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Journal of Experimental and Theoretical Physics

Generalized Jordan-Wigner transformations and the Ising-Onsager problem

Martin S. Kochman'ski

Published May 1, 1997• 2 citations
Publisher

Abstract

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.

DOI
10.1134/1.558232

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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.
Recently, an algebraic generalization of the Jordan-Wigner transformation was introduced and applied to one- and two-dimensional systems. This transformation is composed of the interactions ηi that appear in the Hamiltonian H as H=∑i=1NJiηi, where Ji are coupling constants. In this short note, it is derived that operators that are composed of ηi, or its n-state clock generalizations, satisfy the Dolan-Grady condition and hence obey the Onsager algebra which was introduced in the original solution of the rectangular Ising model and appears in some integrable models.
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.
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 $(T\ensuremath{\ne}0).$
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.
We introduce a new spin-fermion mapping, for arbitrary spin S generating the SU(2) group algebra, that constitutes a natural generalization of the Jordan-Wigner transformation for S = 1/2. The mapping, valid for regular lattices in any spatial dimension d, serves to unravel hidden symmetries. We illustrate the power of the transformation by finding exact solutions to lattice models previously unsolved by standard techniques. We also show the existence of the Haldane gap in S = 1 bilinear nearest-neighbor Heisenberg spin chains and discuss the relevance of the mapping to models of strongly correlated electrons. Moreover, we present a general spin-anyon mapping for the case d < or = 2.