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Geometric Orthogonal Metals: Hidden Antiferromagnetism and the Pseudogap from Fluctuating Stripes

Henning Schlömer1,2,*, Annabelle Bohrdt2,3, and Fabian Grusdt1,2,†

  • *Contact author: h.schloemer@physik.uni-muenchen.de
  • †Contact author: fabian.grusdt@physik.uni-muenchen.de
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PRX Quantum 6, 030342 – Published 4 September, 2025

DOI: https://doi.org/10.1103/5sq4-r7dk

Abstract

One of the key features of hole-doped cuprates is the presence of an extended pseudogap phase, whose microscopic origin has been the subject of intense investigation since its discovery and is believed to be crucial for understanding high-temperature superconductivity. Various explanations have been proposed for the pseudogap, including links to symmetry-breaking orders such as stripes or pairing, and the emergence of novel fractionalized Fermi liquid (FL∗) and orthogonal metal (OM) phases. The topological nature of FL∗ and OM phases has been identified as scenarios compatible with a small Fermi surface without symmetry breaking, as suggested experimentally. With recent experimental and numerical studies supporting an intricate relationship between stripe order and the pseudogap phase, we here propose an alternative scenario: an orthogonal metal with a geometric origin (GOM) driven by fluctuating domain walls. The essential mechanism behind our proposal is hidden order, where the proliferation of domain walls stabilized by charge fluctuations obscures the underlying long-range antiferromagnetic order in real space, but order is preserved in the reference frame of the background spins. As a result, well-defined fermionic quasiparticles in the form of magnetic polarons exist, which couple to Z2 topological excitations of the domain-wall string-net condensate in the ground state and constitute a small Fermi surface. At a critical doping value, we argue that hidden order is lost, driving a transition to a regular Fermi liquid at a hidden quantum critical point featuring quantum critical transport properties. Our GOM framework establishes a deep connection between the antiferromagnetic, stripe, and pseudogap phases, and suggests a possible unification of superconductivity in (electron- and hole-) doped cuprates and heavy fermion compounds.

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Popular Summary

One of the central mysteries of hole-doped cuprates is the pseudogap phase, whose unusual properties are believed to be essential for understanding high-temperature superconductivity. While a broad variety of theoretical proposals have been put forward in the past decades, a unified view connecting the pseudogap to other observed phases, like antiferromagnetic (AFM) and stripe phases, has remained elusive. Here we propose a scenario in which these phases all have one common origin: the spins in the material form an ordered AFM background, on top of which fluctuating domain walls exist that can disrupt and obscure long-range order.

We suggest that these fluctuating domain walls are at the heart of the pseudogap phase. They hide magnetic order in real space, leaving only short-range AFM correlations detectable in experiments. Additionally, we propose that these domain-wall fluctuations develop topological order (forming an odd Z2 spin liquid), which explains the formation of a small Fermi surface coexisting with short-range magnetism, aligning with experimental observations. At a critical point, the hidden AFM order fully dissolves, restoring spin symmetry without a divergent correlation length but showing quantum critical scaling, in line with experiments.

Our theory provides a unified view on key observations in cuprates and could shed light on superconductivity across hole- and electron-doped cuprates and heavy fermion compounds, where in the latter, magnetic fluctuations may act as the pairing mechanism. Ultracold-atom simulators, which can measure nonlocal correlations, offer a powerful platform to test our theory directly, which can bring us closer to unraveling the pseudogap’s microscopic origin and its role in high-temperature superconductivity.

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References (176)

  1. J. G. Bednorz and K. A. Müller, Possible high Tc superconductivity in the Ba-La-Cu-O system, Z. Phys. B: Condens. Matter 64, 189 (1986).
  2. P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006).
  3. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179 (2015).
  4. T. Timusk and B. Statt, The pseudogap in high-temperature superconductors: An experimental survey, Rep. Progr. Phys. 62, 61 (1999).
  5. M. R. Norman, D. Pines, and C. Kallin, The pseudogap: Friend or foe of high Tc? Adv. Phys. 54, 715 (2005).
  6. D. Chowdhury and S. Sachdev, in Quantum Criticality in Condensed Matter (World Scientific, Singapore), p. 1.
  7. K. M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, and Z.-X. Shen, Nodal quasiparticles and antinodal charge ordering in Ca2−xNaxCuO2Cl2, Science 307, 901 (2005).
  8. A. Kanigel, M. R. Norman, M. Randeria, U. Chatterjee, S. Souma, A. Kaminski, H. M. Fretwell, S. Rosenkranz, M. Shi, T. Sato, T. Takahashi, Z. Z. Li, H. Raffy, K. Kadowaki, D. Hinks, L. Ozyuzer, and J. C. Campuzano, Evolution of the pseudogap from Fermi arcs to the nodal liquid, Nat. Phys. 2, 447 (2006).
