Nonextensive thermodynamics of the two-site Hubbard model

TL;DR

This paper explores the thermodynamic properties of the two-site Hubbard model within nonextensive statistics, analyzing how different temperature definitions affect specific heat and susceptibility, and comparing canonical and grand-canonical ensembles.

Contribution

It introduces two methods for relating temperature to Lagrange multipliers in nonextensive statistics applied to the Hubbard model, providing new analytical expressions for susceptibility and comparing ensemble results.

Findings

Temperature dependence of specific heat and susceptibility calculated for q between 1 and 2.

Derived expressions for Curie constant in different methods, matching free spin model results.

Comparison shows differences between canonical and grand-canonical ensemble outcomes.

Abstract

Thermodynamical properties of canonical and grand-canonical ensembles of the half-filled two-site Hubbard model have been discussed within the framework of the nonextensive statistics (NES). For relating the physical temperature $T$ to the Lagrange multiplier $\beta$, two methods have been adopted: $T=1/k_B \beta$ in the method A [Tsallis {\it et al.} Physica A {\bf 261} (1998) 534], and $T=c_q/k_B \beta$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281} (2001) 126], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, $p_i$ the probability distribution of the $i$th state, and $q$ the entropic index. Temperature dependences of specific heat and magnetic susceptibility have been calculated for $1 \lleq q \lleq 2$, the conventional Boltzman-Gibbs statistics being recovered in the limit of $q = 1$. The Curie constant $\Gamma_q$ of the susceptibility in the atomic and low-temperature limits ($t/U \to 0, T/U \to 0$) is shown to be given by $\Gamma_q=2 q 2^{2(q-1)}$ in the method A, and $\Gamma_q=2 q$ in the method B, where $t$ stands for electron hoppings and $U$ intra-atomic interaction in the Hubbard model. These expressions for $\Gamma_q$ are shown to agree with the results of a free spin model which has been studied also by the NES with the methods A and B. A comparison has been made between the results for canonical and grand-canonical ensembles of the model.