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arXiv:cond-mat/0112236 · Physical review. B, Condensed matter

Exact time autocorrelation function of theN-spin classical Heisenberg equivalent neighbor model

Richard A. Klemm, Marco Ameduri

Published Jun 20, 2002• 7 citations
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Abstract

We reduce the autocorrelation function ${\mathcal{C}}_{11}(t)$ 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 ${\mathcal{C}}_{11}(t)$ is a simple function of ${(\mathrm{JT})}^{1/2}t,$ exhibiting strong frustration, but the ferromagnetic ${\mathcal{C}}_{11}(t)$ oscillates in a single mode, the frequency of which approaches $\mathrm{NJ}$ 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.

DOI
10.1103/physrevb.66.012403

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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 T\ensuremath→\ensuremath∞,C11(t)\ensuremath∝t\ensuremath−N\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},{\mathcal{C}}_{11}(t)\ensuremath{\propto}{t}^{\ensuremath{-}N}T\ensuremath→​\ensuremath∞,C11​(t)\ensuremath∝t\ensuremath−N for Jt\ensuremath≫1.\mathrm{Jt}\ensuremath{\gg}1.Jt\ensuremath≫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 T\ensuremath→0.\stackrel{\ensuremath{\rightarrow}}{T}0.T\ensuremath→​0. We conjecture that as T\ensuremath→\ensuremath∞,\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},T\ensuremath→​\ensuremath∞, the near-neighbor correlation functions of N-spin classical Heisenberg rings are simply obtained from these results.
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.
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 detailed series-extrapolation investigation of some critical properties ($T\ensuremath{\rightarrow}{T}_{c}+$) of the Heisenberg model is presented. Systematic use is made of ratio and Pad\'e-approximant techniques, some of the latter being new. Attention is confined to those nearest-neighbor and order-two equivalent-neighbor models which are based on the simple cubic, body-centered cubic, and face-centered cubic lattices. For the face-centered cubic nearest-neighbor classical Heisenberg model, the following estimates are obtained for familiar critical exponents: $\ensuremath{\alpha}=0, \ensuremath{-}\frac{1}{8}<~{\ensuremath{\alpha}}_{s}<~\ensuremath{-}\frac{1}{16}$, $\ensuremath{\gamma}=1.375\ifmmode\pm\else\textpm\fi{}0.002$, $0.6875<~\ensuremath{\nu}<~0.7125$, $0<~\ensuremath{\eta}<~0.07$. Estimates for $\ensuremath{\gamma}$ are also obtained for the other nearest-neighbor and equivalent-neighbor classical Heisenberg models, and it is conjectured that $\ensuremath{\gamma}=1\frac{3}{8}$ may be the exact result in all the cases considered. For the face-centered cubic order-two equivalent-neighbor spin-\textonehalf{} Heisenberg ferromagnet, it is estimated that $\ensuremath{\gamma}=1.3744\ifmmode\pm\else\textpm\fi{}0.0008$. A similar but less precise result is found for the corresponding simple cubic model, and the spin-\textonehalf{} nearest-neighbor susceptibility series are briefly examined. It is suggested that consideration should be given to the possibility that for ferromagnetic Heisenberg interactions $\ensuremath{\gamma}$ has the same value 1381\frac{3}{8}183​ irrespective of spin and the particular three-dimensional nearest-neighbor or finite-order equivalent-neighbor model considered. Other quantities estimated include various Curie points and singularity amplitudes and the critical values of certain thermodynamic functions. Some attention is given to the simple classical Heisenberg antiferromagnet.
We study the quantum spin-12\frac{1}{2}21​ Heisenberg model in two dimensions, interacting through a nearest-neighbor antiferromagnetic exchange (J)(J)(J) and a ferromagnetic dipolarlike interaction (Jd)({J}_{d})(Jd​), using double-time Green's function, decoupled within the random phase approximation. We obtain the dependence of kBTc∕Jd{k}_{B}{T}_{c}∕{J}_{d}kB​Tc​∕Jd​ as a function of frustration parameter $\ensuremath{\delta}$, where Tc{T}_{c}Tc​ is the ferromagnetic (F) transition temperature and $\ensuremath{\delta}$ is the ratio between the strengths of the exchange and dipolar interaction (i.e., $\ensuremath{\delta}=J∕{J}_{d}$). The transition temperature between the F and paramagnetic phases decreases with $\ensuremath{\delta}$, as expected, but goes to zero at a finite value of this parameter, namely, $\ensuremath{\delta}={\ensuremath{\delta}}_{c}=\ensuremath{\pi}∕8$. At T=0T=0T=0 (quantum phase transition), we analyze the critical parameter ${\ensuremath{\delta}}_{c}(p)$ for the general case of an exchange interaction in the form Jij=Jd∕rijp{J}_{ij}={J}_{d}∕{r}_{ij}^{p}Jij​=Jd​∕rijp​, where ferromagnetic and antiferromagnetic phases are present.
The dynamics of the classical XXZ model with uniform interaction is nonlinear for N≥2 spins and nonintegrable for N≥3. However, the nonlinearities disappear in the thermodynamic limit N→∞, and the spin autocorrelation functions can be determined exactly for infinite temperature. The function 〈Szi(t)Szi〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈Sxi(t)Sxi〉 decays algebraically to zero or like a Gaussian, depending on the type (easy axis or easy plane) and amount of uniaxial anisotropy.