  9. H.-B. Yang, J. D. Rameau, Z.-H. Pan, G. D. Gu, P. D. Johnson, H. Claus, D. G. Hinks, and T. E. Kidd, Reconstructed Fermi surface of underdoped Bi2Sr2CaCu2O8+δ cuprate superconductors, Phys. Rev. Lett. 107, 047003 (2011).
  10. K. Kurokawa, S. Isono, Y. Kohama, S. Kunisada, S. Sakai, R. Sekine, M. Okubo, M. D. Watson, T. K. Kim, C. Cacho, S. Shin, T. Tohyama, K. Tokiwa, and T. Kondo, Unveiling phase diagram of the lightly doped high-Tc cuprate superconductors with disorder removed, Nat. Commun. 14, 4064 (2023).
  11. N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor, Nature 447, 565 (2007).
  12. B. Vignolle, A. Carrington, R. A. Cooper, M. M. J. French, A. P. Mackenzie, C. Jaudet, D. Vignolles, C. Proust, and N. E. Hussey, Quantum oscillations in an overdoped high-Tc superconductor, Nature 455, 952 (2008).
  13. E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison, F. F. Balakirev, B. Dabrowski, and J. R. Cooper, Quantum oscillations in the underdoped cuprate YBa2Cu4O8, Phys. Rev. Lett. 100, 047003 (2008).
  14. B. J. Ramshaw, B. Vignolle, J. Day, R. Liang, W. N. Hardy, C. Proust, and D. A. Bonn, Angle dependence of quantum oscillations in YBa2Cu3O6.59 shows free-spin behaviour of quasiparticles, Nat. Phys. 7, 234 (2011).
  15. Y. Ando, Y. Kurita, S. Komiya, S. Ono, and K. Segawa, Evolution of the Hall coefficient and the peculiar electronic structure of the cuprate superconductors, Phys. Rev. Lett. 92, 197001 (2004).
  16. S. Badoux, W. Tabis, F. Laliberté, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Béard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust, Change of carrier density at the pseudogap critical point of a cuprate superconductor, Nature 531, 210 (2016).
  17. C. Collignon, S. Badoux, S. A. A. Afshar, B. Michon, F. Laliberté, O. Cyr-Choinière, J.-S. Zhou, S. Licciardello, S. Wiedmann, N. Doiron-Leyraud, and L. Taillefer, Fermi-surface transformation across the pseudogap critical point of the cuprate superconductor La1.6−xNd0.4SrxCuO4, Phys. Rev. B 95, 224517 (2017).
  18. S. I. Mirzaei, D. Stricker, J. N. Hancock, C. Berthod, A. Georges, E. van Heumen, M. K. Chan, X. Zhao, Y. Li, M. Greven, N. Barišić, and D. van der Marel, Spectroscopic evidence for Fermi liquid-like energy and temperature dependence of the relaxation rate in the pseudogap phase of the cuprates, Proc. Natl. Acad. Sci. 110, 5774 (2013).
  19. M. K. Chan, M. J. Veit, C. J. Dorow, Y. Ge, Y. Li, W. Tabis, Y. Tang, X. Zhao, N. Barišić, and M. Greven, In-plane magnetoresistance obeys Kohler’s rule in the pseudogap phase of cuprate superconductors, Phys. Rev. Lett. 113, 177005 (2014).
  20. J. M. Luttinger, Fermi surface and some simple equilibrium properties of a system of interacting fermions, Phys. Rev. 119, 1153 (1960).
  21. S. Sachdev, Quantum Phases of Matter (Cambridge University Press, Cambridge, 2023).
  22. G. Xu, G. D. Gu, M. Hücker, B. Fauqué, T. G. Perring, L. P. Regnault, and J. M. Tranquada, Testing the itinerancy of spin dynamics in superconducting Bi2Sr2CaCu2O8+δ, Nat. Phys. 5, 642 (2009).
  23. M. Le Tacon, G. Ghiringhelli, J. Chaloupka, M. M. Sala, V. Hinkov, M. W. Haverkort, M. Minola, M. Bakr, K. J. Zhou, S. Blanco-Canosa, C. Monney, Y. T. Song, G. L. Sun, C. T. Lin, G. M. De Luca, M. Salluzzo, G. Khaliullin, T. Schmitt, L. Braicovich, and B. Keimer, Intense paramagnon excitations in a large family of high-temperature superconductors, Nat. Phys. 7, 725 (2011).
  24. M. P. M. Dean, G. Dellea, R. S. Springell, F. Yakhou-Harris, K. Kummer, N. B. Brookes, X. Liu, Y.-J. Sun, J. Strle, T. Schmitt, L. Braicovich, G. Ghiringhelli, I. Božović, and J. P. Hill, Persistence of magnetic excitations in La2−xSrxCuO4 from the undoped insulator to the heavily overdoped non-superconducting metal, Nat. Mater. 12, 1019 (2013).
