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Bulletin of the American Physical Society

Scaling of electrical and thermal conductivities in an almost integrable chain

Joel E. Moore, Christoph Karrasch, Roni Ilan

Published Mar 22, 2013• 0 citations
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Abstract

Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak nonintegrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless (XXZ-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.

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Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak nonintegrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless (XXZ-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.
Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak nonintegrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless (XXZXXZXXZ-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.
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.
Resta proposed a definition of the electric polarization in one-dimensional systems in terms of the ground-state expectation value of the large gauge transformation operator. Vanishing of the expectation value in the thermodynamic limit implies that the system is a conductor. We study Resta's polarization amplitude (expectation value) in the S=1/2S=1/2S=1/2 XXZ chain and its several generalizations, in the gapless conducting Tomonaga-Luttinger liquid phase. We obtain an analytical expression in the lowest-order perturbation theory about the free fermion point (XY chain) and an exact result for the Haldane-Shastry model with long-range interactions. We also obtain numerical results, mostly using the exact diagonalization method. We find that the amplitude exhibits a power-law scaling in the system size (chain length) and vanishes in the thermodynamic limit. On the other hand, the exponent depends on the model even when the low-energy limit is described by the Tomonaga-Luttinger liquid with the same Luttinger parameter. We find that a change in the exponent occurs when the Umklapp term(s) are eliminated, suggesting the importance of the Umklapp terms.
The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin-1/21/21/2 nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength ${J}_{\ensuremath{\perp}}$ leads to $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}$ as expected from simple golden-rule arguments, we obtain a much larger $\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}$ for a weak next-nearest-neighbor interaction ${J}^{\ensuremath{'}}$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ Heisenberg chain.
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.
Many fundamental one-dimensional lattice models such as the Heisenberg or the Hubbard model are integrable. For these microscopic models, parameters in the Luttinger liquid theory can often be fixed and parameter-free results at low energies for many physical quantities such as dynamical correlation functions obtained where exact results are still out of reach. Quantum integrable models thus provide an important testing ground for low-energy Luttinger liquid physics. They are, furthermore, also very interesting in their own right and show, for example, peculiar transport and thermalization properties. The consequences of the conservation laws leading to integrability for the structure of the low-energy effective theory have, however, not fully been explored yet. I will discuss the connection between integrability and Luttinger liquid theory here, using the anisotropic Heisenberg model as an example. In particular, I will review the methods which allow to fix free parameters in the Luttinger model with the help of the Bethe ansatz solution. As applications, parameter-free results for the susceptibility in the presence of nonmagnetic impurities, for spin transport, and for the spin-lattice relaxation rate are discussed.