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arXiv:0806.2670

Thermal transport in the anisotropic Heisenberg chain with S = 1/2 and nearest-neighbor interactions

David L. Huber

Published Jun 16, 2008• 0 citations
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Abstract

The purpose of this note is to connect early work on thermal transport in spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and nearest-neighbor interactions that was based on a moment analysis of the Fourier transform of the energy density correlation function with subsequent studies that make use of thermal current correlation functions.

DOI
10.48550/arxiv.0806.2670

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The purpose of this note is to connect early work on thermal transport in spin-1/2 Heisenberg chains with uniaxial exchange anisotropy and nearest-neighbor interactions that was based on a moment analysis of the Fourier transform of the energy density correlation function with subsequent studies that make use of thermal current correlation functions.
Spin and energy transport in spin-\textonehalf{} anisotropic magnetic chains with nearest-neighbor interactions are studied in the high-temperature limit. The analysis is based on a calculation of the zeroth, second, and fourth moments of the Fourier transforms of the spin and energy density correlation functions. Spin diffusion is predicted for sufficiently small values of the wave vector, except in the XY\mathrm{XY}XY limit. Energy transport is nondiffusive for all values of the wave vector. Energy diffusion is predicted for chains with isotropic nearest-neighbor interactions and spin greater than \textonehalf{} as well as for spin-\textonehalf{} two-dimensional square lattices and three-dimensional simple and body-centered cubic lattices having isotropic nearest-neighbor interactions.
The thermal conductivity of a one dimensional magnetic lattice is calculated in the thermal current formalism. It is assumed that the spins are coupled by a nearest-neighbor ferromagnetic Heisenberg exchange interaction. The conductivity is shown to be proportional to a six-spin correlation function which is factored into a product of three two-spin correlation functions. The two-spin functions are evaluated in a semi-classical approximation assuming exponential decay in time. The rate of decay is taken to be proportional to the spin diffusion constant, which is obtained from the non-linear formalism of Bennett and Martin. The resulting expression for the conductivity scales as the product of the spin diffusion constant and the specific heat of a classical chain over the entire temperature range 0≪T≪∞. The significance of this result is discussed with reference to the approximations made in the calculation. An alternative treatment which is appropriate at low temperatures is also given. This latter approach leads to a conductivity which vanishes linearly with temperature in the boundary scattering region.
The thermal conductivity of a one dimensional magnetic lattice is calculated in the thermal current formalism. It is assumed that the spins are coupled by a nearest-neighbor ferromagnetic Heisenberg exchange interaction. The conductivity is shown to be proportional to a six-spin correlation function which is factored into a product of three two-spin correlation functions. The two-spin functions are evaluated in a semi-classical approximation assuming exponential decay in time. The rate of decay is taken to be proportional to the spin diffusion constant, which is obtained from the non-linear formalism of Bennett and Martin. The resulting expression for the conductivity scales as the product of the spin diffusion constant and the specific heat of a classical chain over the entire temperature range 0≪T≪∞. The significance of this result is discussed with reference to the approximations made in the calculation. An alternative treatment which is appropriate at low temperatures is also given. This latter approach leads to a conductivity which vanishes linearly with temperature in the boundary scattering region.
Measurements of the spin-lattice relaxation rate 1∕T11∕{T}_{1}1∕T1​ by nuclear magnetic resonance for the one-dimensional Heisenberg antiferromagnet Sr2CuO3{\mathrm{Sr}}_{2}\mathrm{Cu}{\mathrm{O}}_{3}Sr2​CuO3​ have provided evidence for a diffusionlike contribution at finite temperature and small wave vector. By analyzing real-time data for the auto- and nearest-neighbor spin-spin correlation functions obtained by the density-matrix renormalization group I show that such a contribution indeed exists for temperatures T>JT>JT>J, where JJJ is the coupling constant, but that it becomes exponentially suppressed for T⪡JT⪡JT⪡J. I present evidence that the frequency dependence of 1∕T11∕{T}_{1}1∕T1​ in the Heisenberg case is smoothly connected to that in the free fermion case where the exponential suppression of the diffusionlike contribution is easily understood.
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.