  25. M. Guarise, B. D. Piazza, H. Berger, E. Giannini, T. Schmitt, H. M. Rønnow, G. A. Sawatzky, J. van den Brink, D. Altenfeld, I. Eremin, and M. Grioni, Anisotropic softening of magnetic excitations along the nodal direction in superconducting cuprates, Nat. Commun. 5, 5760 (2014).
  26. V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Stripe phases in high-temperature superconductors, Proc. Natl. Acad. Sci. 96, 8814 (1999).
  27. S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, How to detect fluctuating stripes in the high-temperature superconductors, Rev. Mod. Phys. 75, 1201 (2003).
  28. M. Vojta, Lattice symmetry breaking in cuprate superconductors: Stripes nematics, and superconductivity, Adv. Phys. 58, 699 (2009).
  29. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Evidence for stripe correlations of spins and holes in copper oxide superconductors, Nature 375, 561 (1995).
  30. J. M. Tranquada, J. D. Axe, N. Ichikawa, Y. Nakamura, S. Uchida, and B. Nachumi, Neutron-scattering study of stripe-phase order of holes and spins in La1.48Nd0.4Sr0.12CuO4, Phys. Rev. B 54, 7489 (1996).
  31. P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A. Sawatzky, and D. L. Feng, Spatially modulated “Mottness” in La2−xBaxCuO4, Nat. Phys. 1, 155 (2005).
  32. C. Howald, H. Eisaki, N. Kaneko, M. Greven, and A. Kapitulnik, Periodic density-of-states modulations in superconducting Bi2Sr2CaCu2O8+δ, Phys. Rev. B 67, 014533 (2003).
  33. M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Local ordering in the pseudogap state of the high-Tc superconductor Bi2Sr2CaCu2O8+δ, Science 303, 1995 (2004).
  34. C. V. Parker, P. Aynajian, E. H. da Silva Neto, A. Pushp, S. Ono, J. Wen, Z. Xu, G. Gu, and A. Yazdani, Fluctuating stripes at the onset of the pseudogap in the high-Tc superconductor Bi2Sr2CaCu2O8+x, Nature 468, 677 (2010).
  35. T. Wu, H. Mayaffre, S. Krämer, M. Horvatić, C. Berthier, W. N. Hardy, R. Liang, D. A. Bonn, and M.-H. Julien, Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy, Nature 477, 191 (2011).
  36. J. Chang, N. B. Christensen, C. Niedermayer, K. Lefmann, H. M. Rønnow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess, M. Boehm, R. Mottl, S. Pailhés, N. Momono, M. Oda, M. Ido, and J. Mesot, Magnetic-field-induced soft-mode quantum phase transition in the high-temperature superconductor La1.855Sr0.145CuO4: An inelastic neutron-scattering study, Phys. Rev. Lett. 102, 177006 (2009).
  37. M. Vojta, Stripes and electronic quasiparticles in the pseudogap state of cuprate superconductors, Physica C 481, 178 (2012).
  38. M. Frachet, I. Vinograd, R. Zhou, S. Benhabib, S. Wu, H. Mayaffre, S. Krämer, S. K. Ramakrishna, A. P. Reyes, J. Debray, T. Kurosawa, N. Momono, M. Oda, S. Komiya, S. Ono, M. Horio, J. Chang, C. Proust, D. LeBoeuf, and M.-H. Julien, Hidden magnetism at the pseudogap critical point of a cuprate superconductor, Nat. Phys. 16, 1064 (2020).
  39. I. Vinograd, R. Zhou, H. Mayaffre, S. Krämer, S. K. Ramakrishna, A. P. Reyes, T. Kurosawa, N. Momono, M. Oda, S. Komiya, S. Ono, M. Horio, J. Chang, and M.-H. Julien, Competition between spin ordering and superconductivity near the pseudogap boundary in La2−xSrxCuO4: Insights from NMR, Phys. Rev. B 106, 054522 (2022).
  40. A. Missiaen, H. Mayaffre, S. Krämer, D. Zhao, Y. Zhou, T. Wu, X. Chen, S. Pyon, T. Takayama, H. Takagi, D. LeBoeuf, and M.-H. Julien, Spin-stripe order tied to the pseudogap phase in La1.8−xEu0.2SrxCuO4, Phys. Rev. X 15, 021010 (2025).
  41. F. Šimkovic, R. Rossi, A. Georges, and M. Ferrero, Origin and fate of the pseudogap in the doped Hubbard model, Science 385, eade9194 (2024).
  42. H. Xu, H. Shi, E. Vitali, M. Qin, and S. Zhang, Stripes and spin-density waves in the doped two-dimensional Hubbard model: Ground state phase diagram, Phys. Rev. Res. 4, 013239 (2022).
  43. A. J. Millis and M. R. Norman, Antiphase stripe order as the origin of electron pockets observed in 1/8-hole-doped cuprates, Phys. Rev. B 76, 220503 (2007).
