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arXiv:0909.4751 · Physical Review Letters

Temperature Correlations of Quantum Spins

Alexander Rudol'fovich Its, Anatoli G. Izergin, V. E. Korepin, N. A. Slavnov

Published Apr 12, 1993• 24 citations
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Abstract

This is a historical note. In 1993 we calculated space, time and temperature dependent correlation function in isotropic version of one dimensional XY spin chain. The correlation function decays exponentially with time and space separation. The rate of exponential decay was evaluated explicitly. Since that time similar results were obtained in other models: Bose gas with delta interaction, Ising model and strongly correlated electrons.

DOI
10.1103/physrevlett.70.2357

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This is a historical note. In 1993 we calculated space, time and temperature dependent correlation function in isotropic version of one dimensional XY spin chain. The correlation function decays exponentially with time and space separation. The rate of exponential decay was evaluated explicitly. Since that time similar results were obtained in other models: Bose gas with delta interaction, Ising model and strongly correlated electrons.
The long-time and long-distance asymptotic behavior of the xxx spin correlations at finite temperature in an anisotropic spin-1/2 XY chain is determined numerically. The decay of the correlations is exponential in both space and time. Similar exponential decay of correlations was already found earlier in the special case of the isotropic model, where analytical expressions for the decay rates could be derived via a mapping to a different model. While no such mapping is known for the anisotropic model, the asymptotic correlations can be very well approximated by a natural generalization of the known analytic results for the isotropic case.
The long-time and long-distance asymptotic behavior of the xxx spin correlations at finite temperature in an anisotropic spin-1/2 XY chain is determined numerically. The decay of the correlations is exponential in both space and time. Similar exponential decay of correlations was already found earlier in the special case of the isotropic model, where analytical expressions for the decay rates could be derived via a mapping to a different model. While no such mapping is known for the anisotropic model, the asymptotic correlations can be very well approximated by a natural generalization of the known analytic results for the isotropic case.
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.
We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.
The infinite-temperature space- and time-dependent spin-correlation functions grx(t){g}_{r}^{x}(t)grx​(t) are studied for the one-dimensional XY\mathrm{XY}XY model. Numerical calculations are performed to obtain the exact autocorrelation function g0x(t){g}_{0}^{x}(t)g0x​(t) for chains containing 5, 7, and 9 spins (S=12S=\frac{1}{2}S=21​). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that g0x(t){g}_{0}^{x}(t)g0x​(t) for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that grx(t){g}_{r}^{x}(t)grx​(t) for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for g0x(t){g}_{0}^{x}(t)g0x​(t) is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.