Coarse-grained Entanglement and Operator Growth in Anomalous Dynamics

Abstract

In two-dimensional Floquet systems, many-body localized dynamics in the bulk may give rise to a chaotic evolution at the one-dimensional edges that is characterized by a nonzero chiral topological index. Such anomalous dynamics is qualitatively different from local-Hamiltonian evolution. Here we show how the presence of a nonzero index affects entanglement generation and the spreading of local operators, focusing on the coarsegrained description of generic systems. We tackle this problem by analyzing exactly solvable models of random quantum cellular automata (QCA) which generalize random circuits. We find that a nonzero index leads to asymmetric butterfly velocities with different diffusive broadening of the light cones, and to a modification of the order relations between the butterfly and entanglement velocities. We propose that these results can be understood via a generalization of the recently-introduced entanglement membrane theory, by allowing for a spacetime entropy current, which in the case of a generic QCA is fixed by the index. We work out the implications of this current on the entanglement "membrane tension" and show that the results for random QCA are recovered by identifying the topological index with a background velocity for the coarse-grained entanglement dynamics.

Bounds in Nonequilibrium Quantum Dynamics

“…While v tri ≤ v E is also observed in the random QCA models, 230 we expect that this is not always the case and can be violated by some normal (ind = 0) dynamics with v…”

“…Very recently, the entanglement-membrane theory has been generalized to anomalous dynamics which are still locality-preserving but are not Hamiltonian evolution or quantum circuits. 230 Examples include discrete dynamics generated by nontrivial QCA, as discussed in Sec. 5.2.2, and edge dynamics of chiral many-body localized Floquet phases in 2D.…”

“…where from A on the input side is scrambled into C and D on the output side, i.e., the information that cannot be retrieved by individually observing C and D. At least for two different types of Haar random QCA in Ref. 230, the tripartite-information velocity v tri ≡ − lim t→∞ t −1 I A:C:D (t) is found to be the difference between the operator-entanglement velocity and the scaled index:…”

Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond

“…( 9), it follows that B2 p = 1, satisfying property 1. Moreover, the relations Bp , Bp = Fp , Bp = Fp , Bp = 0 (22) follow from [ Bp , Bp ] = 0 and Eq. ( 14), verifying property 2.…”

“…1 They originally arose in the context of quantum simulation and as a model for quantum computation [1][2][3][4][5]. However, in recent years, QCAs have seen wide-ranging applications from discretized quantum field theories [6,7] to the classification of Floquet phases [8][9][10][11][12][13] and tensor network unitary operators [14][15][16][17][18], entanglement growth in quantum dynamics [19][20][21][22], and the construction of symmetry-protected topological states and their anomalous boundaries [23,24]. Beyond such applications, QCAs represent a fundamental class of mathematical objects in the quantum many-body setting, meshing the notions of unitarity and locality.…”