Critically Slow Hilbert-Space Ergodicity in Quantum Morphic Drives

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.

Figures (3)

• Figure 1: (a) The Thue-Morse drive involves applying a pair of unitaries $A$ and $B$ according to the Thue-Morse word. We ask if, under such dynamics, a quantum state uniformly covers the Hilbert space over time. (b) Trace distance between the temporal distribution [dotted spheres in (a)] and the uniform distribution (dark sphere) for a representative initial state. One sees a nonsmooth decay involving many intervening long-lived plateaus, each lasting for astronomically long times. It is not clear whether the trace distance keeps decreasing (CHSE) or eventually saturates at a nonzero value.
• Figure 2: Trace distance $\Delta_{T=2^n}^{(k)}$ for $k$$=$$2$ between temporal and Haar distribution for the TMD (each line is a different realization of Haar-randomly sampled basic unitaries $A$$,$$B$). (a) Trace distances for various $d$-level systems. A power-law decay $T^{-1/2}$ is shown in a dashed line for reference, which is tracked for large $d$. However, plateaus appear for small $d$. (b)-(d) Different instances of trace distance for $d$$=$$2$ (green line) and distance $\xi_n$ (gray line). Orange vertical lines indicate times at which $\xi_n>-1.$ (e) Distribution of first-return steps of the quasirandom walk on $\xi_n$, over random initial $\theta_1$ and $\xi_1$$=$$-1/2$. The orange dotted line is $\propto$$n^{-3/2}$. (f) Plot of step-length function $f(\theta)$$=$$\log(4\sin(\theta)^2)$.
• Figure 3: Trace distance $\Delta_T^{(k)}$ between the Haar distribution and temporal ensemble generated by quantum morphic drives associated with substitution rules labeled in panel (b), for Haar-randomly chosen starting unitaries $A$$,$$B$ (same for all drives). (a) $d$$=$$2$$,$$k$$=$$2$, (b) $d$$=$$5$$,$$k$$=$$1$.

Theorems & Definitions (3)

• Lemma 1
• Lemma 2
• Theorem 1