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Physical review. E

Energy and spin diffusion in the one-dimensional classical Heisenberg spin chain at finite and infinite temperatures

Nianbei Li

Published Dec 2, 2019• 15 citations
Publisher

Abstract

The energy and spin diffusion behaviors in the one-dimensional classical Heisenberg spin chain have been systematically investigated using the equilibrium diffusion method. The spatiotemporal autocorrelation functions for energy and spin are calculated at finite and infinite temperatures. As conserved quantities, the spreading of excess energy and spin can be used to determine their actual diffusion behaviors. At low temperatures, the energy diffusion shows almost ballistic behavior, and spin shows superdiffusion behavior for finite chain size. For energy diffusion, normal diffusion behavior can be obtained when the temperature is higher than 0.75. For spin diffusion, normal diffusion behavior is observed at infinite temperature.

DOI
10.1103/physreve.100.062104

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The energy and spin diffusion behaviors in the one-dimensional classical Heisenberg spin chain have been systematically investigated using the equilibrium diffusion method. The spatiotemporal autocorrelation functions for energy and spin are calculated at finite and infinite temperatures. As conserved quantities, the spreading of excess energy and spin can be used to determine their actual diffusion behaviors. At low temperatures, the energy diffusion shows almost ballistic behavior, and spin shows superdiffusion behavior for finite chain size. For energy diffusion, normal diffusion behavior can be obtained when the temperature is higher than 0.75. For spin diffusion, normal diffusion behavior is observed at infinite temperature.
The problem of spin diffusion is studied numerically in the one-dimensional classical Heisenberg model using a deterministic odd-even spin precession dynamics. We demonstrate that spin diffusion in this model is normal in the infinite-temperature limit and one obtains a long-time diffusive tail in the decay of the autocorrelation function. Some variations of the model with different coupling schemes and anisotropy are also studied and we find normal diffusion in all of them. A systematic finite-size analysis also suggests normal diffusive spreading of spin fluctuations, contrary to previous claims of anomalous diffusion. We also briefly discuss spin diffusion in this model at finite temperatures.
We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of DS{D}_{S}DS​ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures kBT∣J∣=1.0and0.5\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5∣J∣kB​T​=1.0and0.5. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.
The maximum Lyapunov exponent (MLE) is evaluated as a function of temperature in the isotropic Heisenberg chain. At low temperatures the MLE varies almost quadratically with temperature; it corresponds closely with ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$, the rate at which the local self-correlation decays to half its initial value. At higher temperatures, the MLE saturates at a value close to 1/2 in the limit of large chain lengths. The strong stochasticity threshold (defined by the change of slope of the MLE) parallels closely the transition from predominantly ballistic to predominantly diffusive behavior of the self-correlation and the concomitant steep increase in ${\mathrm{\ensuremath{\tau}}}_{1\mathrm{/}2}^{\mathrm{\ensuremath{-}}1}$. The complete Lyapunov spectrum has been derived for a chain of 18 spins; deviations from linearity occurring at infinite temperature suggest that the chaoticity of the system is incomplete. Finally, it is suggested that a systematic study of finite-size effects might be useful in deciding the issue of anomalous versus conventional spin diffusion.
We investigate transport in several translationally invariant spin-12\frac{1}{2}21​ chains in the limit of high temperatures. We concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. Our approach is essentially based on a connection between the evolution of the variance of an inhomogeneous nonequilibrium density and the current-autocorrelation function at finite times. Although this relationship is not restricted to the case of diffusive transport, it allows to extract a quantitative value for the diffusion constant in that case. By means of numerically exact diagonalization we indeed observe diffusive behavior in the considered spin chains for a range of model parameters and confirm the diffusion coefficients which were obtained for these systems from nonequilibrium bath scenarios.
We study the spin and energy dynamics in one-dimensional spin-12\frac{1}{2}21​ systems induced by local quantum quenches at finite temperatures using a time-dependent density matrix renormalization group method. System sizes are chosen large enough to ensure that the time-dependent data for the accessible time scales represent the behavior in the thermodynamic limit. As a main result, we observe a ballistic spreading of perturbations of the energy density in the integrable spin-12\frac{1}{2}21​ XXZ chain for all temperatures and exchange anisotropies, related to the divergent thermal conductivity in this model and the exact conservation of the energy current. In contrast, the spin dynamics is ballistic in the massless phase, but shows a diffusive behavior at high temperatures in the easy-axis phase in the case of a vanishing background spin density. We extract a quantitative estimate for the spin-diffusion constant from the time dependence of the spatial variance of the spin density, which agrees well with values obtained from current-current correlation functions using an Einstein relation. Interestingly, the diffusion constant approaches a constant value deep in the easy-axis regime. As an example for nonintegrable models, we consider two-leg ladders, for which we observe indications of diffusive energy and spin dynamics. The relevance of our results for recent experiments with quantum magnets and bosons in optical lattices is discussed.