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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

Y. M. Vilk1,*,† and A.-M. S. Tremblay2

  • *Retired.
  • †Contact author: Yury.Vilk2@gmail.com
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Phys. Rev. X 16, 031034 – Published 11 August, 2026

DOI: https://doi.org/10.1103/7v3y-tbbr

Abstract

Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a simulator of the Hubbard model with nearest-neighbor hoppings [L. H. Kendrick et al., arXiv:2509.18075.] showed that the isothermal compressibility κ(δ) has a maximum as a function of doping δ. Here, we use the two-particle self-consistent plus approach to explain these experiments and connect the maximum in κ(δ) to the transformation of the single-particle spectrum from the pseudogapped to the metallic regime. This elucidates the nature of the pseudogap. Specifically, the maximum in κ(δ) practically coincides with the doping at which the precursor of the lower (π,π) spin-density-wave (SDW) band at the antinodal point crosses the zero-frequency ω=0. The Knight shift χsp(0,0)(δ), as a function of doping, should also have a maximum. In addition, we demonstrate that two-particle self-consistent plus correctly predicts a maximum in the temperature dependence of the Knight shift χsp(0,0)(T), consistent with recent ultracold-atom experiments [T. Chalopin et al., Proc. Natl. Acad. Sci. U.S.A. 123, e2525539123 (2026).]. The maxima in both quantities should exist, at sufficiently low temperatures (T), in both the intermediate U≈UMott and weak U<UMott interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at δmax(T) can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at δ=δmax. We predict that, at low T, the multiple peaks in the spin susceptibility in the incommensurate case lead to more than two SDW precursor peaks in the spectral function and density of states. By allowing access to parameter regimes relevant to cuprates—including further-neighbor hopping (t′,t′′) and low temperatures, our work provides a high-impact tool for further studies by the broader community.

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Popular Summary

Identifying the boundary of the pseudogap regime is a central challenge in understanding strongly correlated materials because clear thermodynamic signatures for this electronic transition have been difficult to pinpoint. We addressed this challenge by using the two-particle self-consistent plus method to analyze Hubbard model simulations, revealing that a distinct maximum in the isothermal compressibility serves as a definitive thermodynamic marker for the boundary. Our analysis shows that this peak occurs when thermal spin-density-wave fluctuations alter the single-particle spectrum, causing a precursor band to cross the zero-frequency level at the antinodal Van Hove point. We found that these critical magnetic fluctuations drive universal behaviors across weak and intermediate interaction strengths, yielding fresh predictions such as a doping-dependent maximum in the Knight shift and multipeak spectral structures at lower temperatures. These results provide an accessible tool for interpreting cold-atom quantum simulations under conditions that closely mimic real copper oxide superconductors. Our work establishes a rigorous theoretical foundation that guides the community in mapping phase boundaries and searching for unconventional superconductivity in correlated systems.

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Phys. Rev. X 16, 031034– Published 11 August, 2026
  • Received 7 February 2026
  • Revised 24 April 2026
  • Accepted 23 June 2026
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