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)Δ=cos(γ) with \gamma = \pi n/mγ=πn/m, n\leq mn≤m integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general \gamma=\pi n/mγ=πn/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.
Highlights
The Hamiltonian of the XXZ chain is given by NH=J σlx σlx+1 + σly σly+1 + ∆σzl σzl+1 − 2h σzl, (1)l =1 l =1 where σx,y,z are Pauli matrices, ∆ = cos(γ) is the anisotropy, h the applied magnetic field, and we use periodic boundary conditions
Starting from (37) it is straightforward to show that our result is identical to the one obtained 20 years ago based on the Kohn formula and using the thermodynamic Bethe ansatz (TBA) to calculate the curvature of energy levels [4,5]
Using the Mazur bound and assuming that it becomes exhausted if one considers the full TBA particle content we derived a closed-form expression for the spin Drude weight
Summary
An obvious question is if the TBA formulas for the current and current-charge expectation values obtained in this way are exact To answer this question we will present in this paper a fourth approach where we derive current and current-charge correlators exactly starting from a generalized free energy and the operator expression for the spin current, without using the GHD conjecture. Where the rapidity density ∂θ p = 2πσ (ρ + ρh) and effective velocity v ≡ ∂θ /∂θ p are defined by the dressed derivatives with respect to the spectral parameter of the energy and the momentum This formula agrees with the conjectured general current formula used in GHD and appearing in Ref. This result agrees with Eq (13) and with Eq (26) provided we replace q0 with q1
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