- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Unrestricted Hartree-Fock analysis of the Hubbard model on single-layer graphene and Bernal bilayer graphene

Robin Scholle1 and Laura Classen1,2

PDF

Phys. Rev. B 113, 085132 – Published 19 February, 2026

DOI: https://doi.org/10.1103/drn2-4s3q

Abstract

We perform unrestricted Hartree-Fock calculations on the two-dimensional Hubbard model on a honeycomb and bilayer honeycomb lattice at both zero and finite temperatures. Finite-size real-space calculations are supplemented with random-phase approximation calculations in the thermodynamic limit. Our motivation comes from high doping levels achieved in graphene and Bernal bilayer graphene by intercalation. We present phase diagrams in doping and temperature for a moderate Hubbard interaction. The magnetic states we find are classified systematically based on the dominant Fourier components of their spin patterns, their average magnetization and spin incommensurabilities. The dominant spin patterns are Néel order and various types of stripes. Around Van Hove filling, we resolve the competition between stripe and chiral spin density waves in the symmetry-broken regime. We also investigate the effect of an applied external displacement field on the spin patterns of Bernal bilayer graphene.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (94)

  1. D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys. 13, 239 (2022).
  2. M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
  3. B. I. Shraiman and E. D. Siggia, Spiral phase of a doped quantum antiferromagnet, Phys. Rev. Lett. 62, 1564 (1989).
  4. B. I. Shraiman and E. D. Siggia, Excitation spectrum of the spiral state of a doped antiferromagnet, Phys. Rev. B 46, 8305 (1992).
  5. A. V. Chubukov and D. M. Frenkel, Renormalized perturbation theory of magnetic instabilities in the two-dimensional Hubbard model at small doping, Phys. Rev. B 46, 11884 (1992).
  6. A. V. Chubukov and K. A. Musaelian, Magnetic phases of the two-dimensional Hubbard model at low doping, Phys. Rev. B 51, 12605 (1995).
  7. T. Dombre, Modulated spiral phases in doped quantum antiferromagnets, J. Phys. France 51, 847 (1990).
  8. R. Frésard, M. Dzierzawa, and P. Wölfle, Slave-Boson approach to spiral magnetic order in the Hubbard model, Europhys. Lett. 15, 325 (1991).
  9. V. N. Kotov and O. P. Sushkov, Stability of the spiral phase in the two-dimensional extended t−J model, Phys. Rev. B 70, 195105 (2004).
  10. P. A. Igoshev, M. A. Timirgazin, A. A. Katanin, A. K. 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).
  11. P. A. Igoshev, M. A. Timirgazin, V. F. Gilmutdinov, A. K. Arzhnikov, and V. Y. Irkhin, Spiral magnetism in the single-band Hubbard model: The Hartree–Fock and slave-boson approaches, J. Phys.: Condens. Matter 27, 446002 (2015).
  12. H. Yamase, A. Eberlein, and W. Metzner, Coexistence of Incommensurate Magnetism and Superconductivity in the two-dimensional Hubbard Model, Phys. Rev. Lett. 116, 096402 (2016).
  13. A. Eberlein, W. Metzner, S. Sachdev, and H. Yamase, Fermi surface reconstruction and drop in the hall number due to spiral antiferromagnetism in high-Tc cuprates, Phys. Rev. Lett. 117, 187001 (2016).
  14. J. Mitscherling and W. Metzner, Longitudinal conductivity and Hall coefficient in two-dimensional metals with spiral magnetic order, Phys. Rev. B 98, 195126 (2018).