  44. I. Dimov, P. Goswami, X. Jia, and S. Chakravarty, Competing order, Fermi surface reconstruction, and quantum oscillations in underdoped high-temperature superconductors, Phys. Rev. B 78, 134529 (2008).
  45. M. R. Norman, J. Lin, and A. J. Millis, Lifshitz transition in underdoped cuprates, Phys. Rev. B 81, 180513 (2010).
  46. H. Yao, D.-H. Lee, and S. Kivelson, Fermi-surface reconstruction in a smectic phase of a high-temperature superconductor, Phys. Rev. B 84, 012507 (2011).
  47. C. Castellani, C. Di Castro, and M. Grilli, Singular quasiparticle scattering in the proximity of charge instabilities, Phys. Rev. Lett. 75, 4650 (1995).
  48. S. A. Kivelson, E. Fradkin, and V. J. Emery, Electronic liquid-crystal phases of a doped Mott insulator, Nature 393, 550 (1998).
  49. M. Punk, A. Allais, and S. Sachdev, Quantum dimer model for the pseudogap metal, Proc. Natl. Acad. Sci. 112, 9552 (2015).
  50. Y. Wang, L. Li, and N. P. Ong, Nernst effect in high-Tc superconductors, Phys. Rev. B 73, 024510 (2006).
  51. H. B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Emergence of preformed Cooper pairs from the doped Mott insulating state in Bi2Sr2CaCu2O8+δ, Nature 456, 77 (2008).
  52. A. Kanigel, U. Chatterjee, M. Randeria, M. R. Norman, G. Koren, K. Kadowaki, and J. C. Campuzano, Evidence for pairing above the transition temperature of cuprate superconductors from the electronic dispersion in the pseudogap phase, Phys. Rev. Lett. 101, 137002 (2008).
  53. J. Niu, M. O. Larrazabal, T. Gozlinski, Y. Sato, K. M. Bastiaans, T. Benschop, J.-F. Ge, Y. M. Blanter, G. Gu, I. Swart, et al., Equivalence of pseudogap and pairing energy in a cuprate high-temperature superconductor, ArXiv:2409.15928.
  54. I. Battisti, K. M. Bastiaans, V. Fedoseev, A. de la Torre, N. Iliopoulos, A. Tamai, E. C. Hunter, R. S. Perry, J. Zaanen, F. Baumberger, and M. P. Allan, Universality of pseudogap and emergent order in lightly doped Mott insulators, Nat. Phys. 13, 21 (2017).
  55. Y.-T. Hsu, A. Rydh, M. Berben, C. Duffy, A. de la Torre, R. S. Perry, and N. E. Hussey, Carrier density crossover and quasiparticle mass enhancement in a doped 5d Mott insulator, Nat. Phys. 20, 1596 (2024).
  56. D. Chowdhury and S. Sachdev, Density-wave instabilities of fractionalized Fermi liquids, Phys. Rev. B 90, 245136 (2014).
  57. T. Senthil, S. Sachdev, and M. Vojta, Fractionalized Fermi liquids, Phys. Rev. Lett. 90, 216403 (2003).
  58. N. Read and S. Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66, 1773 (1991).
  59. X. G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991).
  60. R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, Hole dynamics in an antiferromagnet across a deconfined quantum critical point, Phys. Rev. B 75, 235122 (2007).
  61. Y. Qi and S. Sachdev, Effective theory of Fermi pockets in fluctuating antiferromagnets, Phys. Rev. B 81, 115129 (2010).
  62. E. G. Moon and S. Sachdev, Underdoped cuprates as fractionalized Fermi liquids: Transition to superconductivity, Phys. Rev. B 83, 224508 (2011).
  63. J.-W. Mei, S. Kawasaki, G.-Q. Zheng, Z.-Y. Weng, and X.-G. Wen, Luttinger-volume violating Fermi liquid in the pseudogap phase of the cuprate superconductors, Phys. Rev. B 85, 134519 (2012).
  64. M. Punk and S. Sachdev, Fermi surface reconstruction in hole-doped t-J models without long-range antiferromagnetic order, Phys. Rev. B 85, 195123 (2012).
  65. S. Sachdev and D. Chowdhury, The novel metallic states of the cuprates: Topological Fermi liquids and strange metals, Prog. Theor. Exp. Phys. 2016, 12C102 (2016).
  66. G. Kotliar and J. Liu, Superexchange mechanism and d-wave superconductivity, Phys. Rev. B 38, 5142 (1988).
  67. M. Oshikawa, Topological approach to Luttinger’s theorem and the Fermi surface of a Kondo lattice, Phys. Rev. Lett. 84, 3370 (2000).
  68. A. Rüegg, S. D. Huber, and M. Sigrist, Z2-slave-spin theory for strongly correlated fermions, Phys. Rev. B 81, 155118 (2010).
  69. R. Nandkishore, M. A. Metlitski, and T. Senthil, Orthogonal metals: The simplest non-Fermi liquids, Phys. Rev. B 86, 045128 (2012).
