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physica status solidi (b)

One‐Dimensional Spin‐½ XY Models as Test for Methods in the Spin System Theory

Oleg Derzhko, Taras Verkholyak

Published Mar 1, 1997• 5 citations
Publisher

Abstract

The average density of states that determines the thermodynamic properties is calculated analytically and numerically for two random versions of the one-dimensional spin-½ XY model. The exact results obtained are compared with the results derived within some well-known approximate approaches that permits to understand the region of their validity.

DOI
10.1002/1521-3951(199703)200:1<255::aid-pssb255>3.0.co;2-#

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The average density of states that determines the thermodynamic properties is calculated analytically and numerically for two random versions of the one-dimensional spin-½ XY model. The exact results obtained are compared with the results derived within some well-known approximate approaches that permits to understand the region of their validity.
We have considered a numerical scheme for the calculation of the equilibrium properties of spin{ 1 2 XY chains.Within its frames it is necessary to solve in the last resort only the 2N 2N eigenvalue and eigenvector problem but not the 2 N 2 N one as for an arbitrary system consisting of N spins 1 2 .To illustrate the approach we have presented some new results.Namely, the xx dynamic structure factor for the Ising model in transverse eld, the density of states for the isotropic chain with random intersite couplings and transverse elds that linearly depend on the surrounding couplings, and the zz dynamic structure factor for the Ising model in the random transverse eld.The results obtained are hoped to be useful for an interpretation of observable data for one{dimensional spin{ 1 2 XY substances.
We have considered a numerical scheme for the calculation of the equilibrium properties of spin-1/2 XY chains. Within its frames it is necessary to solve in the last resort only the 2N\times 2N eigenvalue and eigenvector problem but not the 2^N\times 2^N one as for an arbitrary system consisting of N spins 1/2. To illustrate the approach we have presented some new results. Namely, the xx dynamic structure factor for the Ising model in transverse field, the density of states for the isotropic chain with random intersite couplings and transverse fields that linearly depend on the surrounding couplings, and the zz dynamic structure factor for the Ising model in random transverse field. The results obtained are hopped to be useful for an interpretation of observable data for one-dimensional spin-1/2 XY substances.
The one dimensional XY model of spin 1/2 is discussed by transforming the spin operators to the Fermi operators. This method can be applied to the XY model with an impurity spin. In that case a pair of localized exciton (or a kind of magnon) levels are obtained. The energy of states can be determined by distributing the excitons over two sets of exciton levels which depend on whether the number of excitons is even or odd. The energy levels of states are calculated by the computer for small rings of 4, 6 and 8 spins. The results can be perfectly understood by the theory.