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

TL;DR

This work uses the TPS C+ framework to study the pseudogap–to–metal crossover in the 2D Hubbard model and explains the recently observed maximum in the isothermal compressibility $\kappa(\delta)$. By incorporating renormalized spin/charge interactions and a self-consistent feedback into single-particle properties, the authors show that the maximum in $\kappa(\delta)$ coincides with the crossing of the precursor lower SDW band at the antinodal point through $\omega=0$, which shifts the Van Hove singularity of the DOS from occupied to unoccupied states. They also predict a maximum in the Knight shift $\chi_{sp}(0,0)(\delta)$ at low temperatures, and demonstrate that both maxima persist in weak-to-intermediate coupling ($U \lesssim U_{Mott}$) due to critical thermal SDW fluctuations, including regimes with incommensurate SDW fluctuations. The findings connect single-particle spectral evolution to thermodynamic response and offer predictive insights for cuprate-like systems, including parameter regimes with further-neighbor hopping $t'$ and $t''$, accessible to cold-atom experiments and future numerical studies.

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, $χ_{sp}(0,0)(δ)$, 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 $U \approx U_{Mott}$ and weak $U < U_{Mott}$ interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at $δ_{max}(T)$ can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at $δ=δ_{max}$. 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 ($t', t''$) and low temperatures, our work provides a high-impact tool for further studies by the broader community.

Figures (16)

• Figure 1: Comparison of the TPSC+ doping dependence of the compressibility $\kappa(\delta)$ with results from experiments on a cold atoms simulator of the Hubbard model. The experiments were conducted on a system of $300$ sites in a circular geometry, while TPSC+ calculations were performed on an $18\times18$ lattice (324 sites) with periodic boundary conditions. The three interaction strengths, $U=(3.69,4.67,7)$, correspond to weak-to-intermediate interactions. The temperature in the experiment is $T \lesssim 0.15$, and the TPSC+ calculations were performed for two temperatures: $T=0.1$ and $T=0.15$. The TPSC+ theory predicts the existence of a well-defined maximum in the compressibility as a function of doping $\kappa(\delta)$ for $T=0.1$ for all considered $U$. By contrast, there is only a broad maximum in $\kappa(\delta)$ for $T=0.15$. In the weaker interaction case $U=3.69$, there is quantitative agreement with the experiment, while in the intermediate interaction case $U=7$, the theory somewhat underestimates $\kappa(\delta)$ around the maximum. This is consistent with other observations indicating that TPSC+ underestimates the antiferromagnetic correlations in the intermediate-interaction limit.
• Figure 2: Effect of the finite system size on the doping dependence of the compressibility $\kappa(\delta)$. The temperature is $T=0.1$, the two interaction strengths are $U=(3.69, 7)$ and the system sizes are $18\times18$ and $64\times64$. We can see that $\kappa(\delta)$ has a significantly sharper maximum in the $18\times18$ system than in the $64\times64$ system. The correlation length for doping close to the maximum $\delta\sim \delta_{max}$ is a few lattice spacings in this case. This affects the result for $18\times18$ but not for the $64\times64$ system. We double checked that increasing linear system size $L$ further than $L=64$ has virtually no effect on the maximum in the $\kappa(\delta)$. The analysis suggests that the finite size effect on the maximum in $\kappa(\delta)$ plays a significant role when about a quarter of the linear system size is smaller than the correlation length $1/4 L < \xi$.
• Figure 3: Comparison of the experimental equal-time spin correlation function $|C(d)|$ as a function of Euclidean distance $d$ with TPSC+ results for two lattices, $18 \times 18$ and $256 \times 256$. The parameters are $U=7$, $T=0.15$ and $\delta=0.04$. The first lattice has approximately the same number of sites as in the experiment but a different geometry (square with periodic boundary conditions versus circular, without such conditions). The second lattice is equivalent to the thermodynamic limit (TDL), having a TPSC+ correlation length of $\xi=9.74$. The finite-size effects are negligible in both small systems for short distances, $d \leq 6 \sim L/3$, but become progressively more important for larger distances. For $d \leq 6$, the TPSC+ results underestimate the spin correlation function by about $27\%$ to $33\%$ for $U=7$. This is consistent with our earlier observation that TPSC+ underestimates spin correlations in the intermediate interaction case. The fits to theoretical and experimental data are described in the main text.
• Figure 4: Dependence of the compressibility $\kappa$ on doping for four different temperatures, $T = (0.15, 0.1, 0.0714,0.025)$, and two different interaction strengths, $U = (3.69, 7)$. The smaller interaction strength represents a weak-interaction case, $U = 3.69 < U_{Mott}$, while the larger interaction strength, $U = 7 \approx U_{Mott}$, corresponds to an intermediate-interaction case. The results are obtained in the thermodynamic limit. The doping-dependent isothermal compressibility $\kappa(\delta)$ shows a maximum for $T = 0.1$. The curves for both interaction strengths $U$ are qualitatively similar, indicating that the same physics is at play. The maximum in $\kappa(\delta)$ signals the crossover from a pseudogapped electronic liquid to a correlated Fermi liquid at $\delta_{max}(T)$. The value of $\delta_{max}$ is significantly lower for smaller $U$ (for $T = 1/14 \approx 0.0714$, we have $\delta_{max} = 0.1$ for $U = 3.69$ while $\delta_{max} = 0.15$ for $U = 7$).
• Figure 5: Dependence of the uniform spin susceptibility $\chi_{sp}(0,0)$ on doping for four different temperatures, $T = (0.15, 0.1, 0.0714,0.025)$, and two different interaction strengths, $U = (3.69, 7)$. The smaller interaction strength represents a weak-interaction case, $U = 3.69 < U_{Mott}$, while the larger interaction strength, $U = 7 \approx U_{Mott}$, corresponds to an intermediate-interaction case. The results are obtained in the thermodynamic limit. The function $\chi_{sp}(0,0)(\delta)$ shows a maximum for all temperatures. The curves for both interaction strengths $U$ are qualitatively similar, indicating that the same physics is at play. The maximum in $\chi_{sp}(0,0)(\delta)$signals the transition from a pseudogapped electronic liquid to a correlated Fermi liquid and lies on the crossover line $\delta_{max}(T)$. The value of $\delta_{max}$ is significantly smaller for smaller $U$ (for $T = 1/14 \approx 0.0714$, $\delta_{max} = 0.1$ for $U = 3.69$ while $\delta_{max} = 0.15$ for $U = 7$).