Nonextensive thermodynamics of the two-site Hubbard model: Canonical ensembles

TL;DR

This paper investigates the thermodynamic properties of Hubbard dimers within nonextensive statistics, comparing two methods for relating temperature to the Lagrange multiplier, and discusses implications for Heisenberg dimers.

Contribution

It introduces a nonextensive statistical approach to Hubbard dimers, compares two temperature definitions, and applies the framework to Heisenberg dimers, highlighting the more reasonable method.

Findings

Temperature dependence of thermodynamic quantities calculated for different cluster sizes.

Method B for relating temperature to the Lagrange multiplier is more consistent.

Nonextensive effects significantly influence the thermodynamics of spin dimers.

Abstract

Canonical ensembles consisting of $M$-unit $Hubbard$ $dimers$ have been studies within the nonextensive statistics (NES). The temperature dependences of the energy, entropy, specific heat and susceptibility have been calculated for the number of dimers, $M = 1, 2, 3$ and $\infty$. We have assumed the relation between the entropic index $q$ and the cluster size $N$ given by $q=1+2/N$ ($N = 2\:M$ for $M$ dimers), which was previously derived by several methods. 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}, 534 (1998)], and $T=c_q/k_B \beta$ in the method B [Abe {\it et al.} Phys. Lett. A {\bf 281}, 126 (2001)], where $k_B$ denotes the Boltzman constant, $c_q= \sum_i p_i^q$, and $p_i$ the probability distribution of the $i$th state. The susceptibility and specific heat of spin dimers ({\it Heisenberg dimers}) described by the Heisenberg model have been discussed also by using the NES with the methods A and B. A comparison between the two methods suggests that the method B may be more reasonable than the method A for nonextensive systems.