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 simulator of the Hubbard model with nearest-neighbor hoppings \cite{kendrick2025pseudogap} 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 (TPSC+) 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 (PG). 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, $χ<em>{sp}(0,0)(δ)$, as a function of doping, should also have a maximum. The maxima in both quantities should exist, at sufficiently low temperatures ($T$), in both the intermediate $U \approx U</em>{Mott}$ and weak $U &lt; U_{Mott}$ interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at $δ<em>{max}(T)$ can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at $δ=δ</em>{max}$. 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 ($t&#39;, t&#39;&#39;$) and low temperatures, our work provides a high-impact tool for further studies by the broader community.