Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
Anderson, P. W. et al. The physics behind high-temperature superconducting cuprates: the plain vanilla version of RVB. J. Phys. Condens. Matter 16, R755–R769 (2004).
Weng, Z.-Y. Superconducting ground state of a doped Mott insulator. New J. Phys. 13, 103039 (2011).
Zhang, F. C. & Rice, T. M. Effective Hamiltonian for the superconducting Cu oxides. Phys. Rev. B 37, 3759–3761 (1988).
Corboz, P., Rice, T. M. & Troyer, M. Competing states in the t–J model: uniform d-wave state versus stripe state. Phys. Rev. Lett. 113, 046402 (2014).
Huang, E. W. et al. Numerical evidence of fluctuating stripes in the normal state of high-Tc cuprate superconductors. Science 358, 1161–1164 (2017).
Jiang, H.-C. & Devereaux, T. P. Superconductivity in the doped Hubbard model and its interplay with next-nearest hopping t′. Science 365, 1424–1428 (2019).
Zheng, B.-X. et al. Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358, 1155–1160 (2017).
Qin, M. et al. Absence of superconductivity in the pure two-dimensional Hubbard model. Phys. Rev. X 10, 031016 (2020).
Zhao, J.-Y., Chen, S. A., Zhang, H.-K. & Weng, Z.-Y. Two-hole ground state: dichotomy in pairing symmetry. Phys. Rev. X 12, 011062 (2022).
Tranquada, J. M., Sternlieb, B. J., Axe, J. D., Nakamura, Y. & Uchida, S. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 375, 561–563 (1995).
Kivelson, S. A., Fradkin, E. & Emery, V. J. Electronic liquid-crystal phases of a doped Mott insulator. Nature 393, 550–553 (1998).
Kohsaka, Y. et al. An intrinsic bond-centered electronic glass with unidirectional domains in underdoped cuprates. Science 315, 1380–1385 (2007).
Hinkov, V. et al. Electronic liquid crystal state in the high-temperature superconductor YBa2Cu3O6.45. Science 319, 597–600 (2008).
Hanaguri, T. et al. A ‘checkerboard’ electronic crystal state in lightly hole-doped Ca2−xNaxCuO2Cl2. Nature 430, 1001–1005 (2004).
Hanaguri, T. et al. Quasiparticle interference and superconducting gap in Ca2−xNaxCuO2Cl2. Nat. Phys. 3, 865–871 (2007).
Kohsaka, Y. et al. Visualization of the emergence of the pseudogap state and the evolution to superconductivity in a lightly hole-doped Mott insulator. Nat. Phys. 8, 534–538 (2012).
Ye, C. et al. Visualizing the atomic-scale electronic structure of the Ca2CuO2Cl2 Mott insulator. Nat. Commun. 4, 1365 (2013).
Li, H., Ye, S., Zhao, J., Jin, C. & Wang, Y. Imaging the atomic-scale electronic states induced by a pair of hole dopants in Ca2CuO2Cl2 Mott insulator. Sci. Bull. 66, 1395–1400 (2021).
Ronning, F. et al. Photoemission evidence for a remnant Fermi surface and a d-wave-like dispersion in insulating Ca2CuO2Cl2. Science 282, 2067–2072 (1998).
Shen, K. M. et al. Missing quasiparticles and the chemical potential puzzle in the doping evolution of the cuprate superconductors. Phys. Rev. Lett. 93, 267002 (2004).
Shen, K. M. et al. Nodal quasiparticles and antinodal charge ordering in Ca2−xNaxCuO2Cl2. Science 307, 901–904 (2005).
Hu, C. et al. Evidence for multiple underlying Fermi surface and isotropic energy gap in the cuprate parent compound Ca2CuO2Cl2. Chin. Phys. Lett. 35, 067403 (2018).
Cai, P. et al. Visualizing the evolution from the Mott insulator to a charge-ordered insulator in lightly doped cuprates. Nat. Phys. 12, 1047–1051 (2016).
Ruan, W. et al. Relationship between the parent charge transfer gap and maximum transition temperature in cuprates. Sci. Bull. 61, 1826–1832 (2016).
Li, H. et al. Mott gap filling by doping electrons through depositing one sub-monolayer thin film of Rb on Ca2CuO2Cl2. Chin. Phys. Lett. 41, 057402 (2024).
Pan, S. H. et al. Imaging the effects of individual zinc impurity atoms on superconductivity in Bi2Sr2CaCu2O8+δ. Nature 403, 746–750 (2000).
Hudson, E. W. et al. Interplay of magnetism and high-Tc superconductivity at individual Ni impurity atoms in Bi2Sr2CaCu2O8+δ. Nature 411, 920–924 (2001).
