The pseudogap -- a partial depletion of the electronic density of states at the Fermi level that opens in the normal state of underdoped cuprate superconductors well above the superconducting transition temperature -- is widely regarded as the central obstacle to understanding the normal state from which high-temperature superconductivity emerges. Angle-resolved photoemission on underdoped Bi-based cuprates showed that the pseudogap at the Fermi-surface hot spots is non-monotonic in temperature: it decreases with increasing temperature up to a characteristic temperature T∗T^*, reaches a sharp minimum there, and increases again above T∗T^*, remaining open at room temperature. This behavior rules out a description by any single mechanism such as superconducting phase fluctuations, and it is mirrored -- in magnitude, momentum anisotropy, and temperature evolution -- by the gap of the incommensurate charge-density-wave state of the transition-metal dichalcogenide 2H-TaSe2, suggesting that the pseudogap is a general phenomenon of two-dimensional metals, governed by incommensurate charge ordering. The problem is to identify the mechanism that produces this behavior and to determine whether the cuprate-dichalcogenide correspondence reflects one common mechanism or analogous but distinct physics. Later work discovered incommensurate charge order in cuprates but separated it from the pseudogap critical point; the mechanism remains unidentified.

Background

High-transition-temperature (high-TcT_c) superconductivity arises in layered copper-oxide compounds (cuprates) upon doping an antiferromagnetic Mott insulator -- a material that band theory would predict to be metallic, but in which strong electron-electron repulsion localizes the electrons and renders it insulating; the metallic 'normal' state out of which superconductivity develops is strongly anomalous and is widely regarded as the main obstacle to a theory of high-TcT_c superconductivity. Its most prominent anomaly is the pseudogap: a partial depletion of the electronic density of states at the Fermi level that opens above the superconducting transition temperature TcT_c throughout the underdoped region of the doping-temperature phase diagram (Ding et al. 1996; Timusk and Statt 1999). In angle-resolved photoemission spectroscopy (ARPES), which measures the probability of removing an electron at a given energy and momentum, the pseudogap appears as a shift of the leading edge of the energy-distribution curve away from the Fermi level; the leading-edge midpoint (LEM) of momentum-integrated spectra serves as an empirical measure of a gap of unknown origin. The pseudogap is strongest at the 'hot spots' -- Fermi-surface points connected by the antiferromagnetic wavevector (π,π)(\pi,\pi) -- and weakest along the nodal direction, truncating the Fermi surface into arcs.

The mechanism of the pseudogap has remained unsettled since its discovery (Norman, Pines and Kallin 2005). The main candidate classes are: (i) precursor superconductivity, in which Cooper pairs form well above TcT_c but lack phase coherence, as emphasized by Emery and Kivelson (1995) on the basis of the small superfluid density of underdoped cuprates; (ii) an order competing with superconductivity, such as a charge- or spin-density wave; and (iii) correlations inherited from the parent Mott insulator, such as short-range antiferromagnetic correlations. Momentum-resolved experiments in the 2000s established a 'two-gap' structure -- the pseudogap and the superconducting gap have different momentum dependences and coexist below TcT_c -- which excludes identifying the pseudogap with a fluctuation-smeared superconducting gap, but does not by itself select among the remaining scenarios.

