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arXiv:cond-mat/0509615 · Physical Review Letters

Transport in Almost Integrable Models: Perturbed Heisenberg Chains

Peter Jung, R. W. Helmes, Achim Rosch

Published Feb 14, 2006• 123 citations
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Abstract

The heat conductivity $\ensuremath{\kappa}(T)$ of integrable models, like the one-dimensional spin-$1/2$ 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.

DOI
10.1103/physrevlett.96.067202

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The heat conductivity \ensuremathκ(T)\ensuremath{\kappa}(T)\ensuremathκ(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⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​ leads to \ensuremathκ(T)\ensuremath∝1/J\ensuremath⊥2\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}_{\ensuremath{\perp}}^{2}\ensuremathκ(T)\ensuremath∝1/J\ensuremath⊥2​ as expected from simple golden-rule arguments, we obtain a much larger \ensuremathκ(T)\ensuremath∝1/J\ensuremath′4\ensuremath{\kappa}(T)\ensuremath{\propto}1/{J}^{\ensuremath{'}4}\ensuremathκ(T)\ensuremath∝1/J\ensuremath′4 for a weak next-nearest-neighbor interaction J\ensuremath′{J}^{\ensuremath{'}}J\ensuremath′. This can be explained by a new approximate conservation law of the J\ensuremath-J\ensuremath′J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}J\ensuremath-J\ensuremath′ Heisenberg chain.
The spin conductivity in the integrable spin-1∕21∕21∕2 XXZXXZXXZ chain is known to be infinite at finite temperatures TTT for anisotropies $\ensuremath{-}1<\ensuremath{\Delta}<1$. Perturbations, which break integrability, e.g., a next-nearest neighbor coupling ${J}^{\ensuremath{'}}$, render the conductivity finite. We construct numerically a nonlocal conserved operator ${J}_{\ensuremath{\parallel}}$ which is responsible for the finite spin Drude weight of the integrable model and calculate its decay rate for small ${J}^{\ensuremath{'}}$. This allows us to obtain a lower bound for the spin conductivity ${\ensuremath{\sigma}}_{s}\ensuremath{\geqslant}c(T)∕{{J}^{\ensuremath{'}}}^{2}$, where c(T)c(T)c(T) is finite for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}0$. We discuss the implication of our result for the general question how nonlocal conservation laws affect transport properties.
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.
We investigate the heat conductivity $\ensuremath{\kappa}$ of the Heisenberg spin-1/21/21/2 ladder at finite temperature covering the entire range of interchain coupling ${J}_{\ensuremath{\perp}}$, by using several numerical methods and perturbation theory within the framework of linear response. We unveil that a perturbative prediction $\ensuremath{\kappa}\ensuremath{\propto}{J}_{\ensuremath{\perp}}^{\ensuremath{-}2}$, based on simple golden-rule arguments and valid in the strict limit ${J}_{\ensuremath{\perp}}\ensuremath{\rightarrow}0$, applies to a remarkably wide range of ${J}_{\ensuremath{\perp}}$, qualitatively and quantitatively. In the large ${J}_{\ensuremath{\perp}}$ limit, we show power-law scaling of opposite nature, namely, $\ensuremath{\kappa}\ensuremath{\propto}{J}_{\ensuremath{\perp}}^{2}$. Moreover, we demonstrate the weak and strong coupling regimes to be connected by a broad minimum, slightly below the isotropic point at ${J}_{\ensuremath{\perp}}={J}_{\ensuremath{\parallel}}$. Reducing temperature TTT, starting from $T=\ensuremath{\infty}$, this minimum scales as $\ensuremath{\kappa}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ down to TTT on the order of the exchange coupling constant. These results provide for a comprehensive picture of $\ensuremath{\kappa}({J}_{\ensuremath{\perp}},T)$ of spin ladders.
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 consider a spin chain given by the XXZ model with a weak next-to-nearest-neighbor perturbation that breaks its exact integrability. We prove that such a system has an ideal metallic behavior (infinite conductivity), by rigorously establishing strict lower bounds on the zero-temperature Drude weight, which are strictly positive. The proof is based on exact renormalization group methods allowing us to prove the convergence of the expansions and to fully take into account the irrelevant terms, which play an essential role in ensuring the correct lattice symmetries. We also prove that the Drude weight verifies the same parameter-free relations as in the absence of the integrability-breaking perturbation.
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.