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arXiv:2111.06755 · Physical Review Research

Measuring postquench entanglement entropy through density correlations

Adrian Del Maestro, Hatem Barghathi, Bernd Rosenow

Published Apr 26, 2022• 12 citations
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Abstract

Following a sudden change of interactions in an integrable system of one-dimensional fermions, we analyze the dependence of the static structure factor on the observation time after the quantum quench.At small waiting times after the quench, we map the system to noninteracting bosons such that we are able to extract their occupation numbers from the Fourier transform of the density-density correlation function, and use these to compute a bosonic entropy from a diagonal ensemble.By comparing this bosonic entropy with the asymptotic steady-state entanglement entropy per fermion computed with exact diagonalization, we find excellent agreement.

DOI
10.1103/physrevresearch.4.l022023

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We analyze the entanglement properties of the asymptotic steady state after a quench from free to hard-core bosons in one dimension. The Rényi and von Neumann entanglement entropies are found to be extensive, and the latter coincides with the thermodynamic entropy of the generalized Gibbs ensemble (GGE). Computing the spectrum of the two-point function, we provide exact analytical results for both the leading extensive parts and the subleading terms for the entropies as well as for the cumulants of the particle-number fluctuations. We also compare the extensive part of the entanglement entropy with the thermodynamic ones, showing that the GGE entropy equals the entanglement one and it is twice the diagonal entropy.
We investigate the time-dependent R\'enyi entanglement entropy after a quantum quench starting from the Mott-insulating and charge-density-wave states in a one-dimensional free boson system. The second R\'enyi entanglement entropy is found to be the negative of the logarithm of the permanent of a matrix consisting of time-dependent single-particle correlation functions. From this relation and a permanent inequality, we obtain rigorous conditions for satisfying the volume-law entanglement growth. We also succeed in calculating the time evolution of the R\'enyi entanglement entropy in extremely large systems by brute-force computations of the permanent. We discuss possible applications of our findings to the real-time dynamics of noninteracting bosonic systems.
The time evolution of the entanglement entropy in non-equilibrium quantum systems provides crucial information about the structure of the time-dependent state. For quantum quench protocols, by combining a quasiparticle picture for the entanglement spreading with the exact knowledge of the stationary state provided by Bethe ansatz, it is possible to obtain an exact and analytic description of the evolution of the entanglement entropy. Here we discuss the application of these ideas to several integrable models. First we show that for non-interacting systems, both bosonic and fermionic, the exact time-dependence of the entanglement entropy can be derived by elementary techniques and without solving the dynamics. We then provide exact results for interacting spin chains that are carefully tested against numerical simulations. Finally, we apply this method to integrable one-dimensional Bose gases (Lieb-Liniger model) both in the attractive and repulsive regimes. We highlight a peculiar behaviour of the entanglement entropy due to the absence of a maximum velocity of excitations.