In this setting, Kordyuk et al. (2009) measured the temperature dependence of the hot-spot pseudogap in underdoped Bi(Pb)2Sr2Ca(Tb)Cu2O8+delta (hole concentration x = 0.11, TcT_c = 77 K). They found the gap to be non-monotonic: it decreases with increasing temperature up to T∗≈170T^* \approx 170 K, where the slope changes sharply and the quasiparticles are most coherent, and it increases again above T∗T^*, remaining clearly open at room temperature -- a behavior they argued rules out any single-mechanism description such as superconducting phase fluctuations, and one that had not been observed in cuprates before. Remarkably, the same non-monotonic evolution -- sharp slope change, linear decrease, non-closure at the transition, and re-increase above it -- characterizes the incommensurate charge-ordering gap of the transition-metal dichalcogenide (TMD) 2H-TaSe2 measured by the same group (Borisenko et al. 2008). A charge-density wave (CDW) is a periodic modulation of the electronic charge density, commensurate or incommensurate with the underlying lattice, that partially gaps those parts of the Fermi surface connected by its wavevector; 2H-TaSe2 enters an incommensurate CDW state at TNICT_{\mathrm{NIC}} = 122 K and a commensurate 3x3 CDW state at TICCT_{\mathrm{ICC}} = 90 K. Beyond the temperature evolution, the momentum anisotropy of the gap around the Fermi surface in the cuprate pseudogap state and in the incommensurate CDW state of TaSe2 are almost identical, and the one-particle spectra agree in character: a partial depletion of spectral weight at the Fermi level with no band backfolding. Kordyuk et al. concluded that the governing contribution to the cuprate pseudogap comes from incommensurate charge ordering, proposed that the pseudogap is a general complex phenomenon of two-dimensional (2D) metals rather than a peculiarity of cuprates, and left the mechanism open -- offering two candidate explanations for the persistence of a gap above T∗T^*: fluctuating incommensurate charge ordering, or momentum-dependent scattering tied to Fermi-surface nesting.

Later developments transformed the context of this problem without resolving it. Resonant X-ray scattering -- X-ray scattering tuned to the absorption edge of a chosen atom, which detects modulations of that atom's valence charge even when short-range -- discovered in 2012 that underdoped cuprates do host short-range incommensurate charge order that competes with superconductivity (Ghiringhelli et al. 2012; Chang et al. 2012; reviewed by Comin and Damascelli 2016) -- but its onset lies well below T∗T^* and its correlation lengths reach only a few nanometers. High-field transport in YBa2Cu3Oy showed that the Fermi-surface reconstruction -- the breaking-up of the large hole Fermi surface into small pockets by a symmetry-breaking order -- caused by this charge order ends at hole doping p≈0.16p \approx 0.16, whereas the pseudogap ends at a distinct critical point p∗≈0.19p^* \approx 0.19 at which the carrier density abruptly changes -- evidence that the pseudogap and charge order are separate phenomena, tied instead to the proximity of the Mott insulator (Badoux et al. 2016; Fang et al. 2022). The origin of the pseudogap remains among the central open questions of the field (Keimer et al. 2015), with short-range antiferromagnetic correlations (Kyung et al. 2006; Boschini et al. 2020; Zhu et al. 2023), preformed pairs, and intertwined multi-order pictures (Kordyuk 2015) all actively pursued. On the TMD side, pseudogap phenomenology is confirmed to be widespread, but the origin of CDWs in TMDs is itself material-dependent and debated (Zhu et al. 2017), and the normal state of TiSe2 was recently established as a phonon-induced pseudogap phase driven by strong electron-phonon coupling (Fragkos et al. 2026) -- a mechanism qualitatively different from the correlation-driven physics of the cuprates.

Problem Statement

Identify the physical mechanism(s) responsible for the non-monotonic temperature dependence of the normal-state pseudogap in underdoped cuprates and for the one-particle spectral character of the cuprate pseudogap state and of the incommensurate charge-ordered state of transition-metal dichalcogenides, and then determine whether the spectral correspondence between the two states shares a single common underlying mechanism, as proposed for incommensurate charge ordering, or arises from distinct mechanisms that produce analogous phenomenology in quasi-two-dimensional metals.

A resolution must account for all of the following, as parts of one coherent mechanistic picture:

  1. The temperature evolution of the hot-spot pseudogap in the normal state: its decrease with increasing temperature up to the pseudogap temperature T∗T^*, the sharp minimum at T∗T^*, and its re-increase above T∗T^*, with the gap persisting far above T∗T^* (observed up to room temperature).

