The spin Drude weight of the XXZ chain and generalized hydrodynamics

TL;DR

This paper derives exact formulas for the spin Drude weight in the XXZ chain using thermodynamic Bethe ansatz and confirms their consistency with generalized hydrodynamics, providing numerical evaluations and high-temperature asymptotics.

Contribution

It provides the first exact derivation of the spin Drude weight formulas from a proper operator treatment, confirming and extending previous conjectures.

Findings

Exact TBA formulas for spin current and Drude weight

Numerical evaluation of Drude weight for specific anisotropies

High-temperature asymptotics match quasi-local charge bounds

Abstract

Based on a generalized free energy we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators, and consequently the Drude weight. These formulas agree with recent conjectures within the generalized hydrodynamics formalism. They follow, however, directly from a proper treatment of the operator expression of the spin current. The result for the Drude weight is identical to the one obtained 20 years ago based on the Kohn formula and TBA. We numerically evaluate the Drude weight for anisotropies $\Delta=\cos(\gamma)$ with $\gamma = n\pi/m$, $n\leq m$ integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general $\gamma=\pi n/m$---obtained by analysis of the quantum transfer matrix eigenvalues---agrees with the bound which has been obtained by the construction of quasi-local charges.