  70. S. Gazit, F. F. Assaad, and S. Sachdev, Fermi surface reconstruction without symmetry breaking, Phys. Rev. X 10, 041057 (2020).
  71. J. Zaanen, O. Y. Osman, H. V. Kruis, Z. Nussinov, and J. Tworzydlo, The geometric order of stripes and Luttinger liquids, Philos. Mag. B 81, 1485 (2001).
  72. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  73. T. Esslinger, Fermi-Hubbard physics with atoms in an optical lattice, Annu. Rev. Condens. Matter Phys. 1, 129 (2010).
  74. I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nat. Phys. 8, 267 (2012).
  75. M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan, K. Wooley-Brown, S. Blatt, and M. Greiner, Site-resolved imaging of fermionic 6Li in an optical lattice, Phys. Rev. Lett. 114, 213002 (2015).
  76. L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein, Quantum-gas microscope for fermionic atoms, Phys. Rev. Lett. 114, 193001 (2015).
  77. E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce, and S. Kuhr, Single-atom imaging of fermions in a quantum-gas microscope, Nat. Phys. 11, 738 (2015).
  78. C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  79. A. Bohrdt, L. Homeier, C. Reinmoser, E. Demler, and F. Grusdt, Exploration of doped quantum magnets with ultracold atoms, Ann. Phys. (N. Y.) 435, 168651 (2021), Special issue on Philip W. Anderson.
  80. D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Short-range quantum magnetism of ultracold fermions in an optical lattice, Science 340, 1307 (2013).
  81. 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 519, 211 (2015).
  82. M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Spin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chains, Science 353, 1257 (2016).
  83. 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 545, 462 (2017).
  84. H.-J. Shao, Y.-X. Wang, D.-Z. Zhu, Y.-S. Zhu, H.-N. Sun, S.-Y. Chen, C. Zhang, Z.-J. Fan, Y. Deng, X.-C. Yao, Y.-A. Chen, and J.-W. Pan, Antiferromagnetic phase transition in a 3D fermionic Hubbard model, Nature 632, 267 (2024).
  85. J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch, and C. Gross, Imaging magnetic polarons in the doped Fermi–Hubbard model, Nature 572, 358 (2019).
  86. C. S. Chiu, G. Ji, A. Bohrdt, M. Xu, M. Knap, E. Demler, F. Grusdt, M. Greiner, and D. Greif, String patterns in the doped Hubbard model, Science 365, 251 (2019).
  87. T. Hartke, B. Oreg, N. Jia, and M. Zwierlein, Doublon-hole correlations and fluctuation thermometry in a Fermi-Hubbard gas, Phys. Rev. Lett. 125, 113601 (2020).
  88. J. Koepsell, D. Bourgund, P. Sompet, S. Hirthe, A. Bohrdt, Y. Wang, F. Grusdt, E. Demler, G. Salomon, C. Gross, and I. Bloch, Microscopic evolution of doped Mott insulators from polaronic metal to Fermi liquid, Science 374, 82 (2021).
  89. G. Ji, M. Xu, L. H. Kendrick, C. S. Chiu, J. C. Brüggenjürgen, D. Greif, A. Bohrdt, F. Grusdt, E. Demler, M. Lebrat, and M. Greiner, Coupling a mobile hole to an antiferromagnetic spin background: Transient dynamics of a magnetic polaron, Phys. Rev. X 11, 021022 (2021).
  90. D. Bourgund, T. Chalopin, P. Bojović, H. Schlömer, S. Wang, T. Franz, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and T. A. Hilker, Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system, Nature 637, 57 (2025).
  91. P. W. Anderson, The resonating valence bond state in La2CuO4 and superconductivity, Science 235, 1196 (1987).
  92. 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).
  93. B. I. Shraiman and E. D. Siggia, Mobile vacancies in a quantum Heisenberg antiferromagnet, Phys. Rev. Lett. 61, 467 (1988).
  94. S. Sachdev, Hole motion in a quantum Néel state, Phys. Rev. B 39, 12232 (1989).
  95. C. L. Kane, P. A. Lee, and N. Read, Motion of a single hole in a quantum antiferromagnet, Phys. Rev. B 39, 6880 (1989).
  96. V. J. Emery, Theory of high-Tc superconductivity in oxides, Phys. Rev. Lett. 58, 2794 (1987).
  97. J. R. Schrieffer, X.-G. Wen, and S.-C. Zhang, Spin-bag mechanism of high-temperature superconductivity, Phys. Rev. Lett. 60, 944 (1988).
  98. P. Béran, D. Poilblanc, and R. B. Laughlin, Evidence for composite nature of quasiparticles in the 2D t-J model, Nucl. Phys. B 473, 707 (1996).
  99. F. Grusdt, M. Kánasz-Nagy, A. Bohrdt, C. S. Chiu, G. Ji, M. Greiner, D. Greif, and E. Demler, Parton theory of magnetic polarons: Mesonic resonances and signatures in dynamics, Phys. Rev. X 8, 011046 (2018).
