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

Significance The XXZ model is a canonical model of quantum magnetism. In one dimension, this model is integrable and has ballistically moving quasiparticles; thus, energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. Here, we show that spin transport is even richer in the presence of a magnetic field: Some fraction of the spin moves out ballistically, but the dynamical spin-structure factor decays with an anomalous, continuously varying exponent. A hitherto-unnoticed phase transition takes place at finite fields, separating ballistic from subballistic decay of local correlations. We analytically derive these results using generalized hydrodynamics and support our findings numerically. These anomalous exponents can readily be measured via the frequency and wavevector dependence of the spin conductivity. 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 cross-over 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.