Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems

We address the hydrodynamics of operator spreading in interacting integrable lattice models. In these models, operators spread through the ballistic propagation of quasiparticles, with an operator front whose velocity is locally set by the fastest quasiparticle velocity. In interacting integrable systems, this velocity depends on the density of the other quasiparticles, so equilibrium density fluctuations cause the front to follow a biased random walk, and therefore to broaden diffusively. Ballistic front propagation and diffusive front broadening are also generically present in nonintegrable systems in one dimension; thus, although the mechanisms for operator spreading are distinct in the two cases, these coarse-grained measures of the operator front do not distinguish between the two cases. We present an expression for the front-broadening rate; we explicitly derive this for a particular integrable model (the “Floquet-Fredrickson-Andersen” model), and argue on kinetic grounds that it should apply generally. Our results elucidate the microscopic mechanism for diffusive corrections to ballistic transport in interacting integrable models.

Figure 1

Geometric picture of quasiparticle diffusion: The worldline of the “tagged” quasiparticle (thick black line), with mean velocity $v_0$, wanders owing to collisions with other quasiparticles (dashed lines). In a time $t$, the tagged quasiparticle collides with quasiparticles moving at velocity $v_i$ so long as those quasiparticles started out in a spatial window of size $|v_i-v_0|t$; their density fluctuations inside this window govern front broadening.

Figure 2

Upper panel: Biased random walk of a single right-moving quasiparticle in the FFA model for a specific initial product state (left) and diffusive broadening when averaged over 1000 product states (right). Lower panel: Variance of front position vs filling for a system of size $L=400$; numerical results (averaged over 1000 random product states) are in good agreement with the analytic formula (5). We emphasize that there are no free parameters. Inset: OTOCs in the FFA model generically fill in behind the front [40].