  100. A. Bohrdt, E. Demler, F. Pollmann, M. Knap, and F. Grusdt, Parton theory of angle-resolved photoemission spectroscopy spectra in antiferromagnetic Mott insulators, Phys. Rev. B 102, 035139 (2020).
  101. P. Bermes, A. Bohrdt, and F. Grusdt, Magnetic polarons beyond linear spin-wave theory: Mesons dressed by magnons, Phys. Rev. B 109, 205104 (2024).
  102. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Fluctuating spin density waves in metals, Phys. Rev. B 80, 155129 (2009).
  103. S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, Spin density wave order, topological order, and Fermi surface reconstruction, Phys. Rev. B 94, 115147 (2016).
  104. S. Chatterjee, S. Sachdev, and M. S. Scheurer, Intertwining topological order and broken symmetry in a theory of fluctuating spin-density waves, Phys. Rev. Lett. 119, 227002 (2017).
  105. S. Sachdev, Topological order, emergent gauge fields, and Fermi surface reconstruction, Rep. Progr. Phys. 82, 014001 (2019).
  106. 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).
  107. Y. Zhang, E. Demler, and S. Sachdev, Competing orders in a magnetic field: Spin and charge order in the cuprate superconductors, Phys. Rev. B 66, 094501 (2002).
  108. K. Machida, Magnetism in La2CuO4 based compounds, Physica C 158, 192 (1989).
  109. 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).
  110. J. Zaanen and O. Gunnarsson, Charged magnetic domain lines and the magnetism of high-Tc oxides, Phys. Rev. B 40, 7391 (1989).
  111. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996).
  112. S. R. White and D. J. Scalapino, Density matrix renormalization group study of the striped phase in the 2D t-J model, Phys. Rev. Lett. 80, 1272 (1998).
  113. S. R. White and D. J. Scalapino, Stripes on a 6-leg Hubbard ladder, Phys. Rev. Lett. 91, 136403 (2003).
  114. 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).
  115. E. W. Huang, C. B. Mendl, H.-C. Jiang, B. Moritz, and T. P. Devereaux, Stripe order from the perspective of the Hubbard model, npj Quantum Mater. 3, 22 (2018).
  116. Y.-F. Jiang, J. Zaanen, T. P. Devereaux, and H.-C. Jiang, Ground state phase diagram of the doped Hubbard model on the four-leg cylinder, Phys. Rev. Res. 2, 033073 (2020).
  117. M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwöck, S. R. White, S. Zhang, Simons Collaboration on the Many-Electron ProblemAbsence of superconductivity in the pure two-dimensional Hubbard model, Phys. Rev. X 10, 031016 (2020).
  118. H. Xu, C.-M. Chung, M. Qin, U. Schollwöck, S. R. White, and S. Zhang, Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science 384, eadh7691 (2024).
  119. R. Preuss, W. Hanke, C. Gröber, and H. G. Evertz, Pseudogaps and their interplay with magnetic excitations in the doped 2D Hubbard model, Phys. Rev. Lett. 79, 1122 (1997).
  120. A. Macridin, M. Jarrell, T. Maier, P. R. C. Kent, and E. D’Azevedo, Pseudogap and antiferromagnetic correlations in the Hubbard model, Phys. Rev. Lett. 97, 036401 (2006).
  121. B. Kyung, S. S. Kancharla, D. Sénéchal, A.-M. S. Tremblay, M. Civelli, and G. Kotliar, Pseudogap induced by short-range spin correlations in a doped Mott insulator, Phys. Rev. B 73, 165114 (2006).
  122. M. Ferrero, P. S. Cornaglia, L. De Leo, O. Parcollet, G. Kotliar, and A. Georges, Pseudogap opening and formation of Fermi arcs as an orbital-selective Mott transition in momentum space, Phys. Rev. B 80, 064501 (2009).
  123. G. Sordi, P. Sémon, K. Haule, and A.-M. S. Tremblay, Strong coupling superconductivity, pseudogap, and Mott transition, Phys. Rev. Lett. 108, 216401 (2012).
  124. W. Wu, M. S. Scheurer, S. Chatterjee, S. Sachdev, A. Georges, and M. Ferrero, Pseudogap and Fermi-surface topology in the two-dimensional Hubbard model, Phys. Rev. X 8, 021048 (2018).
  125. I. Osborne, T. Paiva, and N. Trivedi, Broken Luttinger theorem in the two-dimensional Fermi-Hubbard model, Phys. Rev. B 104, 235122 (2021).
  126. H. Eskes, O. Y. Osman, R. Grimberg, W. van Saarloos, and J. Zaanen, Charged domain walls as quantum strings on a lattice, Phys. Rev. B 58, 6963 (1998).
  127. F. Krüger and S. Scheidl, Nonuniversal ordering of spin and charge in stripe phases, Phys. Rev. Lett. 89, 095701 (2002).