  15. P. M. Bonetti, J. Mitscherling, D. Vilardi, and W. Metzner, Charge carrier drop at the onset of pseudogap behavior in the two-dimensional Hubbard model, Phys. Rev. B 101, 165142 (2020).
  16. H. J. Schulz, Domain walls in a doped antiferromagnet, J. Phys. France 50, 2833 (1989).
  17. J. Zaanen and O. Gunnarsson, Charged magnetic domain lines and the magnetism of high-Tc oxides, Phys. Rev. B 40, 7391 (1989).
  18. K. Machida, Magnetism in La2CuO4 based compounds, Phys. C (Amsterdam, Neth,) 158, 192 (1989).
  19. D. Poilblanc and T. M. Rice, Charged solitons in the Hartree-Fock approximation to the large-U Hubbard model, Phys. Rev. B 39, 9749 (1989).
  20. H. J. Schulz, Incommensurate antiferromagnetism in the two-dimensional Hubbard model, Phys. Rev. Lett. 64, 1445 (1990).
  21. 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).
  22. G. Seibold, C. Castellani, C. Di Castro, and M. Grilli, Striped phases in the two-dimensional Hubbard model with long-range Coulomb interaction, Phys. Rev. B 58, 13506 (1998).
  23. M. Fleck, A. I. Lichtenstein, E. Pavarini, and A. M. Oleś, One-dimensional metallic behavior of the stripe phase in La2−xSrxCuO4, Phys. Rev. Lett. 84, 4962 (2000).
  24. M. Fleck, A. I. Lichtenstein, and A. M. Oleś, Spectral properties and pseudogap in the stripe phases of cuprate superconductors, Phys. Rev. B 64, 134528 (2001).
  25. M. Raczkowski and F. F. Assaad, Melting of stripe phases and its signature in the single-particle spectral function, Phys. Rev. B 82, 233101 (2010).
  26. M. A. Timirgazin, M. A. Arzhnikov, and A. V. Vedyayev, Incommensurate spin-density wave in two-dimensional Hubbard model, Solid State Phenom. 190, 67 (2012).
  27. R. Peters and N. Kawakami, Spin density waves in the Hubbard model: A DMFT approach, Phys. Rev. B 89, 155134 (2014).
  28. K. Matsuyama and J. Greensite, Multiplicity, localization, and domains in the Hartree–Fock ground state of the two-dimensional Hubbard model, Ann. Phys. 442, 168922 (2022).
  29. B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155 (2017).
  30. M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwöck, S. R. White, and S. Zhang (Simons Collaboration on the Many-Electron Problem), Absence of superconductivity in the pure two-dimensional Hubbard model, Phys. Rev. X 10, 031016 (2020).
  31. T. Schäfer, N. Wentzell, F. Šimkovic, Y.-Y. He, C. Hille, M. Klett, C. J. Eckhardt, B. Arzhang, V. Harkov, F. M. Le Régent, et al., Tracking the footprints of spin fluctuations: A multimethod, multimessenger study of the two-dimensional Hubbard model, Phys. Rev. X 11, 011058 (2021).
  32. 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).
  33. 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).
  34. M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero, A. Georges, and S. Sachdev, Topological order in the pseudogap metal, Proc. Natl. Acad. Sci. USA 115, E3665 (2018).
  35. S. Sachdev, Topological order, emergent gauge fields, and Fermi surface reconstruction, Rep. Prog. Phys. 82, 014001 (2019).
  36. P. M. Bonetti and W. Metzner, SU(2) gauge theory of the pseudogap phase in the two-dimensional Hubbard model, Phys. Rev. B 106, 205152 (2022).
  37. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  38. E. McCann and M. Koshino, The electronic properties of bilayer graphene, Rep. Prog. Phys. 76, 056503 (2013).
