Quantum Monte Carlo study of the metal-insulator crossover in the square-lattice Hubbard model

Abstract

The interaction-driven evolution from a Fermi liquid to a Mott insulator is a hallmark of strongly correlated fermion systems. In this work, we present a {\it numerically unbiased} study of such metal-to-insulator crossover in the half-filled square-lattice Hubbard model at finite temperatures, employing auxiliary-field quantum Monte Carlo method. By jointly analyzing thermodynamic and dynamical observables, we establish the crossover diagram of the model in the temperature-interaction ($T$-$U$) plane. With increasing $U$, our numerical results reveal an extended crossover regime, which we refer to as the {\it Bad Metal}, that separates the Fermi liquid and Mott insulator. During the crossover, we also examine the antiferromagnetic spin correlations and observe pronounced nodal-antinodal dichotomy in the momentum-resolved single-particle spectral functions. Furthermore, we investigate the temperature dependence of several commonly used observables in the model. As representative results, we achieve an accurate map of the thermal entropy across the crossover diagram, and identify the parameter regions in which the model exhibits the Pomeranchuk cooling, characterized by an adiabatic cooling with increasing $U$. Beyond offering a more refined understanding of the crossover phenomenon, our work also provides valuable benchmark and guideline for future optical lattice experiments on the square-lattice Hubbard model.

Figures (17)

• Figure 1: Crossover diagram of the half-filled square-lattice Hubbard model in the $T$-$U$ plane, and the characteristic signatures of various physical observables from our AFQMC calculations. (a) Shows the metal-insulator crossover, consisting of a weak-coupling Fermi liquid and a strong-coupling Mott insulator, as well as a Bad metal in between. The onset of bad metal as $U_{\rm BM}$ (green squares, taken as the local maximum location $U_{\rm S1}$ of the thermal entropy) and Mott insulator as $U_{\rm MI}$ (brown triangles) are the crossover boundaries. Moreover, $U_{\rm AF}$ (blue hexagons) plots the peak position of the AFM structure factor $S_{\rm AFM}^{zz}$, and $U_{\rm D}$ (black pentagons) marks the inflection point of double occupancy $D$. In (b), the values of $U_{\rm BM}$, $U_{\rm MI}$, $U_{\rm AF}$ and $U_{\rm D}$ from (a) are summarized as horizontal dashed lines, with error bars represented by their lengths. Signatures from additional observables are also included, namely the disappearance of the quasiparticle coherence peak of $A_{\rm loc}(\omega)$ around $\omega=0$ ($U_{A\rm{loc}}$, light blue diamonds), the local minimum of the thermal entropy ($U_{\rm{S}2}$, purple left triangles), the peak position of fidelity susceptibility ($U_{\rm F}$, yellow right triangles), and the crossing point of $\Im\Sigma(\mathbf{k}_F, i\omega_0)$ and $\Im\Sigma(\mathbf{k}_F, i\omega_1)$ ($U_{\rm \Sigma1}$ for $\mathbf{k}_F=\mathbf{k}_{\rm an}$ and $U_{\rm \Sigma2}$ for $\mathbf{k}_F=\mathbf{k}_{\rm n}$, magenta octagons). The contour line of charge compressibility at a tiny threshold is also shown (pink dashed line), nearly coincident with $U_{\rm MI}$. The vertical gray dashed line marks an additional low-temperature crossover from Slater insulator to Mott insulator at $U^{*}/t\simeq 4.25$, taken from Ref. Borejsza2003. Further discussion about these results is provided in Sec. \ref{['sec:PhaseDiagram']}.
• Figure 2: Local single-particle spectrum $A_{\mathrm{loc}}(\omega)$ as a function of $\omega/t$ with different interaction strengths, at four temperatures $T/t=0.2,0.3,0.5$ and $0.7$. For each $T/t$, with increasing $U/t$, the spectrum $A_{\mathrm{loc}}(\omega)$ around $\omega=0$ evolves from a coherence peak, describing the Fermi Liquid state, to a dip, indicating a bad metal, and eventually approaches zero, suggesting the Mott insulating state. These results are from $L=20$ system for $T/t=0.2$ and $L=16$ for other temperatures, and the residual finite-size effects are negligible (see Appendix. \ref{['sec:AppendixB']}).
• Figure 3: Extrapolations of the spectrum ratio $Q$ [see the definition in Eq. (\ref{['eq:SpecRatioQ']})] versus $U/t$ for $T/t=0.1\sim 0.7$. The dashed lines represent linear fits for the small but nonzero $Q$ results, and the extrapolated interaction strength at $Q=0$ is taken as $U_{A{\rm loc}}$. The horizontal gray dashed line plotting $Q=0.1$ is used for estimating the uncertainty of $U_{A{\rm loc}}$. The results at $T/t=0.1$ are from $L=24$, and the system sizes for $T/t\ge 0.2$ are the same as Fig. \ref{['fig:Fig02Aloc']}.
• Figure 4: (a) Thermal entropy density $\boldsymbol{s}$, and (b) AFM structure factor $S^{zz}_{\mathrm{AFM}}$, versus $U/t$ at four temperatures. For $T/t=0.3,0.4,0.6$, the $\boldsymbol{s}$ results are shifted by $-0.16,-0.20,-0.36$, and the $S^{zz}_{\mathrm{AFM}}$ data are rescaled by factors $\times3,\times4,\times5$, to fit into the plots. In (a), the uncertainty of $\boldsymbol{s}$ is represented by the line thickness. At $T/t=0.2$, the result of $U_{A{\rm loc}}=3.1(2)$ determined from $A_{\mathrm{loc}}(\omega)$ in Figs. \ref{['fig:Fig02Aloc']} and \ref{['fig:Fig03FLtoBM']} is shown as the light blue shading, while the peak position of $S^{zz}_{\mathrm{AFM}}$ as $U_{\rm AF}=5.9(1)$ is depicted as the blue shading. These results are from $L=20$ system for $T/t\le 0.2$ and $L=16$ system for other temperatures.
• Figure 5: The quasiparticle weight $Z_{\mathbf{k}_F}$ versus $U/t$ for temperatures $T/t=0.2,0.4$ and $0.6$, at (a) nodal point ($\mathbf{k}_F=\mathbf{k}_{\rm n}$), and (b) antinodal point ($\mathbf{k}_F=\mathbf{k}_{\rm an}$). Linear fits (dashed lines) are performed at intermediate interactions, and the intercepts associated with extrapolated $Z_{\mathbf{k}_F}=0$ are taken as $U_{\rm MI}$. The insets plot the local spectrum $A_{\rm{loc}}(\omega)$ in the vicinity of $U=U_{\rm MI}$ for $T/t=0.2$ and $0.6$. The system sizes are the same as Fig. \ref{['fig:Fig02Aloc']}.