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

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 Hubbard model simulator with nearest-neighbor hoppings \cite{kendrick2025pseudogap} showed that the isothermal compressibility $\kappa(\delta)$ has a maximum as a function of doping $\delta$. Here we use the two-particle self-consistent plus (TPSC+) approach to explain these experiments and connect the maximum in $\kappa(\delta)$ to the single-particle spectrum transformation from the pseudogapped to the metallic regime, elucidating the nature of the pseudogap (PG). The maximum in $\kappa(\delta)$ practically coincides with the doping where the precursor of the lower $(\pi,\pi)$ spin density wave (SDW) band at the antinodal point crosses zero frequency $\omega=0$. The Knight shift, $\chi_{sp}(0,0)(\delta)$, should also exhibit a maximum versus doping. Additionally, TPSC+ correctly predicts a maximum in the temperature dependence of the Knight shift, $\chi_{sp}(0,0)(T)$, consistent with recent ultracold atom experiments \cite{chalopin2026observation}.These maxima should exist at low temperatures ($T$) in both the intermediate $U \approx U_{Mott}$ and weak $U < U_{Mott}$ interaction limits due to critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at $\delta_{max}(T)$ can be small, controlled by dynamic rather than static critical thermal fluctuations. The SDW fluctuations are incommensurate at $\delta=\delta_{max}$. At low $T$, multiple peaks in the incommensurate spin susceptibility lead to more than two SDW precursor peaks in the spectral function and density of states. By accessing parameter regimes relevant to cuprates, including further-neighbor hopping ($t', t''$) and low temperatures, our work provides a high-impact tool for further studies.