  39. C. Honerkamp, Density waves and Cooper pairing on the honeycomb lattice, Phys. Rev. Lett. 100, 146404 (2008).
  40. M. M. Scherer, S. Uebelacker, and C. Honerkamp, Instabilities of interacting electrons on the honeycomb bilayer, Phys. Rev. B 85, 235408 (2012).
  41. O. Vafek and K. Yang, Many-body instability of Coulomb interacting bilayer graphene: Renormalization group approach, Phys. Rev. B 81, 041401(R) (2010).
  42. R. E. Throckmorton and O. Vafek, Fermions on bilayer graphene: Symmetry breaking for b=0 and ν=0, Phys. Rev. B 86, 115447 (2012).
  43. V. Cvetkovic, R. E. Throckmorton, and O. Vafek, Electronic multicriticality in bilayer graphene, Phys. Rev. B 86, 075467 (2012).
  44. J. Nilsson, A. H. Castro Neto, N. M. R. Peres, and F. Guinea, Electron-electron interactions and the phase diagram of a graphene bilayer, Phys. Rev. B 73, 214418 (2006).
  45. R. Nandkishore and L. Levitov, Quantum anomalous Hall state in bilayer graphene, Phys. Rev. B 82, 115124 (2010).
  46. A. L. Szabó and B. Roy, Extended Hubbard model in undoped and doped monolayer and bilayer graphene: Selection rules and organizing principle among competing orders, Phys. Rev. B 103, 205135 (2021).
  47. Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and A. Muramatsu, Quantum spin liquid emerging in two-dimensional correlated Dirac fermions, Nature (London) 464, 847 (2010).
  48. T. C. Lang, Z. Y. Meng, M. M. Scherer, S. Uebelacker, F. F. Assaad, A. Muramatsu, C. Honerkamp, and S. Wessel, Antiferromagnetism in the Hubbard model on the Bernal-stacked honeycomb bilayer, Phys. Rev. Lett. 109, 126402 (2012).
  49. F. Zhang, Spontaneous chiral symmetry breaking in bilayer graphene, Synth. Met. 210, 9 (2015), in special issue reviews of current advances in graphene science and technology.
  50. M. Raczkowski, R. Peters, T. T. Phùng, N. Takemori, F. F. Assaad, A. Honecker, and J. Vahedi, Hubbard model on the honeycomb lattice: From static and dynamical mean-field theories to lattice quantum Monte Carlo simulations, Phys. Rev. B 101, 125103 (2020).
  51. H. Feldner, Z. Y. Meng, A. Honecker, D. Cabra, S. Wessel, and F. F. Assaad, Magnetism of finite graphene samples: Mean-field theory compared with exact diagonalization and quantum Monte Carlo simulations, Phys. Rev. B 81, 115416 (2010).
  52. S. Sorella and E. Tosatti, Semi-metal-insulator transition of the Hubbard model in the honeycomb lattice, Europhys. Lett. 19, 699 (1992).
  53. S. Pujari, T. C. Lang, G. Murthy, and R. K. Kaul, Interaction-induced Dirac fermions from quadratic band touching in bilayer graphene, Phys. Rev. Lett. 117, 086404 (2016).
  54. R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Chiral superconductivity from repulsive interactions in doped graphene, Nat. Phys. 8, 158 (2012).
  55. R. Nandkishore and A. V. Chubukov, Interplay of superconductivity and spin-density-wave order in doped graphene, Phys. Rev. B 86, 115426 (2012).
  56. R. Nandkishore, R. Thomale, and A. V. Chubukov, Superconductivity from weak repulsion in hexagonal lattice systems, Phys. Rev. B 89, 144501 (2014).
  57. M. L. Kiesel, C. Platt, W. Hanke, D. A. Abanin, and R. Thomale, Competing many-body instabilities and unconventional superconductivity in graphene, Phys. Rev. B 86, 020507(R) (2012).
