Rényi entropies after releasing the Néel state in theXXZspin-chain

Abstract

We study the Rényi entropies in the spin-1/2 anisotropic Heisenberg chain after a quantum quench starting from the Néel state. The quench action method allows us to obtain the stationary Rényi entropies for arbitrary values of the index α as generalised free energies evaluated over a calculable thermodynamic macrostate depending on α. We work out this macrostate for several values of α and of the anisotropy ∆ by solving the thermodynamic Bethe ansatz equations. By varying α different regions of the Hamiltonian spectrum are accessed. The two extremes are α → ∞ for which the thermodynamic macrostate is either the ground state or a low-lying excited state (depending on ∆) and α = 0 when the macrostate is the infinite temperature state. The Rényi entropies are easily obtained from the macrostate as function of α and a few interesting limits are analytically characterised. We provide robust numerical evidence to confirm our results using exact diagonalisation and a stochastic numerical implementation of Bethe ansatz. Finally, using tDMRG we calculate the time evolution of the Rényi entanglement entropies. For large subsystems and for any α, their density turns out to be compatible with that of the thermodynamic Rényi entropies.Rényi entropies after releasing the Néel state in the XXZ spin-chain

Towards a generalized hydrodynamics description of Rényi entropies in integrable systems

“…where we omit the subscript ζ, since we are focusing on ζ = 0. Clearly, from (41), one has that ϑ n (λ) is non-zero only for λ such that v n (λ) > 0. An important consequence of (41) is that ϑ n (λ) = 0 for λ such that v n (λ) < 0.…”

“…Clearly, from (41), one has that ϑ n (λ) is non-zero only for λ such that v n (λ) > 0. An important consequence of (41) is that ϑ n (λ) = 0 for λ such that v n (λ) < 0. The physical interpretation is that quasiparticles with these rapidities that are originated in B do not reach the interface with A, because their group velocity change sign during the dynamics.…”

“…which allows one to replace θ H (v n ) with θ H (e n ) in (41). An important consequence is that the integral equation for the dressed energy e n (cf.…”

“…By using (41) and the fact that for the Néel state in the limit ∆ → ∞ one has that 141 ϑ −∞,1 → 1, one obtains that ϑ 1 becomes a step function. Precisely, one has…”

Entanglement spreading and quasiparticle picture beyond the pair structure

“…Despite not being an ab-initio calculation, the quasi-particle picture is able to exactly reproduce several features of the entanglement growth after a quantum quench. Originally introduced in critical systems [33,45,46], its applicability has been promptly extended first to generic non-interacting models [47][48][49][50][51][52][53][54], then to interacting integrable ones [55][56][57][58][59][60] (with recent developments in non-homogeneous systems [61][62][63]); finally, it has provided important insight even into not strictly-integrable systems [64,65].…”