On the particle-hole symmetry of the fermionic spinless Hubbard model in $D=1$

TL;DR

This paper explores the particle-hole symmetry in the 1D fermionic spinless Hubbard model, linking it to the invariance of the Helmholtz free energy in the related XXZ Heisenberg model, and analyzes thermodynamic similarities and differences.

Contribution

It establishes the connection between particle-hole symmetry in the Hubbard model and Helmholtz free energy invariance in the XXZ model, providing detailed thermodynamic comparisons.

Findings

Specific heat and entropy are identical in particle-hole related regimes.

Internal energy per site and first-neighbor correlations differ between regimes.

The difference in correlations relates to the magnetization of the XXZ model.

Abstract

We revisit the particle-hole symmetry of the one-dimensional ($D=1$) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 $XXZ$ Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature. Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site. Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the $z$-component of magnetization of the $XXZ$ Heisenberg model. The results presented in this paper are valid for any value of the interaction strength parameter $V$, which describes the attractive/null/repulsive interaction of neighboring fermions.