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arXiv:1112.2551 · Journal of Statistical Mechanics Theory and Experiment

Transport in a one-dimensional isotropic Heisenberg model at high temperature

Marko Žnidarič

Published Dec 12, 2011• 49 citations
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Abstract

Magnetization transport in a one-dimensional isotropic spin 1/2 Heisenberg model is studied. It is shown that in a nonequilibrium steady state at high temperature and constant small driving the magnetization current depends on the system length L as ∼ 1/ L 0.5 , meaning that the diffusion constant diverges as ∼ L 0.5 . Spectral properties of a superoperator governing the relaxation towards a nonequilibrium steady state are also discussed.

DOI
10.1088/1742-5468/2011/12/p12008

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Magnetization transport in a one-dimensional isotropic spin 1/2 Heisenberg model is studied. It is shown that in a nonequilibrium steady state at high temperature and constant small driving the magnetization current depends on the system length L as ∼ 1/ L 0.5 , meaning that the diffusion constant diverges as ∼ L 0.5 . Spectral properties of a superoperator governing the relaxation towards a nonequilibrium steady state are also discussed.
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The transport of magnetization is analyzed for the classical Heisenberg chain at and especially above the isotropic point. To this end, the Hamiltonian equations of motion are solved numerically for initial states realizing harmonic-like magnetization profiles of small amplitude and with random phases. Above the isotropic point, the resulting dynamics is observed to be diffusive in a hydrodynamic regime starting at comparatively small times and wave lengths. In particular, hydrodynamic regime and diffusion constant are both found to be in quantitative agreement with close-to-equilibrium results from quantum S =1/2 autocorrelations at high temperatures. At the isotropic point, the resulting dynamics turns out to be non-diffusive at the considered times and wave lengths.
The isotropic Heisenberg chain represents a particular case of an integrable many-body system exhibiting superdiffusive spin transport at finite temperatures. Here, we show that this model has distinct properties also at finite magnetization $m\ensuremath{\ne}0$, even upon introducing the SU(2) invariant perturbations. Specifically, we observe nonmonotonic dependence of the diffusion constant ${\mathcal{D}}_{0}(\mathrm{\ensuremath{\Delta}})$ on the spin anisotropy $\mathrm{\ensuremath{\Delta}}$, with a pronounced maximum at $\mathrm{\ensuremath{\Delta}}=1$. The latter dependence remains true also in the zero magnetization sector, with superdiffusion at $\mathrm{\ensuremath{\Delta}}=1$ that is remarkably stable against isotropic perturbation (at least in finite-size systems), consistent with recent experiments with cold atoms.
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We analytically and numerically study spin transport in a one-dimensional Heisenberg model in linear-response regime at infinite temperature. It is shown that as the anisotropy parameter Δ is varied spin transport changes from ballistic for Δ<1 to anomalous at the isotropic point Δ=1, to diffusive for finite Δ>1, ending up as a perfect isolator in the Ising limit of infinite Δ. Using perturbation theory for large Δ a quantitative prediction is made for the dependence of diffusion constant on Δ.
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.