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  • Open Access

Update of Hartree-Fock theory for Hubbard-like models

E. Langmann1,* and J. Lenells2,†

  • *Contact author: langmann@kth.se
  • †Contact author: jlenells@kth.se
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Phys. Rev. Research 8, 023262 – Published 8 June, 2026

DOI: https://doi.org/10.1103/qyz3-cwmn

Abstract

We show that the standard textbook description of (restricted) Hartree-Fock theory for (Fermi) Hubbard-like models is in need of an update, and we present such an update allowing us to correct basic and established results in the condensed matter physics literature that are qualitatively wrong. Our update amounts to adding a test that reliably checks the thermodynamic stability of solutions of Hartree-Fock equations. This stability test makes it possible to detect, by simple means and with certainty, regions in phase space where the model exhibits mixed phases where two conventional phases coexist and translation invariance is broken in complicated ways; in such a mixed phase, unconventional physics is to be expected. Our results show that mixed phases are ubiquitous in Hubbard-like models in arbitrary dimensions.

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References (62)

  1. P. W. Anderson, Basic Notions of Condensed Matter Physics (CRC Press, Boca Raton, Florida, 2018).
  2. M. C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals, Phys. Rev. Lett. 10, 159 (1963).
  3. J. Hubbard, Electron correlations in narrow energy bands, Proc. R. Soc. London A 276, 238 (1963).
  4. J. Kanamori, Electron correlation and ferromagnetism of transition metals, Prog. Theor. Phys. 30, 275 (1963).
  5. D. R. Penn, Stability theory of the magnetic phases for a simple model of the transition metals, Phys. Rev. 142, 350 (1966).
  6. In the following, we write Hartree-Fock theory short for restricted Hartree-Fock theory, and it is important for us to distinguish this from unrestricted Hartree-Fock theory.
  7. J. E. Hirsch, Two-dimensional Hubbard model: Numerical simulation study, Phys. Rev. B 31, 4403 (1985).
  8. P. W. Anderson, The resonating valence bond state in La2CuO4 and superconductivity, Science 235, 1196 (1987).
  9. M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998).
  10. P. Das, V. Leeb, J. Knolle, and M. Knap, Realizing altermagnetism in Fermi-Hubbard models with ultracold atoms, Phys. Rev. Lett. 132, 263402 (2024).
  11. E. Langmann and M. Wallin, Mean-field approach to antiferromagnetic domains in the doped Hubbard model, Phys. Rev. B 55, 9439 (1997).
  12. E. Langmann and M. Wallin, Mean field magnetic phase diagrams for the two dimensional t–t′–U Hubbard model, J. Stat. Phys. 127, 825 (2007).
  13. V. Bach, E. H. Lieb, and J. P. Solovej, Generalized Hartree-Fock theory and the Hubbard model, J. Stat. Phys. 76, 3 (1994).
  14. V. Bach and J. Poelchau, Hartree-Fock Gibbs states for the Hubbard model, Markov Process. Relat. Fields 2, 225 (1996).
  15. J. Gouveia and R. Dias, Magnetic phase diagram of the Hubbard model in the Lieb lattice, J. Magn. Magn. Mater. 382, 312 (2015).
  16. G. Jana and A. Mukherjee, Frustration effects at finite temperature in the half filled Hubbard model, J. Phys.: Condens. Matter 32, 365602 (2020).
  17. P. Igoshev, M. Timirgazin, A. Katanin, A. Arzhnikov, and V. Y. Irkhin, Incommensurate magnetic order and phase separation in the two-dimensional Hubbard model with nearest-and next-nearest-neighbor hopping, Phys. Rev. B 81, 094407 (2010).
  18. P. Igoshev, A. Zarubin, A. Katanin, and V. Y. Irkhin, Ferromagnetism, spiral magnetic structures and phase separation in the two-dimensional Hubbard model, J. Magn. Magn. Mater. 324, 3601 (2012).
  19. W. Su, Spin polarons in the two-dimensional Hubbard model: A numerical study, Phys. Rev. B 37, 9904 (1988).
  20. W. Su and X. Chen, Magnetic polarons in the two-dimensional Hubbard model, Phys. Rev. B 38, 8879 (1988).
  21. D. Poilblanc and T. Rice, Charged solitons in the Hartree-Fock approximation to the large-U Hubbard model, Phys. Rev. B 39, 9749 (1989).
  22. J. Zaanen and O. Gunnarsson, Charged magnetic domain lines and the magnetism of high-Tc oxides, Phys. Rev. B 40, 7391 (1989).
  23. H. Schulz, Domain walls in a doped antiferromagnet, J. Phys. France 50, 2833 (1989).
  24. K. Machida, Magnetism in La2CuO4 based compounds, Physica C 158, 192 (1989).
  25. M. Kato, K. Machida, H. Nakanishi, and M. Fujita, Soliton lattice modulation of incommensurate spin density wave in two dimensional Hubbard model—A mean field study, J. Phys. Soc. Jpn. 59, 1047 (1990).
  26. R. Scholle, P. M. Bonetti, D. Vilardi, and W. Metzner, Comprehensive mean-field analysis of magnetic and charge orders in the two-dimensional Hubbard model, Phys. Rev. B 108, 035139 (2023).
  27. R. Scholle, W. Metzner, D. Vilardi, and P. M. Bonetti, Spiral to stripe transition in the two-dimensional Hubbard model, Phys. Rev. B 109, 235149 (2024).
