Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).
Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
Qin, M., Schäfer, T., Andergassen, S., Corboz, P. & Gull, E. The Hubbard model: a computational perspective. Annu. Rev. Condens. Matter Phys. 13, 275–302 (2022).
Arovas, D. P., Berg, E., Kivelson, S. A. & Raghu, S. The Hubbard model. Annu. Rev. Condens. Matter Phys. 13, 239–274 (2022).
Norman, M. R., Pines, D. & Kallin, C. The pseudogap: friend or foe of high Tc? Adv. Phys. 54, 715–733 (2005).
Proust, C. & Taillefer, L. The remarkable underlying ground states of cuprate superconductors. Annu. Rev. Condens. Matter Phys. 10, 409–429 (2019).
Xu, M. et al. A neutral-atom Hubbard quantum simulator in the cryogenic regime. Nature 642, 909–915 (2025).
Basov, D. N. & Timusk, T. Electrodynamics of high-Tc superconductors. Rev. Mod. Phys. 77, 721–779 (2005).
Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179–186 (2015).
Pines, D. & Nozières, P. The Theory of Quantum Liquids, Volume I: Normal Fermi Liquids (CRC Press, 1966).
Tinkham, M. Introduction to Superconductivity (McGraw-Hill, 1996).
Monthoux, P. & Pines, D. YBa2Cu3O7: a nearly antiferromagnetic Fermi liquid. Phys. Rev. B 47, 6069–6081 (1993).
Zhang, Y.-H. & Sachdev, S. From the pseudogap metal to the Fermi liquid using ancilla qubits. Phys. Rev. Res. 2, 023172 (2020).
Šimkovic, F. IV, Rossi, R., Georges, A. & Ferrero, M. Origin and fate of the pseudogap in the doped Hubbard model. Science 385, eade9194 (2024).
Schäfer, T. et al. Tracking the footprints of spin fluctuations: a multimethod, multimessenger study of the two-dimensional Hubbard model. Phys. Rev. X 11, 011058 (2021).
Vilk, Y. M. & Tremblay, A.-M. S. Non-perturbative many-body approach to the Hubbard model and single-particle pseudogap. J. Phys. I 7, 1309–1368 (1997).
Šimkovic, F. IV, Rossi, R. & Ferrero, M. Two-dimensional Hubbard model at finite temperature: weak, strong, and long correlation regimes. Phys. Rev. Res. 4, 043201 (2022).
Xu, H., Shi, H., Vitali, E., Qin, M. & Zhang, S. Stripes and spin-density waves in the doped two-dimensional Hubbard model: ground state phase diagram. Phys. Rev. Res. 4, 013239 (2022).
Xiao, B., He, Y.-Y., Georges, A. & Zhang, S. Temperature dependence of spin and charge orders in the doped two-dimensional Hubbard model. Phys. Rev. X 13, 011007 (2023).
Hofstetter, W., Cirac, J. I., Zoller, P., Demler, E. & Lukin, M. D. High-temperature superfluidity of fermionic atoms in optical lattices. Phys. Rev. Lett. 89, 220407 (2002).
Tarruell, L. & Sanchez-Palencia, L. Quantum simulation of the Hubbard model with ultracold fermions in optical lattices. C. R. Phys. 19, 365–393 (2018).
Ku, M. J. H., Sommer, A. T., Cheuk, L. W. & Zwierlein, M. W. Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas. Science 335, 563–567 (2012).
Cocchi, E. et al. Equation of state of the two-dimensional Hubbard model. Phys. Rev. Lett. 116, 175301 (2016).
Pasqualetti, G. et al. Equation of state and thermometry of the 2D SU(N) Fermi-Hubbard model. Phys. Rev. Lett. 132, 083401 (2024).
Gross, C. & Bakr, W. S. Quantum gas microscopy for single atom and spin detection. Nat. Phys. 17, 1316–1323 (2021).
Chalopin, T. et al. Observation of emergent scaling of spin-charge correlations at the onset of the pseudogap. Proc. Natl Acad. Sci. USA 123, e2525539123 (2026).
Devereaux, T. P. & Hackl, R. Inelastic light scattering from correlated electrons. Rev. Mod. Phys. 79, 175–233 (2007).
Sordi, G., Sémon, P., Haule, K. & Tremblay, A.-M. S. Pseudogap temperature as a Widom line in doped Mott insulators. Sci. Rep. 2, 547 (2012).
