Towards a generalized hydrodynamics description of Rényi entropies in integrable systems

Abstract

We investigate the steady-state Rényi entanglement entropies after a quench from a piecewise homogeneous initial state in integrable models. In the quench protocol two macroscopically different chains (leads) are joined together at the initial time, and the subsequent dynamics is studied. We study the entropies of a finite subsystem at the interface between the two leads. The density of Rényi entropies coincides with that of the entropies of the Generalized Gibbs Ensemble (GGE) that describes the interface between the chains. By combining the Generalized Hydrodynamics (GHD) treatment of the quench with the Bethe ansatz approach for the Rényi entropies, we provide exact results for quenches from several initial states in the anisotropic Heisenberg chain (XXZ chain), although the approach is applicable, in principle, to any low-entangled initial state and any integrable model. An interesting protocol that we consider is the expansion quench, in which one of the two leads is prepared in the vacuum of the model excitations. An intriguing feature is that for moderately large anisotropy the transport of bound-state is not allowed. Moreover, we show that there is a "critical" anisotropy, below which bound-state transport is permitted. This is reflected in the steady-state entropies, which for large enough anisotropy do not contain information about the bound states. Finally, we benchmark our results against time-dependent Density Matrix Renormalization Group (tDMRG) simulations. arXiv:1807.01800v2 [cond-mat.stat-mech]

Entanglement evolution and generalised hydrodynamics: interacting integrable systems

“…In the following we show that it is possible to drastically simplify Eq. (58). We begin by changing variables from ζ to φ = Φ α,λ (ζ)…”

Molecular dynamics simulation of entanglement spreading in generalized hydrodynamics

“…A recent interesting direction is to generalize the approach to include diffusive corrections [87][88][89][90][91]. Finally, the GHD approach can be used to study the entanglement dynamics after quenches from piecewise-homogeneous initial states [46,52,54].…”

“…First, the entanglement entropy is expected to grow linearly at short times. Here we refer to the slope of the linear growth as the entanglement production rate [46,52,54]. The entanglement growth is due to the quasiparticles that cross the interface between the two chains.…”

“…The approach of Ref. [32] has been generalized in [45][46][47] to calculate the Rényi entropies in the steady-state, whereas calculating their full-time dynamics is a challenging open problem. Remarkably, the quasiparticle picture allows to obtain the dynamics of the logarithmic negativity [15,48], which is a proper entanglement measure for mixed states.…”

“…These ζ-dependent GGEs can be described analytically within the Generalized Hydrodynamics (GHD) formalism [50,51]. Furthermore, by combining the quasiparticle picture with the GHD approach [46,[52][53][54] it is possible, in principle, to generalize the quasiparticle picture [32] to inhomogeneous settings. However, actually calculating the full-time entanglement dynamics in this case is a demanding task.…”