© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:cond-mat/9807115

On the Lenz-Ising-Onsager Problem in an External Magnetic Field

Martin S. Kochman'ski

Published Jul 8, 1998• 0 citations

Abstract

The Lenz-Ising-Onsager (LIO) problem in an external magnetic field in the second quantization representation is the subject of consideration of the paper. It is shown that the operator $V_h$ in the second quantization representation corresponding to Ising spins interaction with the external magnetic field $H$ can be represented in terms of single-subscript creation and anihilation Fermi operators in such a form that the operator $V_h$ commutes with the operator $\hat{P}\equiv(-1)^{\hat{S}}$, where $\hat{S}= \sum_mβ^†_mβ_m$ is the operator of a total number of Fermions. The possible consequences of such representation with it's relation to the LIO is discussed. In particular, the constructive proof of the Lee-Yang theorem on the absence of phase transition for Ising model in nonzero magnetic field $(\Re h\neq 0)$ is demonstrated.

DOI: 10.48550/arxiv.cond-mat/9807115

Related papers

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

On the two- and three-dimensional Lenz-Ising-Onsager problem in presence of magnetic field

Martin S. Kochman'ski

In this paper a new approach to solving the Ising-Onsager problem in external magnetic field is investigated. The expression for free energy on one Ising spin in external field both for the twodimensional and threedimensional Ising model with interaction of the nearest neighbors are derived. The representations of free energy being expressed by multidimensional integrals of Gauss type with the appropriate dimensionality are shown. Possibility of calculating the integrals and the critical indices on the base of the derived representations for free energy is investigated.

arXiv:cond-mat/9809069•Sep 3, 1998•0 citations

Geometric perspective on the one-dimensional Lenz–Ising model in a non-zero magnetic field

Melchor A Cupatan

Abstract Having been linked to numerous applications, the one-dimensional Lenz–Ising model in a non-zero magnetic field is well-known in graduate- and senior-level statistical physics. The transfer matrix method is the standard way to solve the model. In this line, we reformulate the transfer matrix by subjecting the usual Pauli matrices to rotation. The transfer matrix is then shown as a ‘spin’-representative of a two-dimensional rotation just like the special case when the magnetic field is zero. Consequently, eigenvalues in exponential form are obtained, and the zeroes of the partition function lie on a unit circle in agreement with the Lee–Yang Circle Theorem. The zeroes pinching the positive side of the real axis give the known trivial transition temperature, T = 0 K. Finally, we discuss the significance of the results in teaching the model.

European Journal of Physics•Jan 16, 2020•2 citations

Ansatz for the Two-Dimensional Ising Model in an External Magnetic Field

Moorad Alexanian

An ansatz applied to the two-dimensional Ising model in an external magnetic field h gives rise to an exactly soluble model. The singularity in the magnetization found by Onsager does not survive the presence of the external magnetic field as found earlier by Lee and Yang in 1952. However, the singularity in the heat capacity remains even in the presence of the magnetic field. A surprising result is the presence of negative heat capacity for h > 0.

arXiv:2412.18630•Armenian Journal of Physics•Dec 20, 2024•0 citations

Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

T. D. Schultz, Daniel Charles Mattis, Élliott H. Lieb

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.

Reviews of Modern Physics•Jul 1, 1964•935 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 Lenz-Ising-Onsager (LIO) problem in an external magnetic field in the second quantization representation is the subject of consideration of the paper. It is shown that the operator

V_h

in the second quantization representation corresponding to Ising spins interaction with the external magnetic field

H

can be represented in terms of single-subscript creation and anihilation Fermi operators in such a form that the operator

V_h

commutes with the operator

\hat{P}\equiv(-1)^{\hat{S}}

, where

\hat{S}= \sum_mβ^†_mβ_m

is the operator of a total number of Fermions. The possible consequences of such representation with it's relation to the LIO is discussed. In particular, the constructive proof of the Lee-Yang theorem on the absence of phase transition for Ising model in nonzero magnetic field

(\Re h\neq 0)

is demonstrated.

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 $(T\ensuremath{\ne}0).$

In this paper a new approach to solving the Ising-Onsager problem in external magnetic field is investigated. The expression for free energy on one Ising spin in external field both for the twodimensional and threedimensional Ising model with interaction of the nearest neighbors are derived. The representations of free energy being expressed by multidimensional integrals of Gauss type with the appropriate dimensionality are shown. Possibility of calculating the integrals and the critical indices on the base of the derived representations for free energy is investigated.

Abstract Having been linked to numerous applications, the one-dimensional Lenz–Ising model in a non-zero magnetic field is well-known in graduate- and senior-level statistical physics. The transfer matrix method is the standard way to solve the model. In this line, we reformulate the transfer matrix by subjecting the usual Pauli matrices to rotation. The transfer matrix is then shown as a ‘spin’-representative of a two-dimensional rotation just like the special case when the magnetic field is zero. Consequently, eigenvalues in exponential form are obtained, and the zeroes of the partition function lie on a unit circle in agreement with the Lee–Yang Circle Theorem. The zeroes pinching the positive side of the real axis give the known trivial transition temperature, T = 0 K. Finally, we discuss the significance of the results in teaching the model.

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.