Sordi, G., Haule, K. & Tremblay, A.-M. S. Mott physics and first-order transition between two metals in the normal-state phase diagram of the two-dimensional Hubbard model. Phys. Rev. B 84, 075161 (2011).
Khatami, E. et al. Quantum criticality due to incipient phase separation in the two-dimensional Hubbard model. Phys. Rev. B 81, 201101(R) (2010).
Sinha, A. & Wietek, A. Forestalled phase separation as the precursor to stripe order. Nat. Commun. 16, 10807 (2025).
Luick, N. et al. An ideal Josephson junction in an ultracold two-dimensional Fermi gas. Science 369, 89–91 (2020).
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, 13–125 (1996).
Hayden, S. M. & Tranquada, J. M. Charge correlations in cuprate superconductors. Annu. Rev. Condens. Matter Phys. 15, 215–235 (2024).
Mai, P., Karakuzu, S., Balduzzi, G., Johnston, S. & Maier, T. A. Intertwined spin, charge, and pair correlations in the two-dimensional Hubbard model in the thermodynamic limit. Proc. Natl Acad. Sci. USA 119, e2112806119 (2022).
Huang, E. W. et al. Fluctuating intertwined stripes in the strange metal regime of the Hubbard model. Phys. Rev. B 107, 085126 (2023).
Vilk, Y. M. & Tremblay, A.-M.S. Pseudogap, Fermi liquid, Van Hove singularity and maxima of the compressibility and of the Knight shift as a function of doping in the two-dimensional Hubbard model. Preprint at arxiv.org/abs/2602.06298 (2026).
Jördens, R., Strohmaier, N., Günter, K., Moritz, H. & Esslinger, T. A Mott insulator of fermionic atoms in an optical lattice. Nature 455, 204–207 (2008).
Sacuto, A. et al. Pseudogap in cuprates by electronic Raman scattering. J. Phys. Conf. Ser. 449, 012011 (2013).
Sakai, S. et al. Raman-scattering measurements and theory of the energy-momentum spectrum for underdoped Bi2Sr2CaCuO8+δ superconductors: evidence of an s-wave structure for the pseudogap. Phys. Rev. Lett. 111, 107001 (2013).
Sacuto, A. et al. New insights into the phase diagram of the copper oxide superconductors from electronic Raman scattering. Rep. Prog. Phys. 76, 022502 (2013).
Lin, N., Gull, E. & Millis, A. J. Two-particle response in cluster dynamical mean-field theory: formalism and application to the Raman response of high-temperature superconductors. Phys. Rev. Lett. 109, 106401 (2012).
Bohrdt, A., Demler, E. & Grusdt, F. Spectroscopy of Hubbard-Mott excitons and their ro-vibrational excitations. Preprint at arxiv.org/abs/2406.16854 (2024).
Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 2011).
Kastner, M. A., Birgeneau, R. J., Shirane, G. & Endoh, Y. Magnetic, transport, and optical properties of monolayer copper oxides. Rev. Mod. Phys. 70, 897–928 (1998).
Alloul, H., Ohno, T. & Mendels, P. 89Y NMR evidence for a Fermi-liquid behavior in YBa2Cu3O6+x. Phys. Rev. Lett. 63, 1700–1703 (1989).
Emery, V. J. & Kivelson, S. A. Frustrated electronic phase separation and high-temperature superconductors. Physica C Supercond. 209, 597–621 (1993).
Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium: theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).
Zhou, R. et al. Signatures of two gaps in the spin susceptibility of a cuprate superconductor. Nat. Phys. 21, 97–103 (2025).
Loret, B. et al. Intimate link between charge density wave, pseudogap and superconducting energy scales in cuprates. Nat. Phys. 15, 771–775 (2019).
Mukhopadhyay, S. et al. Evidence for a vestigial nematic state in the cuprate pseudogap phase. Proc. Natl Acad. Sci. USA 116, 13249–13254 (2019).
Parker, C. V. et al. Fluctuating stripes at the onset of the pseudogap in the high-Tc superconductor Bi2Sr2CaCu2O8+x. Nature 468, 677–680 (2010).
McElroy, K. et al. Coincidence of checkerboard charge order and antinodal state decoherence in strongly underdoped superconducting Bi2Sr2CaCu2O8+δ. Phys. Rev. Lett. 94, 197005 (2005).
Xu, M. et al. Frustration- and doping-induced magnetism in a Fermi-Hubbard simulator. Nature 620, 971–976 (2023).