  58. L. Classen, A. V. Chubukov, C. Honerkamp, and M. M. Scherer, Competing orders at higher-order van Hove points, Phys. Rev. B 102, 125141 (2020).
  59. T. Li, Spontaneous quantum Hall effect in quarter-doped Hubbard model on honeycomb lattice and its possible realization in doped graphene system, Europhys. Lett. 97, 37001 (2012).
  60. I. Martin and C. D. Batista, Itinerant electron-driven chiral magnetic ordering and spontaneous quantum Hall effect in triangular lattice models, Phys. Rev. Lett. 101, 156402 (2008).
  61. R. Nandkishore, G.-W. Chern, and A. V. Chubukov, Itinerant half-metal spin-density-wave state on the hexagonal lattice, Phys. Rev. Lett. 108, 227204 (2012).
  62. Y. Kato, I. Martin, and C. D. Batista, Stability of the spontaneous quantum Hall state in the triangular Kondo-lattice model, Phys. Rev. Lett. 105, 266405 (2010).
  63. F. Zhang, J. Jung, G. A. Fiete, Q. Niu, and A. H. MacDonald, Spontaneous quantum Hall states in chirally stacked few-layer graphene systems, Phys. Rev. Lett. 106, 156801 (2011).
  64. R. T. Weitz, M. T. Allen, B. E. Feldman, J. Martin, and A. Yacoby, Broken-symmetry states in doubly gated suspended bilayer graphene, Science 330, 812 (2010).
  65. J. Velasco, L. Jing, W. Bao, Y. Lee, P. Kratz, V. Aji, M. Bockrath, C. N. Lau, C. Varma, R. Stillwell, D. Smirnov, F. Zhang, J. Jung, and A. H. MacDonald, Transport spectroscopy of symmetry-broken insulating states in bilayer graphene, Nat. Nanotechnol. 7, 156 (2012).
  66. F. Freitag, J. Trbovic, M. Weiss, and C. Schönenberger, Spontaneously gapped ground state in suspended bilayer graphene, Phys. Rev. Lett. 108, 076602 (2012).
  67. J. Martin, B. E. Feldman, R. T. Weitz, M. T. Allen, and A. Yacoby, Local compressibility measurements of correlated states in suspended bilayer graphene, Phys. Rev. Lett. 105, 256806 (2010).
  68. D.-K. Ki and A. F. Morpurgo, High-quality multiterminal suspended graphene devices, Nano Lett. 13, 5165 (2013).
  69. A. S. Mayorov, D. C. Elias, M. Mucha-Kruczynski, R. V. Gorbachev, T. Tudorovskiy, A. Zhukov, S. V. Morozov, M. I. Katsnelson, V. I. Fal'ko, A. K. Geim, and K. S. Novoselov, Interaction-driven spectrum reconstruction in bilayer graphene, Science 333, 860 (2011).
  70. B. E. Feldman, J. Martin, and A. Yacoby, Broken-symmetry states and divergent resistance in suspended bilayer graphene, Nat. Phys. 5, 889 (2009).
  71. F. R. Geisenhof, F. Winterer, A. M. Seiler, J. Lenz, T. Xu, F. Zhang, and R. T. Weitz, Quantum anomalous Hall octet driven by orbital magnetism in bilayer graphene, Nature (London) 598, 53 (2021).
  72. S. Yang, S. Fecher, Q. Wang, M. Kühne, and J. H. Smet, Device level reversible potassium intercalation into bilayer graphene, 2D Mater. 9, 025020 (2022).
  73. S. Link, S. Forti, A. Stöhr, K. Küster, M. Rösner, D. Hirschmeier, C. Chen, J. Avila, M. C. Asensio, A. A. Zakharov, T. O. Wehling, A. I. Lichtenstein, M. I. Katsnelson, and U. Starke, Introducing strong correlation effects into graphene by gadolinium intercalation, Phys. Rev. B 100, 121407(R) (2019).
  74. P. Rosenzweig, H. Karakachian, D. Marchenko, K. Küster, and U. Starke, Overdoping graphene beyond the van Hove singularity, Phys. Rev. Lett. 125, 176403 (2020).
  75. S. Link, Intercalation of graphene on SiC(0001): Ultra-high doping levels and new 2D materials, Ph.D. thesis, FAU, 2017.
