Logarithmically Slow Relaxation in Quasi-Periodically Driven Random Spin Chains

TL;DR

This paper investigates the slow, logarithmic relaxation dynamics in a disordered spin chain driven quasi-periodically, revealing a long-lived glassy regime and metastable phases like a time quasi-crystal, with implications for thermalization and effective Hamiltonian descriptions.

Contribution

It introduces a novel simulation approach for long-time dynamics in quasi-periodically driven systems and uncovers a glassy regime with slow relaxation not captured by high-frequency expansions.

Findings

Logarithmic growth of entanglement and decay of correlations

Existence of a metastable time quasi-crystal phase

Breakdown of high-frequency expansion beyond fourth order

Abstract

We simulate the dynamics of a disordered interacting spin chain subject to a quasi-periodic time-dependent drive, corresponding to a stroboscopic Fibonacci sequence of two distinct Hamiltonians. Exploiting the recursive drive structure, we can efficiently simulate exponentially long times. After an initial transient, the system exhibits a long-lived glassy regime characterized by a logarithmically slow growth of entanglement and decay of correlations analogous to the dynamics at the many-body delocalization transition. Ultimately, at long time-scales, which diverge exponentially for weak or rapid drives, the system thermalizes to infinite temperature. The slow relaxation enables metastable dynamical phases, exemplified by a "time quasi-crystal" in which spins exhibit persistent oscillations with a distinct quasi-periodic pattern from that of the drive. We show that in contrast with Floquet systems, a high-frequency expansion strictly breaks down above fourth order, and fails to produce an effective static Hamiltonian that would capture the pre-thermal glassy relaxation.