Mongkolkiattichai, J., Liu, L., Garwood, D., Yang, J. & Schauss, P. Quantum gas microscopy of fermionic triangular-lattice Mott insulators. Phys. Rev. A 108, L061301 (2023).
Downey, P.-O., Gingras, O., Hébert, C.-D., Charlebois, M. & Tremblay, A.-M. S. Doping the Mott insulating state of the triangular-lattice Hubbard model reveals the Sordi transition. Phys. Rev. B 110, L121109 (2024).
Ibarra-García-Padilla, E., Striegel, S., Scalettar, R. T. & Khatami, E. Structural complexity of snapshots of two-dimensional Fermi-Hubbard systems. Phys. Rev. A 109, 053304 (2024).
Miles, C. et al. Correlator convolutional neural networks as an interpretable architecture for image-like quantum matter data. Nat. Commun. 12, 3905 (2021).
Greif, D. et al. Site-resolved imaging of a fermionic Mott insulator. Science 351, 953–957 (2016).
Ferrero, M. et al. Pseudogap opening and formation of Fermi arcs as an orbital-selective Mott transition in momentum space. Phys. Rev. B 80, 064501 (2009).
Ferrero, M. et al. Valence bond dynamical mean-field theory of doped Mott insulators with nodal/antinodal differentiation. Europhys. Lett. 85, 57009 (2009).
Gull, E., Ferrero, M., Parcollet, O., Georges, A. & Millis, A. J. Momentum-space anisotropy and pseudogaps: a comparative cluster dynamical mean-field analysis of the doping-driven metal-insulator transition in the two-dimensional Hubbard model. Phys. Rev. B 82, 155101 (2010).
Gull, E. & Millis, A. J. Superconducting and pseudogap effects on the interplane conductivity and Raman scattering cross section in the two-dimensional Hubbard model. Phys. Rev. B 88, 075127 (2013).
Bruus, H. & Flensberg, K. Many-Body Quantum Theory in Condensed Matter Physics: An Introduction (Oxford Univ. Press, 2004).
Mitra, D. et al. Quantum gas microscopy of an attractive Fermi-Hubbard system. Nat. Phys. 14, 173–177 (2018).
Parsons, M. F. et al. Site-resolved imaging of fermionic 6Li in an optical lattice. Phys. Rev. Lett. 114, 213002 (2015).
Greif, D. G. Quantum Magnetism with Ultracold Fermions in an Optical Lattice. PhD thesis, ETH Zurich (2013).
Zürn, G. et al. Precise characterization of 6Li Feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules. Phys. Rev. Lett. 110, 135301 (2013).
Jiang, S., Scalapino, D. J. & White, S. R. Density matrix renormalization group based downfolding of the three-band Hubbard model: importance of density-assisted hopping. Phys. Rev. B 108, L161111 (2023).
Adlong, H. S., Levinsen, J. & Parish, M. M. Microscopic theory of the Hubbard interaction in low-dimensional optical lattices. Phys. Rev. A 111, 033307 (2025).
Bojović, P. et al. High-fidelity collisional quantum gates with fermionic atoms. Nature 652, 602–608 (2026).
Lenihan, C., Kim, A. J., Šimkovic, F. IV & Kozik, E. Entropy in the non-Fermi-liquid regime of the doped 2D Hubbard model. Phys. Rev. Lett. 126, 105701 (2021).
Brown, P. T. et al. Angle-resolved photoemission spectroscopy of a Fermi-Hubbard system. Nat. Phys. 16, 26–31 (2020).
Troyer, M. & Wiese, U.-J. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005).
Mondaini, R., Tarat, S. & Scalettar, R. T. Quantum critical points and the sign problem. Science 375, 418–424 (2022).
Iglovikov, V. I., Khatami, E. & Scalettar, R. T. Geometry dependence of the sign problem in quantum Monte Carlo simulations. Phys. Rev. B 92, 045110 (2015).
Varney, C. N. et al. Quantum Monte Carlo study of the two-dimensional fermion Hubbard model. Phys. Rev. B 80, 075116 (2009).
Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).
Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl Bur. Stand. 45, 255–282 (1950).
Prelovšek, P. in The Physics of Correlated Insulators, Metals, and Superconductors (eds Pavarini, E. et al.) Ch. 7 (Forschungszentrum Jülich, 2017).
Kendrick, L. et al. Pseudogap in a Fermi-Hubbard quantum simulator. Zenodo https://doi.org/10.5281/zenodo.21053807 (2026).

