
    Hugenholtz, N.: Perturbation theory of large quantum systems. Physica 23, 481 (1957)zbMATHADSMathSciNetCrossRefGoogle Scholar
  109. 109.
    Bickers, N.E., Scalapino, D.J., White, S.R.: Conserving approximations for strongly correlated electron systems: Bethe-Salpeter equation and dynamics for the two-dimensional Hubbard Model. Phys. Rev. Lett. 62, 961 (1989)ADSCrossRefGoogle Scholar
  110. 110.
    Bickers, N.E., Scalapino, D.J.: Conserving approximations for strongly fluctuating electron systems. I. Formalism and calculational approach. Ann. Phys. 193, 206 (1989)ADSCrossRefGoogle Scholar
  111. 111.
    Bulut, N., Scalapino, D.J., White, S.R.: Bethe-Salpeter eigenvalues and amplitudes for the half-filled two-dimensional Hubbard model. Phys. Rev. B 47, 14599 (1993)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hartmut Hafermann
    • 1
    Email author
  • Frank Lechermann
    • 2
  • Alexei N. Rubtsov
    • 3
  • Mikhail I. Katsnelson
    • 4
  • Antoine Georges
    • 5
    • 6
  • Alexander I. Lichtenstein
    • 2
  1. 1.Centre de Physique Théorique CNRSÉcole PolytechniquePalaiseau CedexFrance
  2. 2.I. Institut für Theoretische PhysikUniversität HamburgHamburgGermany
  3. 3.Department of PhysicsMoscow State UniversityMoscowRussia
  4. 4.Institute for Molecules and MaterialsRadboud University NijmegenNijmegenThe Netherlands
  5. 5.Centre de Physique Théorique, CNRSÉcole PolytechniquePalaiseau CedexFrance
  6. 6.Collège de FranceParisFrance

Personalised recommendations