  128. F. D. M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis Néel state, Phys. Rev. Lett. 50, 1153 (1983).
  129. R. Peierls, On Ising’s model of ferromagnetism, Math. Proc. Camb. Philos. Soc. 32, 477 (1936).
  130. F. J. Wegner, Duality in generalized Ising models and phase transitions without local order parameters, J. Math. Phys. 12, 2259 (1971).
  131. S. M. Linsel, A. Bohrdt, L. Homeier, L. Pollet, and F. Grusdt, Percolation as a confinement order parameter in Z2 lattice gauge theories, Phys. Rev. B 110, L241101 (2024).
  132. H. V. Kruis, I. P. McCulloch, Z. Nussinov, and J. Zaanen, Geometry and the hidden order of Luttinger liquids: The universality of squeezed space, Phys. Rev. B 70, 075109 (2004).
  133. M. Ogata and H. Shiba, Bethe-ansatz wave function, momentum distribution, and spin correlation in the one-dimensional strongly correlated Hubbard model, Phys. Rev. B 41, 2326 (1990).
  134. T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll, E. Demler, I. Bloch, and C. Gross, Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators, Science 357, 484 (2017).
  135. F. Grusdt, Z. Zhu, T. Shi, and E. Demler, Meson formation in mixed-dimensional t-J models, SciPost Phys. 5, 057 (2018).
  136. F. Grusdt and L. Pollet, Z2 parton phases in the mixed-dimensional t-Jz model, Phys. Rev. Lett. 125, 256401 (2020).
  137. H. Schlömer, A. Bohrdt, L. Pollet, U. Schollwöck, and F. Grusdt, Robust stripes in the mixed-dimensional t-J model, Phys. Rev. Res. 5, L022027 (2023).
  138. H. Schlömer, T. A. Hilker, I. Bloch, U. Schollwöck, F. Grusdt, and A. Bohrdt, Quantifying hole-motion-induced frustration in doped antiferromagnets by Hamiltonian reconstruction, Commun. Mater. 4, 64 (2023).
  139. Y. Li, V. Balédent, N. Barišić, Y. Cho, B. Fauqué, Y. Sidis, G. Yu, X. Zhao, P. Bourges, and M. Greven, Unusual magnetic order in the pseudogap region of the superconductor HgBa2CuO4+δ, Nature 455, 372 (2008).
  140. A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (N. Y.) 303, 2 (2003).
  141. X.-G. Wen, Quantum Field Theory of Many-Body Systems: From The Origin of Sound to an Origin of Light and Electrons (Oxford University Press, Oxford, England, 2004).
  142. M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005).
  143. S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension, Phys. Rev. Lett. 98, 070602 (2007).
  144. F. Wu, Y. Deng, and N. Prokof’ev, Phase diagram of the toric code model in a parallel magnetic field, Phys. Rev. B 85, 195104 (2012).
  145. S. Sachdev, Where is the quantum critical point in the cuprate superconductors? Phys. Status Solidi B 247, 537 (2010).
  146. N. E. Hussey, Phenomenology of the normal state in-plane transport properties of high-Tc cuprates, J. Phys.: Condens. Matter 20, 123201 (2008).
  147. S. Sachdev and J. Ye, Universal quantum-critical dynamics of two-dimensional antiferromagnets, Phys. Rev. Lett. 69, 2411 (1992).
  148. F. Grusdt, A. Bohrdt, and E. Demler, Microscopic spinon-chargon theory of magnetic polarons in the t-J model, Phys. Rev. B 99, 224422 (2019).
  149. P. Anderson, Resonating valence bonds: A new kind of insulator? Mater. Res. Bull. 8, 153 (1973).
  150. S. R. White, R. M. Noack, and D. J. Scalapino, Resonating valence bond theory of coupled Heisenberg chains, Phys. Rev. Lett. 73, 886 (1994).
  151. F. Mezzacapo, Ground-state phase diagram of the quantum J1-J2 model on the square lattice, Phys. Rev. B 86, 045115 (2012).
  152. W.-J. Hu, F. Becca, A. Parola, and S. Sorella, Direct evidence for a gapless Z2 spin liquid by frustrating Néel antiferromagnetism, Phys. Rev. B 88, 060402 (2013).
  153. L. Wang, D. Poilblanc, Z.-C. Gu, X.-G. Wen, and F. Verstraete, Constructing a gapless spin-liquid state for the spin-1/2J1-J2 Heisenberg model on a square lattice, Phys. Rev. Lett. 111, 037202 (2013).
  154. S.-S. Gong, W. Zhu, D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Plaquette ordered phase and quantum phase diagram in the spin-12J1-J2 square Heisenberg model, Phys. Rev. Lett. 113, 027201 (2014).
  155. H.-C. Jiang, H. Yao, and L. Balents, Spin liquid ground state of the spin-12 square J1-J2 Heisenberg model, Phys. Rev. B 86, 024424 (2012).