  2. The momentum anisotropy of the pseudogap around the Fermi surface and the character of the one-particle spectra in the pseudogap state -- partial depletion of spectral weight at the Fermi level without band backfolding -- in both the cuprates and the incommensurate charge-density-wave state of dichalcogenides such as 2H-TaSe2.

  3. Whether the cuprate-dichalcogenide correspondence in energy, momentum, and temperature reflects a common microscopic origin or distinct mechanisms that produce analogous phenomenology, and, in the latter case, why the mechanisms differ while the spectra nevertheless correspond.

  4. Consistency with the established experimental record, including the two-gap momentum structure of the superconducting state, the onset of short-range incommensurate charge order below T∗T^* in the cuprates, and the distinction between the pseudogap critical point p∗p^* and the end of charge-order-induced Fermi-surface reconstruction.

A resolution must additionally name the microscopic mechanism(s) concretely and state at least one observation that would distinguish the proposed account from the competing candidate mechanisms identified in the Background, so that the account is testable rather than a restatement of the phenomenology.

The problem is posed at the level of mechanism identification and is not restricted to particular compounds, dopings, or experimental probes.

References

  1. A. A. Kordyuk, S. V. Borisenko, V. B. Zabolotnyy, R. Schuster, D. S. Inosov, D. V. Evtushinsky, A. I. Plyushchay, R. Follath, A. Varykhalov, L. Patthey, H. Berger, 'Nonmonotonic pseudogap in high-Tc cuprates', Physical Review B 79, 020504(R) (2009). DOI: 10.1103/PhysRevB.79.020504; preprint (title 'Non-monotonic pseudo-gap in high-Tc cuprates'): arXiv:0801.2546, https://arxiv.org/abs/0801.2546
  2. S. V. Borisenko, A. A. Kordyuk, A. N. Yaresko, V. B. Zabolotnyy, D. S. Inosov, R. Schuster, B. Buechner, R. Weber, R. Follath, L. Patthey, H. Berger, 'Pseudogap and charge density waves in two dimensions', Physical Review Letters 100, 196402 (2008). DOI: 10.1103/PhysRevLett.100.196402; arXiv:0704.1544, https://arxiv.org/abs/0704.1544
  3. H. Ding, T. Yokoya, J. C. Campuzano, T. Takahashi, M. Randeria, M. R. Norman, T. Mochiku, K. Kadowaki, J. Giapintzakis, 'Spectroscopic evidence for a pseudogap in the normal state of underdoped high-Tc superconductors', Nature 382, 51-54 (1996). DOI: 10.1038/382051a0, https://doi.org/10.1038/382051a0
  4. Tom Timusk, Bryan Statt, 'The pseudogap in high-temperature superconductors: an experimental survey', Reports on Progress in Physics 62, 61-122 (1999). DOI: 10.1080/0034-4885/62/1/002, https://doi.org/10.1080/0034-4885/62/1/002
  5. M. R. Norman, D. Pines, C. Kallin, 'The pseudogap: friend or foe of high Tc?', Advances in Physics 54, 715-733 (2005). DOI: 10.1080/00018730500459906, https://doi.org/10.1080/00018730500459906
  6. V. J. Emery, S. A. Kivelson, 'Importance of phase fluctuations in superconductors with small superfluid density', Nature 374, 434-437 (1995). DOI: 10.1038/374434a0, https://doi.org/10.1038/374434a0
  7. A. A. Kordyuk, 'Pseudogap from ARPES experiment: three gaps in cuprates and topological superconductivity', Low Temperature Physics 41, 319-341 (2015). DOI: 10.1063/1.4919371; arXiv:1501.04154, https://arxiv.org/abs/1501.04154
