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arXiv:cond-mat/9710201

Low-temperature asymptotics of free energy of 3D Ising model in an external magnetic field

Martin S. Kochman'ski

Published Oct 20, 1997• 0 citations
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Abstract

The paper presents new method for calculating the low-temperature asymptotics of free energy of the 3D Ising model in external magnetic field $(H\neq 0)$. The results obtained are valid in the wide range of temperature and magnetic field values fulfilling the condition: $[1-\tanh(h/2)]\simε,$ for $ε\ll 1$, where $h=βH$, $β$ - the inverse temperature and $H$ - external magnetic field. For this purpose the method of transfer-matrix, and generalized Jordan-Wigner transformations, in the form introduced by the author in $\cite{mkoch95}$, are applied.

DOI
10.48550/arxiv.cond-mat/9710201

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The paper presents new method for calculating the low-temperature asymptotics of free energy of the 3D Ising model in external magnetic field (H≠0)(H\neq 0)(H=0). The results obtained are valid in the wide range of temperature and magnetic field values fulfilling the condition: [1−tanh⁡(h/2)]∼ε,[1-\tanh(h/2)]\simε,[1−tanh(h/2)]∼ε, for ε≪1ε\ll 1ε≪1, where h=βHh=βHh=βH, βββ - the inverse temperature and HHH - external magnetic field. For this purpose the method of transfer-matrix, and generalized Jordan-Wigner transformations, in the form introduced by the author in \citemkoch95\cite{mkoch95}\citemkoch95, are applied.
The paper presents new method for calculating the low-temperature asymptotics of free energy of the 3D Ising model in external magnetic field (H = 0). The results obtained are valid in the wide range of temperature and magnetic field values fulfilling the condition: [1 − tanh(h/2)] ∼ ε, for ε ≪ 1, where h = βH, β- the inverse temperature and H- external magnetic field. For this purpose the method of transfer-matrix, and generalized Jordan-Wigner transformations, in the form introduced by the author in [1], are applied. PACS number(s): 05.50. +q Typeset using REVTEX 1 I. FORMULATION OF THE PROBLEM As is well known, till now an exact solution for the 2D Ising model in external magnetic field (H = 0) was not found. In the case of the 3D Ising model there does not exist an exact solution forvanishing magnetic field (H = 0), without even mentioning the case with magnetic field. Despite great successes in investigations of the Ising model reachedusing
In this paper an approach to solving the two- (2D) and three-dimensional Ising models in an external magnetic field $H\ensuremath{\ne}0$ is developed. The general formalism for the approach to the problem is presented for the example of the 2D Ising model in the external magnetic field. The paper presents a method obtaining the Onsager solution and computations of asymptotic forms of low-temperature free energy for the 2D Ising model in the external magnetic field (H).(H).(H). The free energy in the limiting case of the magnetic field tending to zero ($H\ensuremath{\rightarrow}0,$ N,N,N, $M\ensuremath{\rightarrow}\ensuremath{\infty}$) at arbitrary temperature is also considered $(T\ensuremath{\ne}0).$
Abstract We introduce approximate expressions for the thermodynamics of the two-dimensional Ising model in an external magnetic field. The external field breaks the system’s symmetry, which complicates the exact calculation of the free energy. To address this, we write the model in its quaternion representation to apply a small angle approximation with respect to a reference frame. The approximation simplifies the transfer matrix to a centrosymmetric matrix for which the largest eigenvalue is known and depends on the external field strength. We therefore write an approximate expression for the system’s free energy and derive the magnetization, susceptibility, internal energy, and specific heat. These results are compared with numerical simulations performed with the BKL algorithm. We find that the approximate expressions and numerical simulations agree in regions where the approximation is valid.