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arXiv:cond-mat/0307578 · AIP conference proceedings

Low Temperature Transport Properties of Strongly Interacting Systems — Thermal Conductivity of Spin-1/2 Chains

N. Andrei

Published Jan 1, 2003• 2 citations
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Abstract

We outline a general approach to the computation of transport properties of interacting systems at low temperetures and frequencies. We show that if the fixed point and the irrelevant operators around it are known, then by studying the structure of the softly violated conserved currents chracterizing the fixed point one may set up an effective calculation in terms of a memory matrix formalism. We apply this approach to the computation of thermal conductivity of spin chains embedded in a matter matrix and interacting with its phonons. The results are found to be in very good agreement with experiment.

DOI
10.1063/1.1639599

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We outline a general approach to the computation of transport properties of interacting systems at low temperetures and frequencies. We show that if the fixed point and the irrelevant operators around it are known, then by studying the structure of the softly violated conserved currents chracterizing the fixed point one may set up an effective calculation in terms of a memory matrix formalism. We apply this approach to the computation of thermal conductivity of spin chains embedded in a matter matrix and interacting with its phonons. The results are found to be in very good agreement with experiment.
We study the low-temperature transport properties of clean one-dimensional spin-12\frac{1}{2}21​ chains coupled to phonons. Due to the presence of approximate conservation laws, the heat current decays very slowly giving rise to an exponentially large heat conductivity, $\ensuremath{\kappa}\ensuremath{\sim}{e}^{{T}^{*}/T}.$ As a result of an interplay of umklapp scattering and spinon-phonon coupling, the characteristic energy scale T∗{T}^{*}T∗ turns out to be of order ${\ensuremath{\Theta}}_{D}/2,$ where ${\ensuremath{\Theta}}_{D}$ is the Debye energy, rather than the magnetic exchange interaction JJJ---in agreement with recent measurements in SrCuO compounds. A large magnetic field h strongly affects the heat transport by two distinct mechanisms. First, it induces a linear spinon-phonon coupling, which alters the nature of the $\stackrel{\ensuremath{\rightarrow}}{T}0$ fixed point: the elementary excitations of the system are composite spinon-phonon objects. Second, the change of the magnetization and the corresponding change of the wave vector of the spinons strongly affects the way in which various umklapp processes can relax the heat current, leading to a characteristic fractal-like spiky behavior of $\ensuremath{\kappa}(T,h)$ as a function of h.
We set up a framework in which controlled lower bounds to conductivities of strongly correlated quantum systems may be calculated perturbatively in situations where the conductivities are infinite in the unperturbed model. This framework is then applied to study the transport properties of almost integrable systems, in general, and heat and spin conductivities in the integrable XXZ Heisenberg spin chain perturbed by various integrability breaking couplings, in particular. Furthermore, we systematically develop a microscopic description of the contribution of spin chains to experimentally measured heat conductivity in spin chain compounds, with an emphasis on the effect of weak disorder.
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The anomalous thermal conductivity in spin chains observed in experiments is studied for the low-temperature regime. In the effective dynamics with most realistic perturbations, the so-called umklapp terms are irrelevant to reduce the mean free path in the energy transport at even finite temperatures. This is consistent with the large conductivities found in recent experiments. The Drude weight which is the prefactor in the divergent conductivity is calculated, and the temperature dependence is discussed.
We study a class of integrable alternating ( S 1 , S 2 ) quantum spin chains with critical ground state properties. Our main result is the description of the thermal Drude weight of the one-dimensional alternating spin chain as a function of temperature. We have identified the thermal current of the model with alternating spins as one of the conserved currents underlying the integrability. This allows for the derivation of a finite set of non-linear integral equations for the thermal conductivity. Numerical solutions to the integral equations are presented for specific cases of the spins S 1 and S 2 . In the low-temperature limit a universal picture evolves where the thermal Drude weight is proportional to temperature T and central charge c .
Motivated by recent investigations of transport properties of strongly correlated 1D models and thermal conductivity measurements of quasi 1D magnetic systems, we present results for the integrable spin-½ XXZ chain. The thermal conductivity κ(ω) of this model has ℜ κ(ω) = δ(ω), i.e. it is infinite for zero frequency ω. The weight of the delta peak is calculated exactly by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy are presented. The low- and high-temperature limits are studied analytically.