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physica status solidi (b)

The ground state for Ising model ±J on the tube

M. Błaszyk

Published Mar 1, 2003• 1 citation
Publisher

Abstract

Abstract The Ising model with exchange integrals ±J for a square lattice spanned over infinite length tubes of the width of 2, 3, or 4 bonds is considered. The paper gives a short description of a method of linear equations allowing a reduction of a physical problem of determination of e.g. energy, entropy, magnetisation for systems with frozen disorder, to a solution of a system of linear equations. The results for energy, entropy and magnetisation possible for a short presentation are given.

DOI
10.1002/pssb.200301708

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Abstract The Ising model with exchange integrals ±J for a square lattice spanned over infinite length tubes of the width of 2, 3, or 4 bonds is considered. The paper gives a short description of a method of linear equations allowing a reduction of a physical problem of determination of e.g. energy, entropy, magnetisation for systems with frozen disorder, to a solution of a system of linear equations. The results for energy, entropy and magnetisation possible for a short presentation are given.
For pt.II see ibid., vol.15, p.1693 (1982). The methods developed by Williams (1981) to describe the ground-state properties of a random-bond infinite Ising chain are extended to apply to a system of finite-length random-bond Ising chains. It is shown how to evaluate the magnetisation, ground-state energy, and entropy as a function of magnetic field. It is shown that the steps in the magnetisation and entropy when B=2J/r behave differently for r even and r odd. The magnetisation and entropy remain finite as B becomes very small.
A. Mi cki ewi cz Uni versi ty U multowsk a 85, 61-61 4 Po zna¥, Pol an d ( Recei ved Ma y 14, 2003) A m et hod of l inear equation s is prop osed allo win g a redu ction of a ph ysical pro blem of determination of energy , entropy , or magnetisati on for the systems w ith fro zen disorder to a mathematical problem of solvin g a set of linear equations .A part from an exact metho d for determination of energy and entropy , a typ e of the mean Ùeld approach is presented, which p ermits a summation over series representing entropy .Moreo ver, it was establishe d that magnetisati on can app ear only w hen the antif erromagnetic integrals occur at the zero probabil ity .PAC S numb ers: 05.50.+ q, 75.10.N r, 75.50.L k
A general technique is presented which allows site-bond disorder problems in Ising systems to be mapped onto a certain class of decorated lattice problems. The specific case of site-bond disorder on the honeycomb lattice is studied in detail through the use of annealed model solutions of the corresponding decoration problems. The critical temperature and magnetisation as functions of bond and site concentrations are calculated. The zero-temperature magnetisation is used to obtain an approximation to the percolocation probability of the system.
Small Ising lattices with both ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) and increasing numbers of spins are studied by means of two independent methods: computational solutions to the Hamiltonian problem and topological counting of frustration paths. Equal magnitudes and concentrations are assumed for both types of bonds. Two different geometries are considered: square lattices (SL's) with coordination number 4 and triangular lattices (TL's) with coordination number 6. Two-dimensional samples with a total number of spins N between 4 and 64 are considered for SL's, while N is varied between 4 and 44 for TL's. They are distributed in two-dimensional arrays where periodic boundary conditions are imposed. After an array is selected, bond distributions (samples) are independently and randomly generated in fixed positions. The physical parameters are then calculated exactly for each sample. The emphasis here is on the ground-state properties and their dependence with size and shape for the two kinds of lattices. All magnitudes correspond to a basic statistics over a large number of samples for each array. The following magnitudes are reported: ground-state energy per bond, frustration segment, abundance of first excited states, remnant entropy, low-temperature specific heat, and site order parameters q, p, and h. Parameters p and h are introduced here, showing advantages over other similar magnitudes. The results are in good correspondence with analytic studies for the thermodynamic limit. This means that the spin site correlation (p) tends to vanish as N grows. However, we have found that the shape dependence modulates the behavior of these systems toward the thermodynamic limit. There is no tendency to vanish for the bond correlation parameter (h). For both kinds of lattices h might be a constant independent of size and shape.