  28. J. Vergés, E. Louis, P. Lomdahl, F. Guinea, and A. Bishop, Holes and magnetic textures in the two-dimensional Hubbard model, Phys. Rev. B 43, 6099 (1991).
  29. E. Langmann and J. Lenells, Mixed phases in a Fermi–Hubbard model describing altermagnetism, arXiv:2506.16503.
  30. D. Bonn, Are high-temperature superconductors exotic? Nat. Phys. 2, 159 (2006).
  31. H. Lin and J. Hirsch, Two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping, Phys. Rev. B 35, 3359 (1987).
  32. Y. Zhang and J. Callaway, Extended Hubbard model in two dimensions, Phys. Rev. B 39, 9397 (1989).
  33. Z. Chen, Y. Wang, S. N. Rebec, T. Jia, M. Hashimoto, D. Lu, B. Moritz, R. G. Moore, T. P. Devereaux, and Z.-X. Shen, Anomalously strong near-neighbor attraction in doped 1D cuprate chains, Science 373, 1235 (2021).
  34. Y. Wang, Z. Chen, T. Shi, B. Moritz, Z.-X. Shen, and T. P. Devereaux, Phonon-mediated long-range attractive interaction in one-dimensional cuprates, Phys. Rev. Lett. 127, 197003 (2021).
  35. M. N. Gastiasoro, J. Ruhman, and R. M. Fernandes, Superconductivity in dilute SrTiO3: A review, Ann. Phys. 417, 168107 (2020).
  36. S. G. Stewart, Heavy-fermion systems, Rev. Mod. Phys. 56, 755 (1984).
  37. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  38. U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch, T. A. Costi, R. Helmes, D. Rasch, and A. Rosch, Metallic and insulating phases of repulsively interacting fermions in a 3D optical lattice, Science 322, 1520 (2008).
  39. R. A. Hart, P. M. Duarte, T.-L. Yang, X. Liu, T. Paiva, E. Khatami, R. T. Scalettar, N. Trivedi, D. A. Huse, and R. G. Hulet, Observation of antiferromagnetic correlations in the Hubbard model with ultracold atoms, Nature (London) 519, 211 (2015).
  40. A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi–Hubbard antiferromagnet, Nature (London) 545, 462 (2017).
  41. For example, an antiferromagnetic and a paramagnetic solution would be qualitatively different, but two ferromagnetic solutions with different magnetization directions would not.
  42. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957).
  43. W. Metzner and D. Vollhardt, Correlated lattice fermions in d = ∞ dimensions, Phys. Rev. Lett. 62, 324 (1989).
  44. V. Bach, E. H. Lieb, and M. V. Travaglia, Ferromagnetism of the Hubbard model at strong coupling in the Hartree–Fock approximation, Rev. Math. Phys. 18, 519 (2006).
  45. E. Langmann and J. Lenells, Universality of mean-field antiferromagnetic order in an anisotropic 3D Hubbard model at half-filling, J. Stat. Phys. 192, 10 (2025).
  46. U. Wolff, Saddle point mean field calculation in the Hubbard model, Nucl. Phys. B 225, 391 (1983).
  47. E. Lieb and D. Mattis, Theory of ferromagnetism and the ordering of electronic energy levels, Phys. Rev. 125, 164 (1962).
  48. B. Farid, Some rigorous results concerning the uniform metallic ground states of single-band Hamiltonians in arbitrary dimensions, arXiv:1305.2089.
  49. C. Charlier, E. Langmann, and J. Lenells, On the Hartree-Fock phase diagram for the two-dimensional Hubbard model, arXiv:2512.20441.
  50. J. Henheik, E. Langmann, A. B. Lauritsen, J. Lenells, and V. Melin, Mixed phases in generalized Hartree–Fock theory for the attractive Hubbard model with an external magnetic field (unpublished).
  51. E. Langmann, J. Lidmar, M. Salmhofer, and M. Wallin, Mean field analysis of a model for superconductivity in an anti-ferromagnetic background, Physica C 296, 119 (1998).
  52. E. Langmann, J. Lenells, and V. Melin, Mean-field phase diagrams of the 2D t-t′-U−V model (unpublished).
  53. K. Sheshadri, D. Malterre, A. Fujimori, and A. Chainani, Connecting the one-band and three-band Hubbard models of cuprates via spectroscopy and scattering experiments, Phys. Rev. B 107, 085125 (2023).
  54. J. Spałek, Effect of pair hopping and magnitude of intra-atomic interaction on exchange-mediated superconductivity, Phys. Rev. B 37, 533 (1988).
  55. J. P. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, and C. A. Jiménez-Hoyos, et al., Solutions of the two-dimensional Hubbard model: Benchmarks and results from a wide range of numerical algorithms, Phys. Rev. X 5, 041041 (2015).
  56. A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221 (1978).
  57. J. V. Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100, 295 (1928).
  58. M. Sion, On general minimax theorems, Pac. J. Math. 8, 171 (1958).
  59. H. Komiya, Elementary proof for Sion's minimax theorem, Kodai Math. J. 11, 5 (1988).
  60. S. J. N. Mosconi, A differential characterisation of the minimax inequality, J. Convex Anal. 19, 185 (2012).
  61. B. Ricceri, On a minimax theorem: An improvement, a new proof and an overview of its applications, arXiv:1612.08191v3.
  62. B. Ricceri, Multiplicity of global minima for parametrized functions, Rend. Lincei Mat. Appl. 21, 47 (2010).

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Phys. Rev. Research 8, 023262– Published 8 June, 2026
  • Received 1 July 2025
  • Accepted 25 April 2026
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