  8. G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. Moretti Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, L. Braicovich, 'Long-range incommensurate charge fluctuations in (Y,Nd)Ba2Cu3O6+x', Science 337, 821-825 (2012). DOI: 10.1126/science.1223532, https://doi.org/10.1126/science.1223532
  9. J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, Ruixing Liang, D. A. Bonn, W. N. Hardy, A. Watenphul, M. v. Zimmermann, E. M. Forgan, S. M. Hayden, 'Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67', Nature Physics 8, 871-876 (2012). DOI: 10.1038/nphys2456, https://doi.org/10.1038/nphys2456
  10. Riccardo Comin, Andrea Damascelli, 'Resonant X-ray scattering studies of charge order in cuprates', Annual Review of Condensed Matter Physics 7, 369-405 (2016). DOI: 10.1146/annurev-conmatphys-031115-011401, https://doi.org/10.1146/annurev-conmatphys-031115-011401
  11. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, J. Zaanen, 'From quantum matter to high-temperature superconductivity in copper oxides', Nature 518, 179-186 (2015). DOI: 10.1038/nature14165, https://doi.org/10.1038/nature14165
  12. S. Badoux, W. Tabis, F. Laliberte, G. Grissonnanche, B. Vignolle, D. Vignolles, J. Beard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, Louis Taillefer, Cyril Proust, 'Change of carrier density at the pseudogap critical point of a cuprate superconductor', Nature 531, 210-214 (2016). DOI: 10.1038/nature16983, https://doi.org/10.1038/nature16983
  13. B. Kyung, S. S. Kancharla, D. Senechal, A.-M. S. Tremblay, M. Civelli, G. Kotliar, 'Pseudogap induced by short-range spin correlations in a doped Mott insulator', Physical Review B 73, 165114 (2006). DOI: 10.1103/PhysRevB.73.165114, https://doi.org/10.1103/PhysRevB.73.165114
  14. Fabio Boschini, Marta Zonno, Elia Razzoli, Ryan P. Day, Matteo Michiardi, Berend Zwartsenberg, Pascal Nigge, Michael Schneider, Eduardo H. da Silva Neto, Andreas Erb, Sergey Zhdanovich, Arthur K. Mills, Giorgio Levy, Claudio Giannetti, David J. Jones, Andrea Damascelli, 'Emergence of pseudogap from short-range spin-correlations in electron-doped cuprates', npj Quantum Materials 5, 6 (2020). DOI: 10.1038/s41535-020-0208-6, https://doi.org/10.1038/s41535-020-0208-6
  15. M. Zhu, D. J. Voneshen, S. Raymond, O. J. Lipscombe, C. C. Tam, S. M. Hayden, 'Spin fluctuations associated with the collapse of the pseudogap in a cuprate superconductor', Nature Physics 19, 99-105 (2023). DOI: 10.1038/s41567-022-01825-3, https://doi.org/10.1038/s41567-022-01825-3
  16. Yawen Fang, Gael Grissonnanche, Anaelle Legros, Simon Verret, Francis Laliberte, Clement Collignon, Amirreza Ataei, Maxime Dion, Jianshi Zhou, David Graf, Michael J. Lawler, Paul A. Goddard, Louis Taillefer, B. J. Ramshaw, 'Fermi surface transformation at the pseudogap critical point of a cuprate superconductor', Nature Physics 18, 558-564 (2022). DOI: 10.1038/s41567-022-01514-1, https://doi.org/10.1038/s41567-022-01514-1
  17. Xuetao Zhu, Jiandong Guo, Jiandi Zhang, E. W. Plummer, 'Misconceptions associated with the origin of charge density waves', Advances in Physics: X 2, 622-640 (2017). DOI: 10.1080/23746149.2017.1343098, https://doi.org/10.1080/23746149.2017.1343098
  18. Sotirios Fragkos, Nina Girotto Erhardt, Evgenia Symeonidou, Hibiki Orio, Dominique Descamps, Stephane Petit, Polychronis Tsipas, Kai Rossnagel, Jakub Schusser, Athanasios Dimoulas, Samuel Beaulieu, Dino Novko, 'Phonon-induced pseudogap phase in TiSe2', arXiv:2606.31298 (2026). https://arxiv.org/abs/2606.31298