Entanglement spreading in non-equilibrium integrable systems

Abstract

These are lecture notes for a short course given at the Les Houches Summer School on ``Integrability in Atomic and Condensed Matter Physics'', in summer 2018. Here, I pedagogically discuss recent advances in the study of the entanglement spreading during the non-equilibrium dynamics of isolated integrable quantum systems. I first introduce the idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics and then join such an idea with the quasiparticle picture for the entanglement spreading to provide quantitive predictions for the time evolution of the entanglement entropy in arbitrary integrable models, regardless of the interaction strength.

Entanglement spreading after local fermionic excitations in the XXZ chain

“…A simple way to quantify the spectral weight of the spinons in the gapless regime is via the energy difference 〈∆E〉 = 〈ψ 0 | (m 1 H m 1 − H) |ψ 0 〉 of the Majorana excitation (equal to that of c † 1 by particle-hole symmetry) measured from the ground state, whereas in the gapped case we replace m 1 →m 1 . Our assumption in both regimes was that one can practically work with single-spinon states, whose energies above the ground state are given by the corresponding dispersions s (q) in (4) and (10), respectively. This yields the simple formula for the energy difference…”

“…where s (q) is the spinon dispersion (10), while c(q) are the overlaps of the domain-wall excitation with the single-spinon states. Note that the momentum of a single spinon satisfies 0 ≤ q ≤ π, however, the total momentum of spinons above the quasidegenerate ground state is shifted by π.…”

“…These quasiparticles transmit entanglement over large distances, contributing to the buildup of an extensive entropy within any given subsystem, which signals the onset of some local thermalization. Specifically, in one-dimensional integrable chains it has been verified that the entanglement entropy accumulated in a subsystem actually plays the role of the thermal entropy as described by the generalized Gibbs ensemble [8][9][10].…”

Subsystem complexity after a global quantum quench

“…where the order of the limits is important and in the last step we used that S GGE is an extensive quantity (see the review [122] and the references therein).…”

Real-time evolution in the Hubbard model with infinite repulsion

“…Therefore, the GGE retains extensive memory about the initial configuration. The exact knowledge of the stationary state allows one to compute stationary values of observables without solving the overwhelmingly complicated many-body dynamics and, moreover, it also gives access the asymptotic dynamics of correlations [11] and entanglement [12][13][14]. Importantly, the occurrence of GGEs has also been observed experimentally [15].…”