Evolution of Entanglement Spectra under Generic Quantum Dynamics

Abstract

We characterize the early stages of the approach to equilibrium in isolated quantum systems through the evolution of the entanglement spectrum. We find that the entanglement spectrum of a subsystem evolves with at least three distinct timescales. First, on an o(1) timescale, independent of system or subsystem size and the details of the dynamics, the entanglement spectrum develops nearest-neighbor level repulsion. The second timescale sets in when the light-cone has traversed the subsystem. Between these two times, the density of states of the reduced density matrix takes a universal, scale-free 1/f form; thus, random-matrix theory captures the local statistics of the entanglement spectrum but not its global structure. The third time scale is that on which the entanglement saturates; this occurs well after the light-cone traverses the subsystem. Between the second and third times, the entanglement spectrum compresses to its thermal Marchenko-Pastur form. These features hold for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum circuit models.that the entanglement spectra behave universally even at relatively early times as demonstrated in Fig. 1, although its arXiv:1811.00029v3 [cond-mat.dis-nn]

Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics

“…While we do not have an analytical proof of the above scalings, we present a physical picture that elucidates the evolution of the ES in the second stage in terms of the operator spreading of the inserted T gates, as depicted in Fig. 1 (right panel) [15,16]. We show that each T gate spreads over a spatial region that grows linearly in time, and hence the operator spreading of T gates should determine the timescale for the saturation to Wigner-Dyson distribution of the wavefunction ES.…”

“…It has been shown that the dynamics of ES is able to distinguish between random unitary circuits of different complexities [7][8][9], as well as thermalization and localization phases of the underlying Hamiltonian [10][11][12][13]. Moreover, the onset of level repulsion in the ES signals the spreading of operator fronts, which serves as an important diagnostic of quantum chaos and information scrambling [14][15][16].…”

Dynamics of spectral correlations in the entanglement Hamiltonian of the Aubry-André-Harper model

“…In the present work, we will study the spectral form factor of the entanglement Hamiltonian. While it is also meaningful to study the spectral correlations of the RDM of the subsystem [23,37], it is easier to consider the entanglement Hamiltonian directly due to its intimate relationship with the RDM, see Eq. (7).…”

“…It has been seen recently that the dynamics of the spectral form factor serves as a useful probe in understanding the various stages of approach to thermalization in chaotic (nonintegrable) models [37] through the study of the correlations in the spectrum of the entanglement Hamiltonian. It has been noticed that there is a certain correspondence between the dynamics of the level repulsion, entanglement entropy [38][39][40], and the development of the spectral form factor.…”

“…It has been noticed that there is a certain correspondence between the dynamics of the level repulsion, entanglement entropy [38][39][40], and the development of the spectral form factor. With the aid of a variety of quantities, including gap ratio and entanglement entropy, Chang et al [37] were able to identify three distinct timescales which were observed in both the chaotic and MBL phases. In the backdrop of such interesting findings in an interacting model, it is of great interest to understand how these different timescales play out within a noninteracting model.…”