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arXiv:cond-mat/9710203 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

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

Martin S. Kochman'ski

Published Oct 1, 1997• 4 citations
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Abstract

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

DOI
10.1103/physreve.56.3831

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In this paper an approach to solving the two- (2D) and three-dimensional Ising models in an external magnetic field H\ensuremath≠0H\ensuremath{\ne}0H\ensuremath=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→0,H\ensuremath{\rightarrow}0,H\ensuremath→0, N,N,N, M\ensuremath→\ensuremath∞M\ensuremath{\rightarrow}\ensuremath{\infty}M\ensuremath→\ensuremath∞) at arbitrary temperature is also considered (T\ensuremath≠0).(T\ensuremath{\ne}0).(T\ensuremath=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.
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 analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature Tc{T}_{c}Tc​ can be regarded as an analytic function in the uuu plane, where $u=\mathrm{exp}(\ensuremath{-}\frac{4J}{{k}_{B}T})$, with singularities at $u=0, \ensuremath{\infty}, \mathrm{and} \mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\frac{4J}{{k}_{B}{T}_{c}})$. But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature---a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field.
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 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 VhV_hVh​ in the second quantization representation corresponding to Ising spins interaction with the external magnetic field HHH can be represented in terms of single-subscript creation and anihilation Fermi operators in such a form that the operator VhV_hVh​ commutes with the operator P^≡(−1)S^\hat{P}\equiv(-1)^{\hat{S}}P^≡(−1)S^, where S^=∑mβm†βm\hat{S}= \sum_mβ^†_mβ_mS^=∑m​βm†​βm​ 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 (ℜh≠0)(\Re h\neq 0)(ℜh=0) is demonstrated.
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.