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Physical review. B, Condensed matter

Simple quantum model for correlations in random one-dimensional Heisenberg antiferromagnets

Hans A. De Raedt, Ad Lagendijk

Published Jul 1, 1981• 4 citations
Publisher

Abstract

Using a simple quantum model we calculate static correlations for the random $S=\frac{1}{2}$ Heisenberg antiferromagnets. The correlations decay exponentially with distance. In addition the second moment of a two-spin correlation function was evaluated. No agreement with experiment could be obtained when we tried to extract the correlation time of this correlation function from its second moment.

DOI
10.1103/physrevb.24.463

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Using a simple quantum model we calculate static correlations for the random S=12S=\frac{1}{2}S=21​ Heisenberg antiferromagnets. The correlations decay exponentially with distance. In addition the second moment of a two-spin correlation function was evaluated. No agreement with experiment could be obtained when we tried to extract the correlation time of this correlation function from its second moment.
We calculate the ground-state two-spin correlation functions of spin-12\frac{1}{2}21​ quantum Heisenberg chains with random exchange couplings using the real-space renormalization-group scheme. We extend the conventional scheme to take account of the contribution of local higher multiplet excitations in each decimation step. This extended scheme can provide highly accurate numerical data for large systems. The random average of staggered spin correlations of the chains with random antiferromagnetic (AF) couplings shows algebraic decay like 1/r2,{1/r}^{2},1/r2, which verifies the Fisher's analytic results. For chains with random ferromagnetic (FM) and AF couplings, the random average of generalized staggered correlations is found to decay more slowly than a power law, in the form close to 1/ln(r).1/\mathrm{ln}(r).1/ln(r). The difference between the distribution functions of the spin correlations of the random AF chains and of the random FM-AF chains is also discussed.
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.
The static two- and four-spin correlations for one-dimensional easy-axis Heisenberg systems are investigated in the classical limit by means of the transfer matrix method. Both in the high- and low-temperature regions they are found to depend exponentially on the distance between sites. Thus generalisations of the Tomita-Mashiyama recursion relations for the isotropic case can be given. Analytic forms for several types of the correlation functions are obtained for asymptotically low temperatures. These are derived by discussing properties of the second frequency moment of the dynamical structure factor. The dynamical structure factor for the axial components, unlike in the easy-plane systems, should be of non-vanishing width at T=0.
We compute the ground-state spin-spin correlation function for the one-dimensional antiferromagnetic Heisenberg model of spin 1/2, 1, and 3/2. The spin-1 chain has a disordered ground state with a correlation length of 6.2 lattice spacings whereas the spin-1/2 and the spin-3/2 chains both have (ln(r)${)}^{\mathrm{\ensuremath{\sigma}}}$/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.
The dynamic-spin-correlation function S(q,\ensuremath{\omega}) is computed for the one-dimensional antiferromagnetic Heisenberg model for S=1/2 and S=1. Imaginary-time correlation functions are evaluated with the quantum Monte Carlo method and the maximum-entropy method is used to continue these results to real frequencies. This technique produces a value for the S=1 Haldane gap that agrees with previous work and also gives a good description of inelastic neutron scatter- ing data for the one-dimensional S=1/2 antiferromagnet CuCl2{\mathrm{CuCl}}_{2}CuCl2​\ensuremath{\cdot}2N(C5{\mathrm{C}}_{5}C5​D5{\mathrm{D}}_{5}D5​).
The decay of (disorder-averaged) static spin correlation functions at T=0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling JΔ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At Δ=0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N=100 sites. At Δ≠0 Lanczos diagonalizations are carried out for chains with up to N=22 sites. The longitudinal correlation function 〈Sz0Szr〉 is found to exhibit a power-law decay with an exponent that varies with Δ and, for nonzero Δ, also with the width of the λi-distribution. The results for the transverse correlation function 〈Sx0Sxr〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on.