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Journal of Physical Studies

Dynamic properties of spin-½ XY chains

Oleg Derzhko, Taras Krokhmalskii

Published Jan 1, 1998• 1 citation
Publisher

Abstract

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.

DOI
10.30970/jps.02.263

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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 paper presents a numerical approach for the calculation of thermodynamics and spin correlations for spin-½ XY chains. To illustrate the approach we performed exact finite-chain calculations of all time-dependent two-spin correlation functions for the spin-½ Ising chain in the transverse field. The computed correlation functions permit to evaluate the frequency-dependent susceptibilities and structure factors for this model. The obtained results are discussed in detail. They may have an application for the interpretation of dynamic experiments on quasi-one-dimensional crystals that can be treated within the frames of the spin-½ transverse Ising model.
Dynamic xx and xy spin pair correlation functions for the isotropic spin-½ XY chain are calculated numerically for long open chains in the presence of a transverse ( z ) magnetic field at finite temperature. From these data the first numerical results for the dynamic structure factors S xx (κ,ω) and S xy (κ,ω) are obtained and compared with the known results for S zz (κ,ω). While S zz (κ,ω) is restricted to certain regions in the (κ,ω)-plane S xx (κ,ω) and S xy (κ,ω) are not. Nevertheless, the numerical results show that the latter structure factors are quite small outside the regions defined by S zz (κ,ω) and moreover, that they may in many circumstances, especially at low temperature, be approximately described by a small number of broadened excitation branches.
One-dimensional quantum spin-1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors).This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions.A number of dynamic quantities (e.g.related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.We consider the spin-1/2 XY chain in a transverse ( z) magnetic field with the Hamiltonianand calculate the dynamic structure factorsis the dimer operator.The results for the dynamic transverse structure factor S zz (κ, ω) and for the dynamic dimer structure factor S DD (κ, ω) are known, whereas the analysis of the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been reported so far.We compare different two-fermion dynamic quantities contrasting their generic and specific features.