Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Acta Physica Polonica B

Low-Temperature Asymptotics of Free Energy for 3D Ising Model in an External Magnetic Field

Martin S. Kochman'ski

Published Aug 1, 1998• 0 citations
OpenAlex

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 = 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

Related papers

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

Martin S. Kochman'ski

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.

arXiv:cond-mat/9710201•Oct 20, 1997•0 citations

Lenz-Ising-Onsager problem in an external field as a soluble problem of many fermions

Martin S. Kochman'ski

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 $(H).$ The free energy in the limiting case of the magnetic field tending to zero ($H\ensuremath{\rightarrow}0,$ $N,$ $M\ensuremath{\rightarrow}\ensuremath{\infty}$) at arbitrary temperature is also considered $(T\ensuremath{\ne}0).$

arXiv:cond-mat/9710203•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1997•4 citations

Exact solution of the two-dimensional (2D) Ising model at an external magnetic field

Zhang Zhi-dong

The exact solution of the two-dimensional (2D) Ising model at an external magnetic field is derived by a modified Clifford algebraic approach. At first, the transfer matrices are analyzed in three representations, i.e., Clifford algebraic representation, transfer tensor representation and schematic representation, to inspect nonlocal effects in this many-body interacting system. It is ensured that nontrivial topological structures exist in this system, which is analogous to (but different with) those in the three-dimensional (3D) Ising model at zero magnetic field. Therefore, the approaches developed for the 3D Ising models are modified to be appropriable for solving analytically the solution of the 2D Ising model at a magnetic field. An additional rotation, serving as a topological Lorentz transformation, is applied for dealing with the topological problems in the present system. The rotation angle for the transformation is determined by Yang-Baxter relations and a subsequent average of rotation angles treating the linear change of the topological actions. Application of a magnetic field increases the magnetization, shifting the critical point to higher temperatures. At the temperature above the critical point, the magnetization keeps zero until a critical field at which it jumps rapidly as a first-order magnetization process. The partition function and the magnetization obtained are helpful for understanding the physical properties, in particular, the magnetization processes of the 2D magnetic materials.

arXiv:2512.16935•Dec 12, 2025•0 citations

Approximate thermodynamics of the two-dimensional Ising model in an external magnetic field

Mathieu Gaudreault

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.

arXiv:2607.16735•Journal of Physics A Mathematical and Theoretical•Apr 15, 2025•0 citations

Construction of an approximate solution for an Ising magnet in an external magnetic field

S. V. Semkin, Виктор Павлович Смагин, Elena G. Lavrushina

A method of approximate calculation of the magnetization of an Ising magnet located in an external magnetic field is proposed. This method is based on using an exact or approximate expression for the spontaneous magnetization of an Ising magnet on the same lattice. The method uses the ratio of effective fields for clusters of one and two lattice nodes. Using the proposed method for a magnet located in an external field, the dependence of magnetization on temperature and the magnitude of the external field is calculated. The proposed method is applied in the work to the solution in the mean field approximation, to the solution in the Beta approximation and to the exact solution on a square lattice; for all these solutions, critical indices characterizing the behavior of a magnet in an external field are calculated. Keywords: the Ising model, phase transitions, the influence of an external field.

Physics of the Solid State•Jan 1, 2023•0 citations

Generalized Jordan-Wigner transformations and the Ising-Onsager problem

Martin S. Kochman'ski

Another possible method for obtaining Onsager’s solution for the 2 D Ising model is presented. In contrast to previous methods, the method proposed here makes it possible to study the problem in a weak external magnetic field. Generalized Jordan-Wigner transformations in the form introduced by Kochmański [J. Tech. Phys. 36 , 485 (1995)] are employed.

Journal of Experimental and Theoretical Physics•May 1, 1997•2 citations
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
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 $\cite{mkoch95}$, are applied.
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 $(T\ensuremath{\ne}0).$
The exact solution of the two-dimensional (2D) Ising model at an external magnetic field is derived by a modified Clifford algebraic approach. At first, the transfer matrices are analyzed in three representations, i.e., Clifford algebraic representation, transfer tensor representation and schematic representation, to inspect nonlocal effects in this many-body interacting system. It is ensured that nontrivial topological structures exist in this system, which is analogous to (but different with) those in the three-dimensional (3D) Ising model at zero magnetic field. Therefore, the approaches developed for the 3D Ising models are modified to be appropriable for solving analytically the solution of the 2D Ising model at a magnetic field. An additional rotation, serving as a topological Lorentz transformation, is applied for dealing with the topological problems in the present system. The rotation angle for the transformation is determined by Yang-Baxter relations and a subsequent average of rotation angles treating the linear change of the topological actions. Application of a magnetic field increases the magnetization, shifting the critical point to higher temperatures. At the temperature above the critical point, the magnetization keeps zero until a critical field at which it jumps rapidly as a first-order magnetization process. The partition function and the magnetization obtained are helpful for understanding the physical properties, in particular, the magnetization processes of the 2D magnetic materials.
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.
A method of approximate calculation of the magnetization of an Ising magnet located in an external magnetic field is proposed. This method is based on using an exact or approximate expression for the spontaneous magnetization of an Ising magnet on the same lattice. The method uses the ratio of effective fields for clusters of one and two lattice nodes. Using the proposed method for a magnet located in an external field, the dependence of magnetization on temperature and the magnitude of the external field is calculated. The proposed method is applied in the work to the solution in the mean field approximation, to the solution in the Beta approximation and to the exact solution on a square lattice; for all these solutions, critical indices characterizing the behavior of a magnet in an external field are calculated. Keywords: the Ising model, phase transitions, the influence of an external field.