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Journal of Applied Physics

The classical equivalent-neighbor X X Z model: Exact results for dynamic correlation functions

Jian-Min Liu, Gerhard H. Muller

Published May 1, 1990• 13 citations
Publisher

Abstract

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.

DOI
10.1063/1.345860

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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.
The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and \ensuremath{\infty} are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​(t)Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.
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.
In studies of dynamic correlation functions the focus is, in general, on their long-time asymptotic behavior and the singularity structure of the associated spectral densities. It turns out that the analysis of the same dynamic correlation functions from a quite different perspective is equally useful and revealing. The focus of our study is on the properties of spectral densities at high frequencies, specifically their decay law, expressible asΦ0■ω■ ∼exp■ − ω2■λ■, in terms of a characteristic exponent λ. That decay law governs the growth rate of the sequence of recurrents which determine the relaxation function in the continued-fraction representation. The value of λ contains valuable information on the underlying dynamical processes taking place in the system, which, in some sense, is complementary to that inferred from the singularity structure of spectral densities. A detailed study of the dynamics of various quantum and classical spin models has prompted us to introduce the concept of ‘‘universality class’’ for a classification of dynamical behavior on the basis of the characteristic exponent λ. For the equivalent-neighbor XYZ model1 we are able to demonstrate four different prototype universality classes: λ=0 (compact support), λ=1 (Gaussian decay), λ=2 (exponential decay), λ=3 (stretched exponential decay), which can be interpreted in terms of basic notions of classical dynamics. Finally, we present the exact T=∞ dynamic correlation functions for a two-sublattice Heisenberg antiferromagnet with uniform intersublattice interaction and zero intrasublattice interaction.
The dynamics of the classical two-sublattice XYZ model with uniform intersublattice interaction and zero intrasublattice interaction is completely integrable for arbitrary system sizes. This makes the system amenable to an exact analysis of dynamic correlation functions. Here we present some exact results for the case with isotropic interaction (XXX model). The dynamical properties of the two-sublattice XYZ model are compared with those of the equivalent-neighbor XYZ model and categorized into universality classes of dynamical behavior.
Dynamic spin correlation functions 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​〉 for the one-dimensional S=1/2 XX model H=-Jtsumi{\mathit{Jtsum}}_{\mathit{i}}Jtsumi​{Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Si+1x{\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}Si+1x​+Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Si{\mathit{S}}_{\mathit{i}}Si​ $_{+1}^{\mathit{y}}\mathrm{}}$ are calculated exactly for finite open chains with up to N=10 000 spins. Over a certain time range the results are free of finite-size effects and thus represent correlation functions of an infinite chain (bulk regime) or a semi-infinite chain (boundary regime). In the bulk regime, the long-time asymptotic decay as inferred by extrapolation is Gaussian at T=\ensuremath{\infty}, exponential at 0T\ensuremath{\infty}, and power law (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$) at T=0, in agreement with exact results. In the boundary regime, a power-law decay is obtained at all temperatures; the characteristic exponent is universal at T=0 (\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$) and at 0T\ensuremath{\infty}(\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$), but is site dependent at T=\ensuremath{\infty}. In the high-temperature regime (T/J\ensuremath{\gg}1) and in the low-temperature regime (T/J\ensuremath{\ll}1), crossovers between different decay laws can be observed in 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Sjx{\mathit{S}}_{\mathit{j}}^{\mathit{x}}Sjx​〉. Additional crossovers are found between bulk-tpe and boundary-type decay for i=j near the boundary, and between spacelike and timelike behavior for i\ensuremath{\ne}j.
Time-dependent spin-autocorrelation functions at T=\ensuremath{\infty} and (in particular) their spectral densities for the bulk spin and the boundary spin of the semi-infinite spin-1/2 XXZ model (with exchange parameters Jx{\mathit{J}}_{\mathit{x}}Jx​=Jy{\mathit{J}}_{\mathit{y}}Jy​\ensuremath{\equiv}J, Jz{\mathit{J}}_{\mathit{z}}Jz​) are investigated on the basis of (i) rigorous bounds in the time domain and (ii) a continued-fraction analysis in the frequency domain. We have found strong numerical evidence for spin diffusion in quantum spin models. For Jz{\mathit{J}}_{\mathit{z}}Jz​/J increasing from zero, the results of the short-time expansion indicate a change of the bulk-spin xx-autocorrelation function from Gaussian decay to exponential decay. The continued-fraction analysis of the same dynamic quantity signals a change from exponential decay to power-law decay as Jz{\mathit{J}}_{\mathit{z}}Jz​/J approaches unity and back to a more rapid decay upon further increase of that parameter. By contrast, the change in symmetry at Jz{\mathit{J}}_{\mathit{z}}Jz​/J=1 has virtually no impact on the bulk-spin zz-autocorrelation function (as expected). Similar contrasting properties are observable in the boundary-spin autocorrelation functions.