Critically Slow Hilbert-Space Ergodicity in Quantum Morphic Drives

TL;DR

This paper proves that certain aperiodic quantum drives, including Thue-Morse, eventually achieve full ergodicity in Hilbert space, but only after extremely long times, revealing a new class of slow ergodic dynamics.

Contribution

It rigorously demonstrates that Thue-Morse and similar morphic sequence drives lead to eventual full quantum ergodicity, resolving previous apparent contradictions.

Findings

Thue-Morse drive achieves strong quantum ergodicity in the long-time limit.

Dynamics approximate a Floquet drive over finite periods, leading to scale-free ergodic behavior.

Many morphic sequence drives exhibit critically slow, but eventual, Hilbert-space ergodicity.

Abstract

The maximum entropy principle is foundational for statistical analyses of complex dynamics. This principle has been challenged by the findings of a previous work [arXiv:1701.07596], where it was argued that a quantum system driven in time by a certain aperiodic sequence without any explicit symmetries, dubbed the Thue-Morse drive, gives rise to emergent nonergodic steady states which are underpinned by effective conserved quantities. Here, we resolve this apparent tension. We rigorously prove that the Thue-Morse drive achieves a very strong notion of quantum ergodicity in the long-time limit: The time evolution of any initial state uniformly visits every corner of its Hilbert space. On the other hand, we find the dynamics also approximates a Floquet drive for arbitrarily long albeit finite periods of time with no characteristic timescale, resulting in a scale-free ergodic dynamics we call critically slow complete Hilbert-space ergodicity. Furthermore, numerical studies reveal that critically slow complete Hilbert-space ergodicity is not specific to the Thue-Morse drive and is, in fact, exhibited by many other aperiodic drives derived from morphic sequences, i.e., words derived from repeatedly applying substitution rules on basic characters. Our work presents a new class of dynamics in time-dependent quantum systems where full ergodicity is eventually attained, but only after astronomically long times.