McElroy, K., Lee, J., Slezak, J. A., Uchida, S. & Davis, J. C. Atomic-scale sources and mechanism of nanoscale electronic disorder in Bi2Sr2CaCu2O8+δ. Science 309, 1048–1052 (2005).
Zeljkovic, I. et al. Imaging the impact of single oxygen atoms on superconducting Bi2+ySr2−yCaCu2O8+x. Science 337, 320–323 (2012).
Erdenemunkh, U. et al. Suppression of superfluid density and the pseudogap state in the cuprates by impurities. Phys. Rev. Lett. 117, 257003 (2016).
Massee, F., Huang, Y. K., Golden, M. S. & Aprili, M. Noisy defects in the high-Tc superconductor Bi2Sr2CaCu2O8+x. Nat. Commun. 10, 544 (2019).
Mulliken, R. S. Spectroscopy, molecular orbitals, and chemical bonding. Science 157, 13–24 (1967).
Kimouche, A. et al. Ultra-narrow metallic armchair graphene nanoribbons. Nat. Commun. 6, 10177 (2015).
Repp, J., Meyer, G., Stojković, S. M., Gourdon, A. & Joachim, C. Molecules on insulating films: scanning-tunneling microscopy imaging of individual molecular orbitals. Phys. Rev. Lett. 94, 026803 (2005).
Sierda, E. et al. Quantum simulator to emulate lower-dimensional molecular structure. Science 380, 1048–1052 (2023).
Kohsaka, Y. et al. Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several GPa. J. Am. Chem. Soc. 124, 12275–12278 (2002).
Emery, V. J., Kivelson, S. A. & Zachar, O. Spin-gap proximity effect mechanism of high-temperature superconductivity. Phys. Rev. B 56, 6120–6147 (1997).
Ruan, W. et al. Visualization of the periodic modulation of Cooper pairing in a cuprate superconductor. Nat. Phys. 14, 1178–1182 (2018).
Li, X. et al. Evolution of charge and pair density modulations in overdoped Bi2Sr2CuO6+δ. Phys. Rev. X 11, 011007 (2021).
Sakurai, Y. et al. Imaging doped holes in a cuprate superconductor with high-resolution Compton scattering. Science 332, 698–702 (2011).
Li, P. et al. Bound states in doped charge transfer insulators. Preprint at https://arxiv.org/abs/2408.00576 (2024).
Xia, N., Guo, Y. & Yang, S. Unveiling stripe-shaped charge density modulations in doped Mott insulators. Phys. Rev. Lett. 135, 116504 (2025).
Ye, S. et al. The emergence of global phase coherence from local pairing in underdoped cuprates. Nat. Phys. 19, 1301–1307 (2023).
Wang, Z., Engelbrecht, J. R., Wang, S., Ding, H. & Pan, S. H. Inhomogeneous d-wave superconducting state of a doped Mott insulator. Phys. Rev. B 65, 064509 (2002).
Hamidian, M. H. et al. Atomic-scale electronic structure of the cuprate d-symmetry form factor density wave state. Nat. Phys. 12, 150–156 (2016).
Da Silva Neto, E. H. et al. Ubiquitous interplay between charge ordering and high-temperature superconductivity in cuprates. Science 343, 393–396 (2014).
Comin, R. et al. Symmetry of charge order in cuprates. Nat. Mater. 14, 796–800 (2015).
McMahon, C. et al. Orbital symmetries of charge density wave order in YBa2Cu3O6+x. Sci. Adv. 6, eaay0345 (2020).
Fujita, K. et al. Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates. Proc. Natl Acad. Sci. USA 111, E3026–E3032 (2014).
Jiang, H.-C. & Kivelson, S. A. Stripe order enhanced superconductivity in the Hubbard model. Proc. Natl Acad. Sci. USA 119, e2109406119 (2022).
Peng, Y. et al. Disappearance of nodal gap across the insulator–superconductor transition in a copper-oxide superconductor. Nat. Commun. 4, 2459 (2013).
Fan, J.-Q. et al. Direct observation of nodeless superconductivity and phonon modes in electron-doped copper oxide Sr1−xNdxCuO2. Natl Sci. Rev. 9, nwab225 (2022).
Xu, H. et al. Coexistence of superconductivity with partially filled stripes in the Hubbard model. Science 384, eadh7691 (2024).
Lawler, M. J. et al. Intra-unit-cell electronic nematicity of the high-Tc copper-oxide pseudogap states. Nature 466, 347–351 (2010).
Choubey, P., Kreisel, A., Berlijn, T., Andersen, B. M. & Hirschfeld, P. J. Universality of scanning tunneling microscopy in cuprate superconductors. Phys. Rev. B 96, 174523 (2017).