Nonextensive thermodynamics of the two-site Hubbard model

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 β , two methods have been adopted: T = 1 / k B β in method A [Tsallis et al., Physica A 261 (1998) 534], and T = c q / k B β in method B [Abe et al., Phys. Lett. A 281 (2001) 126], where k B denotes the Boltzman constant, c q = ∑ i p i q , p i the probability distribution of the ith state, and q the entropic index. Temperature dependences of specific heat and magnetic susceptibility have been calculated for 1 ⩽ q ⩽ 2 , the conventional Boltzman–Gibbs statistics being recovered in the limit of q = 1 . The Curie constant Γ q of the susceptibility in the atomic and low-temperature limits ( t / U → 0 , T / U → 0 ) is shown to be given by Γ q = 2 q 2 2 ( q - 1 ) in method A, and Γ q = 2 q in method B, where t stands for electron hoppings and U intra-atomic interaction in the Hubbard model. These expressions for Γ q are shown to agree with the results of a free spin model which have been studied also by the NES with methods A and B. A comparison has been made between the results for canonical and grand-canonical ensembles of the model.