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arXiv:cond-mat/9906441

Finite temperature transport in integrable quantum many body systems

X. Zotos, Félix Naef, Peter Prelovšek

Published Jun 30, 1999• 1 citation
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Abstract

Recent developments in the analysis of finite temperature dissipationless transport in integrable quantum many body problems are presented. In particular, we will discuss: (i) the role played by the conservation laws in systems as the spin 1/2 Heisenberg chain and the one-dimensional Hubbard model, (ii) exact results obtained using the Bethe ansatz method on the long time decay of current correlations.

DOI
10.48550/arxiv.cond-mat/9906441

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Recent developments in the analysis of finite temperature dissipationless transport in integrable quantum many body problems are presented. In particular, we will discuss: (i) the role played by the conservation laws in systems as the spin 1/2 Heisenberg chain and the one-dimensional Hubbard model, (ii) exact results obtained using the Bethe ansatz method on the long time decay of current correlations.
In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the XXZ\mathit{XXZ}XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the XXZ\mathit{XXZ}XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.
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.
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.
In this Chapter, we present recent theoretical developments on the finite temperature transport of one dimensional electronic and magnetic quantum systems as described by a variety of prototype models. In particular, we discuss the unconventional transport and dynamic - spin, electrical, thermal - properties implied by the integrability of models as the spin-1/2 Heisenberg chain or Hubbard. Furthermore, we address the implication of these developments to experimental studies and theoretical descriptions by low energy effective theories.
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.
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.