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arXiv:hep-th/0106270 · Journal of Experimental and Theoretical Physics

The magnetic susceptibility of two-dimensional Ising model on a finite-size lattice

A. I. Bugrij, O. Lisovyy

Published Jun 1, 2002• 4 citations
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Abstract

The form factor representation of the correlation function of the 2D Ising model on a cylinder is generalized to the case of arbitrary arrangement of correlating spins. The magnetic susceptibility on a lattice, one of whose dimensions ( N ) is finite, is calculated in both the ferromagnetic and the paramagnetic region of the parameters of the model. The structure of singularities of susceptibility in the complex temperature plane at finite values of N and the transition to the thermodynamic limit N →∞ are discussed.

DOI
10.1134/1.1493166

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The form factor representation of the correlation function of the 2D Ising model on a cylinder is generalized to the case of arbitrary arrangement of correlating spins. The magnetic susceptibility on a lattice, one of whose dimensions ( N ) is finite, is calculated in both the ferromagnetic and the paramagnetic region of the parameters of the model. The structure of singularities of susceptibility in the complex temperature plane at finite values of N and the transition to the thermodynamic limit N →∞ are discussed.
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
The correlation function of two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor representation are obtained both for the ferromagnetic and paramagnetic regions of coupling constant. The peculiarities caused by the finite size of the system are discussed. The asymptotic behaviour of the corresponding quantities as functions of size is calculated. Some generalization of the obtained result is conjectured.
. A two-dimensional Ising model with nearest neighbour exchange interactions among electrons, in the field of a finite intensity, is investigated. By including interactions along a given direction into the structure of the transfer matrix, we pass over from the direct lattice space to the pseudo-spin space. A field dependent unitary transformation in the pseudo-spin space is applied in order to evaluate the partition function exactly. There are two important consequences of the general field dependence of the partition function: (a) The order parameter undergoes a finite discontinuity at the transition temperature; and (b) The induced susceptibility discovers a logarithmic singularity just below the transition temperature. Neither consequence seems to conform with an existing theory of a two-dimensional ferromagnet.
We investigate the complex-temperature singularities of the susceptibility of the 2D Ising model on a square lattice. From an analysis of low-temperature series expansions, we find evidence that, as one approaches the point u=u s =-1 (where u=e '4K ) from within the complex extensions of the FM or AFM phases, the susceptibility has a divergent singularity of the form X approximately A s '(1+u)(- gamma s ') with exponent gamma s '=3/2. The critical amplitude A s ' is calculated. Other critical exponents are found to be alpha s '= sigma s =0 and beta s = 1/4 , so that the scaling relation alpha s '+2 beta s + gamma s ' is satisfied. However, using exact results for beta s on the square, triangular, and honeycomb lattices, we show that universality is violated at this singularity: beta s is lattice-dependent. Finally, from an analysis of spin-spin correlation functions, we demonstrate that the correlation length and hence susceptibility are finite as one approaches the point u=-1 from within the symmetric phase. This is confirmed by an explicit study of high-temperature series expansions.
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.