Variational cluster approach to thermodynamic properties of interacting fermions at finite temperatures: A case study of the two-dimensional single-band Hubbard model at half filling

TL;DR

This paper develops a finite-temperature variational cluster approach for the Hubbard model, enabling detailed analysis of thermodynamic properties, phase transitions, and spectral features, with applications to antiferromagnetism and Mott transitions.

Contribution

It introduces a novel finite-temperature scheme for the variational cluster approximation, applicable to exact-diagonalization clusters, and demonstrates its effectiveness in studying thermodynamic and spectral properties.

Findings

Identified the crossover between Slater and Mott insulators in the phase diagram.

Showed the Mott transition is discontinuous with a coexistence region persisting to zero temperature.

Calculated thermodynamic quantities and spectral functions consistent with known physics.

Abstract

We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion $U$ and temperature $T$. We also calculate the $T$ dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. On the basis of the thermodynamic properties, we obtain a crossover diagram in the $(U,T)$-plane which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA) and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the $(U,T)$-plane. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature.