Anomalous relaxation and the high-temperature structure factor of XXZ spin chains

Abstract

We compute the spin structure factor of XXZ spin chains in the Heisenberg and gapped (Ising) regimes in the high-temperature limit for nonzero magnetization, within the framework of generalized hydrodynamics including diffusive corrections. The structure factor shows a hierarchy of timescales in the gapped phase, owing to s-spin magnon bound states ("strings") of various sizes. Although short strings move ballistically, long strings move primarily diffusively as a result of their collisions with short strings. The interplay between these effects gives rise to anomalous power-law decay of the spin structure factor, with continuously varying exponents, at any fixed separation in the late-time limit. We elucidate the crossover to diffusion (in the gapped phase) and to superdiffusion (at the isotropic point) in the half-filling limit. We verify our results via extensive matrix product operator calculations. arXiv:1904.01039v1 [cond-mat.stat-mech]

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

“…Our results are consistent with a crossover time t c ∼ |m| 3 , in agreement with the prediction of Ref. [40].…”

“…In the following, we present an analysis of our numerical data that goes beyond the analysis shown in Ref. [40] and thereby confirm some of their analytic predictions. Short and intermediate times.…”

Kardar-Parisi-Zhang universality from soft gauge modes

“…Our model also predicts a crossover to a regime (σ/J) 1/2 < µ ≪ 1 in which the torsional mode becomes ballistic, with a propagation speed varying linearly in the initial magnetization, as v ball ∼ µ. This result was obtained from generalized hydrodynamics in the spin-1/2 Heisenberg chain 35 ; here we argue that the same phenomenon will arise in all quantum and classical integrable magnets with isotropic rotational symmetry. Indeed, this may explain the slow ballistic modes observed in very recent classical simulations 22 .…”

“…We emphasize that our derivation is "mesoscopic", in the sense that it fixes a particular coarse-graining length scale, l, and therefore excludes the recently discovered anomalous transport phenomena in the quantum spin-1/2 XXZ chain 35,36 , that emerge from linear fluctuating hydrodynamics in the limit of infinite coarse-graining length, l → ∞.…”

Universal spin dynamics in infinite-temperature one-dimensional quantum magnets

“…Unexpectedly, a numerical study observed a third behavior at the isotropic (Heisenberg) point of this model [22,23]: spin dynamics at infinite temperature were characterized by superdiffusion with the same dynamical critical exponent z = 3/2, defined below, that appears in the classical, stochastic Kardar-Parisi-Zhang (KPZ) universality class [24]. This led to additional studies that explained how the diffusion constant must become infinite at the Heisenberg point [25] and showed agreement with the full KPZ scaling function [23,26]. Note that this emergence of superdiffusion and KPZ universality from quantum models is different from the su-perdiffusion with z = 1 that emerges in systems with momentum conservation [27,28] or the variable dynamical critical exponent at low temperatures in Luttinger liquids [29].…”