Diffusion in generalized hydrodynamics and quasiparticle scattering

Abstract

We extend beyond the Euler scales the hydrodynamic theory for quantum and classical integrable models developed in recent years, accounting for diffusive dynamics and local entropy production. We review how the diffusive scale can be reached via a gradient expansion of the expectation values of the conserved fields and how the coefficients of the expansion can be computed via integrated steadystate two-point correlation functions, emphasising that PT -symmetry can fully fix the inherent ambiguity in the definition of conserved fields at the diffusive scale. We develop a form factor expansion to compute such correlation functions and we show that, while the dynamics at the Euler scale is completely determined by the density of single quasiparticle excitations on top of the local steady state, diffusion is due to scattering processes among quasiparticles, which are only present in truly interacting systems. We then show that only two-quasiparticle scattering processes contribute to the diffusive dynamics. Finally we employ the theory to compute the exact spin diffusion constant of a gapped XXZ spin−1/2 chain at finite temperature and half-filling, where we show that spin transport is purely diffusive.

High-temperature spin dynamics in the Heisenberg chain: Magnon propagation and emerging Kardar-Parisi-Zhang scaling in the zero-magnetization limit

“…ings for the spin correlator. On the other hand, it is expected that, for more generic non-equilibrium situations, diffusive dynamics [7] will eventually dominate over the dispersive t 1/3 scaling in the long-time limit. This expectation is indeed confirmed by our numerical results: Upon increasing m, we observe that the features of F (y) close to the light-cone are increasingly washed out.…”

“…For anisotropy ∆ > 1, the spin dynamics at vanishing magnetization m = 0 is known to be normal diffusive [7]. In the long-time limit, it is therefore expected that ∂ t ∆x 2 (t) t→∞ = 2Dt with D denoting the diffusion constant.…”

“…They feature kinetic equations for generalized phase-space distributions, akin to the theory of classical soliton gases [4,5]. Diffusive corrections and entropy-production due to quasi-particle scattering have been incorporated recently [6,7]. Moreover, quantum hydrodynamics for one-dimensional systems at zero temperature developed earlier [8,9], formulated in terms of density and velocity fields, was shown to be reproduced by GHD in the corresponding limit [10].…”

Current operators in integrable spin chains: lessons from long range deformations

“…The mean values of the current operators in the eigenstates are not affected by this transformation, but the off-diagonal matrix elements do change. This has some consequences for the intermediate computations for the diffusive corrections in Generalized Hydrodynamics, which is discussed in detail in [28]. The general form of the charge densities and the relation (3.14) imply that the current operators also enjoy the SU (2)-invariance and they can also be expressed using exchange operators.…”

Locally quasi-stationary states in noninteracting spin chains

“…It has however some limitations, and it was often blamed to be valid only in limits in which the system behaves essentially in a classical way (indeed, does not appear in first-order GHD). In addition, the theory is unable to capture diffusive behaviour in interacting systems, a subject that has been attracting more and more attention [66,67,70,71,[78][79][80]. In order to overcame this weakness, there have been proposals to go beyond the lowest order (of an implicit asymptotic expansion in the limit of low inhomogeneity) in interacting integrable systems [78,79,81,82], but, at the same time, some issues in noninteracting spin chains were uncovered.…”