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Physical Review B

Thermal conductivity of one-dimensional spin- 1∕2 systems coupled to phonons

Kim Louis, Peter Prelovšek, X. Zotos

Published Dec 22, 2006• 25 citations
Publisher

Abstract

Within the memory function approach we study the thermal conductivity of the one-dimensional spin-$1∕2$ Heisenberg, as well as $XY$ model coupled to optic and acoustic phonons using exact diagonalization with the Lanczos technique and analytical evaluation, respectively. We find that the thermal conductivity of the two models is fairly similar, the enhancement found in the Heisenberg model mainly due to the larger energy current correlations. We also observe that the Wiedemann-Franz law is not obeyed in the $XY$ model, presumably due to the inelastic scattering processes.

DOI
10.1103/physrevb.74.235118

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We study a one-dimensional anisotropic XXZ Heisenberg spin-12\frac{1}{2}21​ chain with weak random fields hizSiz{h}_{i}^{z}{S}_{i}^{z}hiz​Siz​ by means of Jordan-Wigner transformation to spinless Luttinger liquid with disorder and bosonization technique. First, we reinvestigate the phase diagram of the system in terms of dimensionless disorder $\ensuremath{\gamma}=\ensuremath{\langle}{h}^{2}\ensuremath{\rangle}/{J}^{2}\ensuremath{\ll}1$ and anisotropy parameter $\mathrm{\ensuremath{\Delta}}={J}_{z}/{J}_{xy}$, we find the range of these parameters where disorder is irrelevant in the infrared limit and spin-spin correlations are described by power laws, and compare it with previously obtained numerical and analytical results. Then we use the diagram technique in terms of plasmon excitations to study the low-temperature $(T\ensuremath{\ll}J)$ behavior of heat conductivity $\ensuremath{\kappa}$ and spin conductivity $\ensuremath{\sigma}$ in this power-law phase. The obtained Lorentz number $L\ensuremath{\equiv}\ensuremath{\kappa}/\ensuremath{\sigma}T$ differs from the value derived earlier by means of the memory function method. We argue also that in the studied region inelastic scattering is strong enough to suppress quantum interference in the low-temperature limit.
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.
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Using numerical diagonalization techniques, we explore the effect of local and bond disorder on the finite-temperature spin and thermal conductivities of the one-dimensional anisotropic spin-1/2 Heisenberg model. High-temperature results for local disorder show that the dc conductivities are finite, apart from the uncorrelated XYXYXY case where dc transport vanishes. Moreover, at strong disorder, we find finite dc conductivities at all temperatures TTT except at T=0T=0T=0. The low-frequency conductivities are characterized by a nonanalytic cusp shape. Similar behavior is found for bond disorder.