  156. R. Haghshenas and D. N. Sheng, U(1)-symmetric infinite projected entangled-pair states study of the spin-1/2 square J1-J2 Heisenberg model, Phys. Rev. B 97, 174408 (2018).
  157. B. Michon, C. Girod, S. Badoux, J. Kačmarčik, Q. Ma, M. Dragomir, H. A. Dabkowska, B. D. Gaulin, J. S. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Verret, N. Doiron-Leyraud, C. Marcenat, L. Taillefer, and T. Klein, Thermodynamic signatures of quantum criticality in cuprate superconductors, Nature 567, 218 (2019).
  158. R. Arpaia, L. Martinelli, M. M. Sala, S. Caprara, A. Nag, N. B. Brookes, P. Camisa, Q. Li, Q. Gao, X. Zhou, M. Garcia-Fernandez, K.-J. Zhou, E. Schierle, T. Bauch, Y. Y. Peng, C. Di Castro, M. Grilli, F. Lombardi, L. Braicovich, and G. Ghiringhelli, Signature of quantum criticality in cuprates by charge density fluctuations, Nat. Commun. 14, 7198 (2023).
  159. N. P. Armitage, P. Fournier, and R. L. Greene, Progress and perspectives on electron-doped cuprates, Rev. Mod. Phys. 82, 2421 (2010).
  160. P. Gegenwart, Q. Si, and F. Steglich, Quantum criticality in heavy-fermion metals, Nat. Phys. 4, 186 (2008).
  161. G. R. Stewart, Superconductivity in iron compounds, Rev. Mod. Phys. 83, 1589 (2011).
  162. D. Jerome and C. Bourbonnais, Quasi one-dimensional organic conductors: From Fröhlich conductivity and Peierls insulating state to magnetically-mediated superconductivity, a retrospective, C. R. Phys. 25, 17 (2024).
  163. S. Paschen, T. Lühmann, S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman, and Q. Si, Hall-effect evolution across a heavy-fermion quantum critical point, Nature 432, 881 (2004).
  164. P. Monthoux, D. Pines, and G. G. Lonzarich, Superconductivity without phonons, Nature 450, 1177 (2007).
  165. L. Taillefer, Scattering and pairing in cuprate superconductors, Annu. Rev. Condens. Matter Phys. 1, 51 (2010).
  166. C. Proust and L. Taillefer, The remarkable underlying ground states of cuprate superconductors, Annu. Rev. Condens. Matter Phys. 10, 409 (2019).
  167. K. Miyake, S. Schmitt-Rink, and C. M. Varma, Spin-fluctuation-mediated even-parity pairing in heavy-fermion superconductors, Phys. Rev. B 34, 6554 (1986).
  168. D. J. Scalapino, E. Loh, and J. E. Hirsch, d-wave pairing near a spin-density-wave instability, Phys. Rev. B 34, 8190 (1986).
  169. W. P. Su, Spin polarons in the two-dimensional Hubbard model: A numerical study, Phys. Rev. B 37, 9904 (1988).
  170. L. Homeier, H. Lange, E. Demler, A. Bohrdt, and F. Grusdt, Feshbach hypothesis of high-Tc superconductivity in cuprates, Nat. Commun. 16, 314 (2025).
  171. B. Fauqué, Y. Sidis, V. Hinkov, S. Pailhès, C. T. Lin, X. Chaud, and P. Bourges, Magnetic order in the pseudogap phase of high-TC superconductors, Phys. Rev. Lett. 96, 197001 (2006).
  172. J. Xia, E. Schemm, G. Deutscher, S. A. Kivelson, D. A. Bonn, W. N. Hardy, R. Liang, W. Siemons, G. Koster, M. M. Fejer, and A. Kapitulnik, Polar Kerr-effect measurements of the high-temperature YBa2Cu3O6+x superconductor: Evidence for broken symmetry near the pseudogap temperature, Phys. Rev. Lett. 100, 127002 (2008).
  173. F. Laliberté, J. Chang, N. Doiron-Leyraud, E. Hassinger, R. Daou, M. Rondeau, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, S. Pyon, T. Takayama, H. Takagi, I. Sheikin, L. Malone, C. Proust, K. Behnia, and L. Taillefer, Fermi-surface reconstruction by stripe order in cuprate superconductors, Nat. Commun. 2, 432 (2011).
  174. M. Xu, L. H. Kendrick, A. Kale, Y. Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat, and M. Greiner, A neutral-atom Hubbard quantum simulator in the cryogenic regime, Nature 642, 909 (2025).
  175. R. H. Wilke, H. Schlömer, S. M. Linsel, A. Bohrdt, and F. Grusdt, Topology meets symmetry breaking: Hidden order, intrinsically gapless topological states and finite-temperature topological transitions, ArXiv:2506.03146.
  176. E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D 19, 3682 (1979).

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PRX Quantum 6, 030342– Published 4 September, 2025
  • Received 1 November 2024
  • Accepted 29 July 2025
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