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arXiv:cond-mat/0302007 · Journal of Physics A Mathematical and General

Exact time correlation functions forNclassical Heisenberg spins in the squashed equivalent neighbour model

Marco Ameduri, Richard A. Klemm

Published Jan 9, 2004• 2 citations
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Abstract

We present exact integral representations of the time-dependent spin-spin correlation functions for the classical Heisenberg N-spin `squashed' equivalent neighbor model, in which one spin is coupled via the Heisenberg exchange interaction with strength $J_1$ to the other N-1 spins, each of which is coupled via the Heisenberg exchange coupling with strength $J_2$ to the remaining N-2 spins. At low temperature T we find that the N spins oscillate in four modes, one of which is a central peak for a semi-infinite range of the values of the exchange coupling ratio. For the N=4 case of four spins on a squashed tetrahedron, detailed numerical evaluations of these results are presented. As $T\to\infty$, we calculate exactly the long-time asymptotic behavior of the correlation functions for arbitrary N, and compare our results with those obtained for three spins on an isosceles triangle.

DOI
10.1088/0305-4470/37/4/001

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We present exact integral representations of the time-dependent spin-spin correlation functions for the classical Heisenberg N-spin `squashed' equivalent neighbor model, in which one spin is coupled via the Heisenberg exchange interaction with strength J1J_1J1​ to the other N-1 spins, each of which is coupled via the Heisenberg exchange coupling with strength J2J_2J2​ to the remaining N-2 spins. At low temperature T we find that the N spins oscillate in four modes, one of which is a central peak for a semi-infinite range of the values of the exchange coupling ratio. For the N=4 case of four spins on a squashed tetrahedron, detailed numerical evaluations of these results are presented. As T→∞T\to\inftyT→∞, we calculate exactly the long-time asymptotic behavior of the correlation functions for arbitrary N, and compare our results with those obtained for three spins on an isosceles triangle.
We reduce the autocorrelation function C11(t){\mathcal{C}}_{11}(t)C11​(t) of the equivalent neighbor model of N classical spins exhibiting Heisenberg dynamics and exchange coupling J to quadrature. As the temperature $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},{\mathcal{C}}_{11}(t)\ensuremath{\propto}{t}^{\ensuremath{-}N}$ for $\mathrm{Jt}\ensuremath{\gg}1.$ At low T, the antiferromagnetic C11(t){\mathcal{C}}_{11}(t)C11​(t) is a simple function of (JT)1/2t,{(\mathrm{JT})}^{1/2}t,(JT)1/2t, exhibiting strong frustration, but the ferromagnetic C11(t){\mathcal{C}}_{11}(t)C11​(t) oscillates in a single mode, the frequency of which approaches NJ\mathrm{NJ}NJ as $\stackrel{\ensuremath{\rightarrow}}{T}0.$ We conjecture that as $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},$ the near-neighbor correlation functions of N-spin classical Heisenberg rings are simply obtained from these results.
Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function G(t)G(t)G(t) for a linear chain of NNN spins (S=12S=\frac{1}{2}S=21​) interacting by nearest-neighbor Heisenberg exchange for N=5,7,9and11N=5, 7, 9 \mathrm{and} 11N=5,7,9and11 by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for M22{M}_{22}M22​ to M30{M}_{30}M30​ of the frequency autocorrelation function for the infinite chain are provided. Excellent agreement is obtained with some recent results obtained by Morita, who, however, gives terms only up to M10{M}_{10}M10​. G(t)G(t)G(t) does not show a simple monotonic diffusive ($\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}$) behavior within $\frac{4\ensuremath{\hbar}}{J}$, the time domain up to which we believe our result for N=11N=11N=11 to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at t=0t=0t=0.
A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.
The two-time Green's function method is applied to the modified quantum Heisenberg model with the following exchange interactions: The interactions decay as inverse square of the distance between sites and have alternate positive and negative signs with increasing distance between sites; moreover, they work well only along the coordinate axes. The critical temperature and zero-temperature spin reduction are calculated for the one-, two- and three-dimensional cases. The temperature dependence of spin reductions in two and three dimensions is estimated in the low-temperature region. The critical behavior of long-range order and susceptibility is also investigated. Lastly, it is attempted to estimate the long-distance behavior of static correlation function in one dimension.
Various static spin-correlation functions and the weights of the spin excitations, or collective normal modes, in the van Hove response function, observed in neutron scattering experiments, are calculated as a function of temperature for a model quantum-spin chain. The spins interact through an isotropic Heisenberg exchange that extends to nearest-neighbour spins. In our model, spins of magnitude 1/2 form a dimerized chain with alternating exchange interactions. We explore the static and dynamic properties for ferromagnetic and antiferromagnetic exchange interactions. Results for the various properties are shown to be exact in the limit of a high temperature, and we argue that the results are very good at low temperatures. Unlike the case for linear spin-wave theory, the results are also exact in the limit of strong dimerization, i.e. non-interacting coupled pairs of spins.
We focus on the transition from quantum to classical behavior in thermodynamic functions and time correlation functions of a system consisting of three identical quantum spins s that interact via isotropic Heisenberg exchange. The partition function and the zero-field magnetic susceptibility are readily shown to adopt their classical forms with increasing s. The behavior of the spin autocorrelation function (ACF) is more subtle. Unlike the classical Heisenberg trimer where the ACF approaches a unique non-zero limit for long times, for the quantum trimer the ACF is periodic in time. We present exact values of the time average over one period of the quantum trimer for s⩽7 and for infinite temperature. These averages differ from the long-time limit, (940)ln3+(730), of the corresponding classical trimer by terms of order 1/s2. However, upon applying the Levin u-sequence acceleration method to our quantum results we can reproduce the classical value to six significant figures.