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arXiv:2109.10446 · Journal of Statistical Mechanics Theory and Experiment

Thermodynamics of an Ising-like XXZ chain in a longitudinal magnetic field in the framework of the quantum transfer matrix approach

P. N. Bibikov

Published Feb 1, 2022• 0 citations
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Abstract

Abstract Taking the Ising chain as a reference model we have derived a perturbative expression for the free energy density of the Heisenberg–Ising chain with strong easy-axis anisotropy. All calculations are performed on the ground of the quantum transfer matrix approach. The obtained result agrees with the direct high-temperature expansion. It also agrees with the low-temperature cluster expansion in the special subregime when quantum fluctuations are weak against thermodynamical ones.

DOI
10.1088/1742-5468/ac4e83

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Abstract Taking the Ising chain as a reference model we have derived a perturbative expression for the free energy density of the Heisenberg–Ising chain with strong easy-axis anisotropy. All calculations are performed on the ground of the quantum transfer matrix approach. The obtained result agrees with the direct high-temperature expansion. It also agrees with the low-temperature cluster expansion in the special subregime when quantum fluctuations are weak against thermodynamical ones.
We report analytical calculations of the Helmholtz free energy of non-integrable anisotropic quantum XXZ chains in the high-temperature regime for several values of the spin S . Single-ion anisotropy and interaction with an external magnetic field are taken into account. The seven lowest-order terms in the high-temperature expansion of Helmholtz free energy are obtained. Our results contribute to the existing literature on high-temperature expansions and numerical studies of those models by discussing the effects of anisotropy upon their high-temperature thermodynamic properties, such as the average energy per site, the specific heat and magnetic susceptibility.
The authors derive the thermodynamics and the static correlation functions for the classical Heisenberg chain with the anisotropies within a low-temperature steepest-descent approach. As regards the thermodynamics, they find agreement with transfer matrix and 'ideal-gas' phenomenological results. For the longitudinal correlation function the authors find an antiferromagnetic-type behaviour with a correlation length inversely proportional to the domain wall density. The transverse in-plane correlation function exhibits a damped oscillatory behaviour due to domain-wall-spin interference. The transverse out-of-plane correlations, on the other hand, are exponentially damped with the correlation length increasing as the easy-axis limit is approached.
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A magnetic property of the one dimensional spin 5/2 Ising model under the magnetic field has been investigated by means of transfer matrix method. Thermodynamic response functions are also obtained for varying values of scaling temperature and scaling magnetic field. Entropy and heat capacity of the system were calculated by benefiting from the temperature dependencies of Helmholtz free energy. Our simulation results demonstrate that as the strength of the magnetic field is increased, heat capacity tend to shift to the relatively higher temperature regions, and these findings are consistent with previous results for the low spin values in one dimensional Ising systems.