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arXiv:cond-mat/9911447 · Physical review. B, Condensed matter

Spin-correlation functions and susceptibilities in the easy-plane XXZ chain

C. Schindelin, Holger Fehske, H. Büttner, Dieter Ihle

Published Nov 1, 2000• 9 citations
arXivHTMLPDF

Abstract

We present a Green's-function theory of magnetic short-range order in the $S=1/2$ easy-plane $\mathrm{XXZ}$ chain based on the projection method for the dynamic spin susceptibility and a decoupling of three-spin operator products introducing vertex parameters. The longitudinal and transverse static susceptibilities and two-point correlation functions of arbitrary range are calculated self-consistently for all wave numbers, temperatures, and anisotropy parameters $\ensuremath{-}1<~\ensuremath{\Delta}<~1.$ In the easy-plane ferromagnetic region $(\ensuremath{\Delta}<0),$ the longitudinal correlators of spins at distance n change sign at a finite temperature ${T}_{0}(n,\ensuremath{\Delta}),$ in reasonable agreement with recent data obtained by finite-chain diagonalizations. The temperature dependence of the uniform static susceptibilities exhibits a maximum which is explained as an effect of magnetic short-range order which decreases with increasing temperature.

DOI
10.1103/physrevb.62.12141

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Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over two, three and four lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime −1<Δ<1. It turns out that the new formulae are numerically efficient and allow us to obtain the correlation functions over the full parameter range with arbitrary precision.
Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime −1 < ∆ < 1. It turns out that the new formulae are numerically efficient and allow us to obtain the correlations functions over the full parameter range with arbitrary precision.
The one-dimensional spin-1/2 Ising-Heisenberg ferromagnets are studied by the Green's function method at finite temperatures. The Green's function is decoupled in terms of the correlation functions of 〈S0z{\mathit{S}}_{0}^{\mathit{z}}S0z​Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉 and 〈S0+{\mathit{S}}_{0}^{+}S0+​${\mathit{S}}_{\mathit{n}}^{\mathrm{\ensuremath{-}}}$〉 under the constraint of 〈Sz{\mathit{S}}^{\mathit{z}}Sz〉=0, from which a set of self-consistent equations are derived and solved numerically. The thermodynamic properties such as internal energy, specific heat, and zero-field parallel susceptibility are calculated. Our results predict the correct behavior for the correlation functions and thermodynamic quantities in the entire temperature region, in agreement both with the exact numerical results on the finite chain and the experimental data.
The static two- and four-spin correlations for the one-dimensional easy-plane Heisenberg spin system with a ferromagnetic coupling, like CsNiF 3 , are investigated in the classical limit by means of the transfer matrix method. They are found to consist essentially of two components which both depend exponentially on the distance between sites. Both in the high- and low-temperature regions, only one component remains and this component still dominates at intermediate temperatures. Analytic forms for the correlation functions are given for asymptotically low temperatures. These are obtained by calculating the second-frequency moment of the dynamical structure factor in three different ways, and by requiring the T=0 limit of the moment to coincide with the square of the spin wave frequency. Generalisations of the Tomita-Mashiyama recursion relations (1972) for the isotropic case are given, valid for low temperatures, Monte Carlo calculations of the correlation functions have also been performed and some results of these are presented as well.
We calculate the spin-spin correlations and the magnetic susceptibility of the quantum one- and two-dimensional XY\mathrm{XY}XY models with S=1,S=1,S=1, using the two-time Green's function method and performing a decoupling proposed by Kondo and Yamaji for cases with no long-range order. A set of self-consistent equations of the correlation functions are derived and solved numerically. We present results for the correlation functions, susceptibilities, and specific heat for the whole range of temperature.