Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).

Article 
ADS 
CAS 

Google Scholar
 

Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

Arovas, D. P., Berg, E., Kivelson, S. A. & Raghu, S. The Hubbard model. Annu. Rev. Condens. Matter Phys. 13, 239–274 (2022).

Article 
ADS 

Google Scholar
 

Norman, M. R., Pines, D. & Kallin, C. The pseudogap: friend or foe of high Tc? Adv. Phys. 54, 715–733 (2005).

Article 
ADS 
CAS 

Google Scholar
 

Proust, C. & Taillefer, L. The remarkable underlying ground states of cuprate superconductors. Annu. Rev. Condens. Matter Phys. 10, 409–429 (2019).

Article 
ADS 
CAS 

Google Scholar
 

Xu, M. et al. A neutral-atom Hubbard quantum simulator in the cryogenic regime. Nature 642, 909–915 (2025).

Article 
ADS 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

Basov, D. N. & Timusk, T. Electrodynamics of high-Tc superconductors. Rev. Mod. Phys. 77, 721–779 (2005).

Article 
ADS 
CAS 

Google Scholar
 

Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 
CAS 

Google Scholar
 

Zhang, Y.-H. & Sachdev, S. From the pseudogap metal to the Fermi liquid using ancilla qubits. Phys. Rev. Res. 2, 023172 (2020).

Article 
CAS 

Google Scholar
 

Šimkovic, F. IV, Rossi, R., Georges, A. & Ferrero, M. Origin and fate of the pseudogap in the doped Hubbard model. Science 385, eade9194 (2024).

Article 
ADS 
MathSciNet 
PubMed 

Google Scholar
 

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).


Google Scholar
 

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).

CAS 

Google Scholar
 

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).

Article 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

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).

CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Tarruell, L. & Sanchez-Palencia, L. Quantum simulation of the Hubbard model with ultracold fermions in optical lattices. C. R. Phys. 19, 365–393 (2018).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Cocchi, E. et al. Equation of state of the two-dimensional Hubbard model. Phys. Rev. Lett. 116, 175301 (2016).

Article 
ADS 
PubMed 

Google Scholar
 

Pasqualetti, G. et al. Equation of state and thermometry of the 2D SU(N) Fermi-Hubbard model. Phys. Rev. Lett. 132, 083401 (2024).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Gross, C. & Bakr, W. S. Quantum gas microscopy for single atom and spin detection. Nat. Phys. 17, 1316–1323 (2021).

Article 
CAS 

Google Scholar
 

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).

Article 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

Devereaux, T. P. & Hackl, R. Inelastic light scattering from correlated electrons. Rev. Mod. Phys. 79, 175–233 (2007).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

Khatami, E. et al. Quantum criticality due to incipient phase separation in the two-dimensional Hubbard model. Phys. Rev. B 81, 201101(R) (2010).

Article 
ADS 

Google Scholar
 

Sinha, A. & Wietek, A. Forestalled phase separation as the precursor to stripe order. Nat. Commun. 16, 10807 (2025).

Article 
ADS 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

Luick, N. et al. An ideal Josephson junction in an ultracold two-dimensional Fermi gas. Science 369, 89–91 (2020).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 
MathSciNet 
CAS 

Google Scholar
 

Hayden, S. M. & Tranquada, J. M. Charge correlations in cuprate superconductors. Annu. Rev. Condens. Matter Phys. 15, 215–235 (2024).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
MathSciNet 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

Huang, E. W. et al. Fluctuating intertwined stripes in the strange metal regime of the Hubbard model. Phys. Rev. B 107, 085126 (2023).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 

Google Scholar
 

Sacuto, A. et al. Pseudogap in cuprates by electronic Raman scattering. J. Phys. Conf. Ser. 449, 012011 (2013).

Article 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 

Google Scholar
 

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).

Article 
ADS 
CAS 

Google Scholar
 

Alloul, H., Ohno, T. & Mendels, P. 89Y NMR evidence for a Fermi-liquid behavior in YBa2Cu3O6+x. Phys. Rev. Lett. 63, 1700–1703 (1989).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Emery, V. J. & Kivelson, S. A. Frustrated electronic phase separation and high-temperature superconductors. Physica C Supercond. 209, 597–621 (1993).

Article 
ADS 
CAS 

Google Scholar
 

Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium: theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).

Article 
ADS 
CAS 

Google Scholar
 

Zhou, R. et al. Signatures of two gaps in the spin susceptibility of a cuprate superconductor. Nat. Phys. 21, 97–103 (2025).

Article 
CAS 

Google Scholar
 

Loret, B. et al. Intimate link between charge density wave, pseudogap and superconducting energy scales in cuprates. Nat. Phys. 15, 771–775 (2019).

Article 
CAS 

Google Scholar
 

Mukhopadhyay, S. et al. Evidence for a vestigial nematic state in the cuprate pseudogap phase. Proc. Natl Acad. Sci. USA 116, 13249–13254 (2019).

Article 
ADS 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Xu, M. et al. Frustration- and doping-induced magnetism in a Fermi-Hubbard simulator. Nature 620, 971–976 (2023).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

Miles, C. et al. Correlator convolutional neural networks as an interpretable architecture for image-like quantum matter data. Nat. Commun. 12, 3905 (2021).

Article 
ADS 
PubMed 
PubMed Central 
CAS 

Google Scholar
 

Greif, D. et al. Site-resolved imaging of a fermionic Mott insulator. Science 351, 953–957 (2016).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

Ferrero, M. et al. Valence bond dynamical mean-field theory of doped Mott insulators with nodal/antinodal differentiation. Europhys. Lett. 85, 57009 (2009).

Article 
ADS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

Parsons, M. F. et al. Site-resolved imaging of fermionic 6Li in an optical lattice. Phys. Rev. Lett. 114, 213002 (2015).

Article 
ADS 
PubMed 

Google Scholar
 

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).

Article 
ADS 
PubMed 

Google Scholar
 

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).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
ADS 
MathSciNet 
CAS 

Google Scholar
 

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).

Article 
ADS 
PubMed 
CAS 

Google Scholar
 

Brown, P. T. et al. Angle-resolved photoemission spectroscopy of a Fermi-Hubbard system. Nat. Phys. 16, 26–31 (2020).

Article 
CAS 

Google Scholar
 

Troyer, M. & Wiese, U.-J. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005).

Article 
ADS 
PubMed 

Google Scholar
 

Mondaini, R., Tarat, S. & Scalettar, R. T. Quantum critical points and the sign problem. Science 375, 418–424 (2022).

Article 
ADS 
MathSciNet 
PubMed 
CAS 

Google Scholar
 

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).

Article 
ADS 

Google Scholar
 

Varney, C. N. et al. Quantum Monte Carlo study of the two-dimensional fermion Hubbard model. Phys. Rev. B 80, 075116 (2009).

Article 
ADS 

Google Scholar
 

Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).

Article 
ADS 
CAS 

Google Scholar
 

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).

Article 
MathSciNet 

Google Scholar
 

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).