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Physical review. B, Solid state

Spin dynamics for the one-dimensional XY model at infinite temperature

Amit Sur, David Jasnow, Irving J. Lowe

Published Nov 1, 1975• 113 citations
Publisher

Abstract

The infinite-temperature space- and time-dependent spin-correlation functions ${g}_{r}^{x}(t)$ are studied for the one-dimensional $\mathrm{XY}$ model. Numerical calculations are performed to obtain the exact autocorrelation function ${g}_{0}^{x}(t)$ for chains containing 5, 7, and 9 spins ($S=\frac{1}{2}$). 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 ${g}_{0}^{x}(t)$ for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that ${g}_{r}^{x}(t)$ for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for ${g}_{0}^{x}(t)$ is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.

DOI
10.1103/physrevb.12.3845

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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 exp(\ensuremath−J2t2)4\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}4exp(\ensuremath−J2t2)​. We show that grx(t){g}_{r}^{x}(t)grx​(t) for r\ensuremath≠0r\ensuremath{\ne}0r\ensuremath=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∞S=\ensuremath{\infty}S=\ensuremath∞) chain.
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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.
We present new results on the time-dependent correlation functions ${\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉$, $\ensuremath{\xi}=x,y$, at zero temperature of the one-dimensional S=12S=\frac{1}{2}S=21​ isotropic XY\mathrm{XY}XY model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$, m=0,1,2,3,…m=0,1,2,3,\dots{}m=0,1,2,3,…, where ${\ensuremath{\omega}}_{0}=J$ for the XY\mathrm{XY}XY model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the XY\mathrm{XY}XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the XY\mathrm{XY}XY-like substances Cs2{\mathrm{Cs}}_{2}Cs2​CoCl4{\mathrm{Cl}}_{4}Cl4​ and PrCl3{\mathrm{Cl}}_{3}Cl3​ and the S=12S=\frac{1}{2}S=21​ Ising-like substance CsCoCl3{\mathrm{Cl}}_{3}Cl3​${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.
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