  76. J. L. McChesney, A. Bostwick, T. Ohta, T. Seyller, K. Horn, J. González, and E. Rotenberg, Extended van Hove singularity and superconducting instability in doped graphene, Phys. Rev. Lett. 104, 136803 (2010).
  77. H. Zhou, L. Holleis, Y. Saito, L. Cohen, W. Huynh, C. L. Patterson, F. Yang, T. Taniguchi, K. Watanabe, and A. F. Young, Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene, Science 375, 774 (2022).
  78. A. M. Seiler, F. R. Geisenhof, F. Winterer, K. Watanabe, T. Taniguchi, T. Xu, F. Zhang, and R. T. Weitz, Quantum cascade of correlated phases in trigonally warped bilayer graphene, Nature (London) 608, 298 (2022).
  79. A. M. Seiler, M. Statz, I. Weimer, N. Jacobsen, K. Watanabe, T. Taniguchi, Z. Dong, L. S. Levitov, and R. T. Weitz, Interaction-driven quasi-insulating ground states of gapped electron-doped bilayer graphene, Phys. Rev. Lett. 133, 066301 (2024).
  80. T. O. Wehling, E. Şaşıoğlu, C. Friedrich, A. I. Lichtenstein, M. I. Katsnelson, and S. Blügel, Strength of effective Coulomb interactions in graphene and graphite, Phys. Rev. Lett. 106, 236805 (2011).
  81. J. A. Vergés, E. Louis, P. S. Lomdahl, F. Guinea, and A. R. Bishop, Holes and magnetic textures in the two-dimensional Hubbard model, Phys. Rev. B 43, 6099 (1991).
  82. P. Novelli, F. Taddei, A. K. Geim, and M. Polini, Failure of conductance quantization in two-dimensional topological insulators due to nonmagnetic impurities, Phys. Rev. Lett. 122, 016601 (2019).
  83. A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed. (Cambridge University Press, Cambridge, England, 2010).
  84. P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, Cambridge, UK, 2015).
  85. S. Maiti and A. V. Chubukov, Superconductivity from repulsive interaction, AIP Conf. Proc. 1550, 3 (2013)
  86. S. Sachdev, H. D. Scammell, M. S. Scheurer, and G. Tarnopolsky, Gauge theory for the cuprates near optimal doping, Phys. Rev. B 99, 054516 (2019).
  87. O. Zachar, S. A. Kivelson, and V. J. Emery, Landau theory of stripe phases in cuprates and nickelates, Phys. Rev. B 57, 1422 (1998).
  88. B. V. Fine and T. Egami, Phase separation in the vicinity of quantum-critical doping concentration: Implications for high-temperature superconductors, Phys. Rev. B 77, 014519 (2008).
  89. S. V. Kokanova, P. A. Maksimov, A. V. Rozhkov, and A. O. Sboychakov, Competition of spatially inhomogeneous phases in systems with nesting-driven spin-density wave state, Phys. Rev. B 104, 075110 (2021).
  90. A. O. Sboychakov, A. V. Rozhkov, A. L. Rakhmanov, and F. Nori, Antiferromagnetic states and phase separation in doped AA-stacked graphene bilayers, Phys. Rev. B 88, 045409 (2013).
  91. T. M. Rice, Band-structure effects in itinerant antiferromagnetism, Phys. Rev. B 2, 3619 (1970).
  92. K. Maki and T. Tsuneto, Pauli paramagnetism and superconducting state, Prog. Theor. Phys. 31, 945 (1964).
  93. L. Classen and J. J. Betouras, High-order van Hove singularities and their connection to flat bands, Annu. Rev. Condens. Matter Phys. 16 (2025).
  94. The code is publicly available at https://github.com/RoScholle/BilayerGraphene.

Outline

Information

Phys. Rev. B 113, 085132– Published 19 February, 2026
  • Received 23 January 2025
  • Revised 13 January 2026
  • Accepted 27 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation