The maximum entropy principle (MEP) is a powerful tenet in physics, guiding predictions about the limiting behavior of a complex system without the need for a full, microscopic solution [1]. It posits that at equilibrium, a system is well described by a state which maximizes a suitable notion of entropy, constrained only by globally conserved quantities. For example, the MEP predicts that the steady state achieved in thermalization is a Gibbs state, characterized solely by a few macroscopic properties like temperature and chemical potential.

The MEP has also guided recent studies of quantum ergodicity concerning the temporal ensemble of wave functions of a closed quantum system [2]–[9]. These works inquire about the limiting distribution formed by the collection of pure states arising in evolution and go beyond traditional notions of quantum ergodicity involving statistics of energy eigenvalues or eigenstates [10]. For example, when dynamics possesses a time-translation symmetry (e.g., generated by a time-independent or periodic Hamiltonian), the MEP predicts that the random phase ensemble emerges — a collection of states wherein energy eigenstates are superposed with random phases [5], [6]. In contrast, in the absence of time-translation and other internal symmetries, the MEP suggests the emergence of the maximally ergodic, uniform (Haar) distribution, a phenomenon dubbed complete Hilbert-space ergodicity (CHSE) [3], [4]. Indeed, Ref. [3] rigorously showed that CHSE arises in the discrete time-quasiperiodic Fibonacci drive, which is derived from the Fibonacci word  [11]–[17]. More generally, the same universal maximally ergodic behavior is expected to hold for generic quantum systems driven aperiodically in time, which do not feature any explicit conserved quantities.

However, a previous study [18] of the equilibration dynamics in the quantum Thue-Morse drive (TMD) [15], [17], [19]–[25], derived from the aperiodic Thue-Morse word [Fig. 1 (a)], challenges this expectation. Reference [18] argued that the TMD has a self-similar structure and, thus, exhibits an emergent (Floquet) time-translation symmetry, precluding CHSE — see Fig. 1 (b) for an example of seemingly nonergodic dynamics. If these predictions are correct, this represents a surprising exception to the MEP and potentially presents new routes to stabilize driven quantum systems against the deleterious effects of heating.

In this work, we carefully analyze the dynamics of the TMD and through it unravel a new class of quantum ergodic dynamics which we dub critically slow complete Hilbert-space ergodicity (CS-CHSE). Specifically, we rigorously prove that a qubit driven by the TMD generically displays emergent time-translation symmetries with arbitrarily good accuracy over arbitrary durations of time; however, these symmetries are finitely lived, resulting in the system achieving CHSE in the strict infinite time limit. Our finding resolves the apparent tension between the purported nonergodicity claimed by Ref. [18] in the TMD and the maximally ergodic predictions of the MEP. Furthermore, our analysis uncovers that the exploration of the full Hilbert space occurs in a scale-free manner — precisely, the distribution of timescales that the emergent time-translation symmetries emerge for obeys a power law and has an unbounded mean, hence the term “critically slow."

Motivated by these findings, we numerically explore other aperiodic driving sequences and find that the phenomenology of CS-CHSE holds similarly for drives derived from certain morphic words — words derived from repeatedly applying substitution rules on basic characters [26]. Thus, CS-CHSE represents a new class of quantum ergodic dynamics in which full ergodicity is eventually achieved — as predicted by the MEP —, but over timescales astronomically larger than those natural to the system. This adds to recent studies on quantum dynamics emphasizing how there can be a large separation of timescales between the asymptotic late-time dynamics and a transient but potentially physically more-relevant long-time dynamics, such as two-step [27] and slow thermalization [28]–[32].

Thue-Morse drive and CHSE.— We begin by defining the quantum TMD. It is derived from the so-called Thue-Morse word (TMW) [33], an infinite word on two characters \(\{0,1\}\) constructed by the following simple concatenation rules [26]. Starting from the base word \(W_0\) \(=\) \(0\), form the word \(W_{n+1}\) at the \((n+1)\)-th level by concatenating \(W_n\) with its bitwise negation \(\overline{W_{n}}\) (i.e., interchanging \(0\) and \(1\)): \[\label{eq:TMWparts} W_{n+1}=W_{n} \overline{W_{n}}.\tag{1}\] For example, \(W_1\) \(=\) \(01\) yields \(W_2\) \(=\) \(0110\), then \(W_3\) \(=\) \(01101001\), etc. The infinite-level word \(W_\infty\) is the TMW, which is known to be aperiodic, i.e. never eventually repeats, but is not quasiperiodic [34] ¹.

The quantum TMD for a \(d\)-dimensional quantum system is realized by applying two fixed unitaries \(A\) \(,\) \(B\in \operatorname{SU}(d)\) according to the characters of \(W_\infty\). Specifically, at integer time \(t\) \(=\) \(n\), apply \(A\) \((B)\) if the \(n\)th character of \(W_\infty\) is \(0\) \((1)\). Thus, time evolution of an initial state \(\ket{\psi(0)}\) is \(\ket{\psi(n)}\) \(=\) \(U(n)\ket{\psi(0)}\), where \[\label{eq:TMD} U(n)=\mathop{\prod^\leftarrow}\limits^{n}_{j=1}\begin{cases}A&\text{if the jth character of W_\infty is 0} \\ B&\text{if it is 1}\end{cases},\tag{2}\] is the unitary time-evolution operator. Above, the product is ordered from right to left. For example, \(\ket{\psi(8)}\) \(=\) \(BAABABBA\ket{\psi(0)}\).

Recent previous works have studied the TMD, for instance on prethermalization timescales in the high-frequency regime [20], [22], [23], and in defining so-called “time-rondeau crystals” [24]. As mentioned, a work most relevant for us is Ref. [18], which studied the equilibration dynamics of free-fermionic spin chains under the TMD, and argued for the emergence of nontrivial steady states. In our context, this amounts to a violation of CHSE, which we define next.

CHSE refers to the property of a sequence of pure states \(\{\psi(t)\}_{t\in\{0,1,2,\dots \}}\) generated in dynamics, called the temporal ensemble [6], uniformly covering the Hilbert space [3], [4] ². Formally, we mean that the empirical distribution of the temporal ensemble asymptotically follows the Haar ensemble, a collection of states \(\{V \dyad{\psi(0)} V^\dagger\}_V\) where \(V\) is a Haar-distributed unitary on \(\operatorname{SU}(d)\). This can be probed through the trace distance ³ \[\label{eq:trace-distance} \Delta_T^{(k)}\mathrel{\vcenter{:}}= \frac{1}{2}\norm{\rho_{T}^{(k)} - \rho_{\text{Haar}}^{(k)}}_1,\tag{3}\] of the \(k\)th temporal moment of the finite-time ensemble \(\rho_T^{(k)}\) \(=\) \(\frac{1}{T}\sum_{t=0}^{T-1}(U(t)\dyad{\psi}U(t)^\dagger)^{\otimes k}\) to the corresponding Haar moment \(\rho_{\text{Haar}}^{(k)}\) \(=\) \(\require{physics} \int\dd{V}(V\dyad{\psi}V^\dagger)^{\otimes k}\), and inquiring if at large times \(\lim_{T\to \infty}\Delta_T^{(k)}=0\) for all \(k\) ⁴. In quantum information parlance, one asks if the temporal ensemble forms a state design.

Here, we will actually study the stronger notion of ergodicity of the temporal ensemble of unitaries themselves, namely ask if the statistical properties of the time-evolution operators \(\{U(t)\}_t\) of the TMD are close to those of Haar random unitaries. Specifically we consider the \(k\)th-moment time-averaging channel \(\mathcal{N}^{(k)}_T[\,\cdot\,]\) \(=\) \(\frac{1}{T}\sum_{t=0}^{T-1} U(t)^{\otimes k}[\,\cdot\,]U(t)^{\dagger \otimes k}\) and compare its similarity to the corresponding \(k\)th-moment Haar averaging channel \(\mathcal{N}_{\mathrm{Haar}}^{(k)}[\, \cdot\,]\) \(=\) \(\require{physics} \int\dd V V^{\otimes k}[\,\cdot\,]V^{\dagger \otimes k}\), for example through the Hilbert-Schimdt distance \[\begin{align} \label{eq:defintionofHSD} D_T=\norm{\mathcal{N}^{(k)}_{T}-\mathcal{N}^{(k)}_{\mathrm{Haar}}}_2. \end{align}\tag{4}\] If \(\lim_{T\to \infty} D_T\) \(=\) \(0\) for all \(k\), then this property of the unitary ensemble is dubbed complete unitary ergodicity (CUE) [4]. In quantum information parlance, this is asking whether the ensemble of time-evolution operators forms a unitary design. Since the \(k\)th moment of the finite-time state ensemble can be recovered via \(\rho_T^{(k)}=\mathcal{N}^{(k)}_T[ |\psi(0)\rangle \langle \psi (0)|^{\otimes k}]\) and similarly for the Haar moment, it is clear CUE implies CHSE.

Transient time-translation symmetries in the TMD.— First, we illustrate the basic phenomenology seen in numerical simulations of the TMD [35] where the basic unitaries \(A,B\) \(\in\) \(\operatorname{SU}(d)\) are chosen Haar randomly. Figure 2 (a) shows the trace distance \(\Delta_T^{(k)}\) for \(k\) \(=\) \(2\). For larger dimension \(d\), there is a power-law behavior \(\sim\) \(1/\sqrt{T}\), which provides strong numerical evidence for CHSE. However, for small \(d\), and particularly for \(d\) \(=\) \(2\) [Figs. 2 (b)-(d)], the decay is not smooth, and, in fact, there are surprisingly many intermediate long-lived plateaus, some up to times \(2^{600}\) (!), the limit of our finite-time numerics. Despite the astronomically large times reached by our simulations (see [35] for details), it is unclear if these plateaus survive indefinitely or if there is eventual decay at even later times, such that CHSE is achieved.

Dynamics of unitaries.—We now turn to our rigorous analysis of the asymptotic behavior of the TMD. A starting observation is that we may simplify the analysis of the TMD by concentrating on exponential times \(t\) \(=\) \(2^n\), i.e., study the sequence of unitaries \(A_n \mathrel{\vcenter{:}}= U(2^n)\). We find \(A_n\) may be generated by repeated applications of a dynamical map \((A_n,B_n)\) \(=\) \(\Phi(A_{n-1},B_{n-1})\) where \(B_n\) is a partner sequence with all \(A\)s swapped with \(B\)s, which we call the ‘complementary TMD’. Specifically, define the map \(\Phi\colon\operatorname{SU}(d)\) \(\times\) \(\operatorname{SU}(d)\) \(\to\) \(\operatorname{SU}(d)\) \(\times\) \(\operatorname{SU}(d)\) where \(\Phi(A,B)\) \(:=\) \((BA\) \(,\) \(AB)\); then \[\label{eq:AnBn} (A_n,B_n)=\Phi^n(A,B),\tag{5}\] which is equivalent to the recursion relations \(A_{n+1}\) \(=\) \(B_nA_n\) and \(B_{n+1}\) \(=\) \(A_nB_n\) [18], [35].

Using this map, the origin of the plateauing structure of the trace distance can be intuitively understood by analyzing the iterates \(A_n\) \(,\) \(B_n\). Suppose (for some reason) \(A_n\approx B_n\) within a range \(n\) \(\in\) \([n_*,N_*]\). Then, defining \(V_*\) \(=\) \(A_{n_*}\), we have \(\Phi(A_{n_*},B_{n_*})\) \(\approx\) \((V_*^2,V_*^2)\) and subsequently, the evolution operator satisfies \(U(T=2^{n})\) \(=\) \(A_n\) \(\approx\) \(V_*^{2^n}\). In other words, between times \(2^{n_*}\) and \(2^{N_*}\), the system behaves like a Floquet system, evolving with the repeated application of \(V_*\). This emergent discrete time-translation symmetry would halt the further exploration of the Hilbert space, due to quasienergy conservation [4].

This preliminary analysis suggests that the emergent Floquet dynamics seen in numerics can be understood by showing \(A_n\) \(\approx\) \(B_n\) for a large range of \(n\). Thus, the distance \(\xi_n\) \(\mathrel{\vcenter{:}}=\) \(\xi(A_n,B_n)\) \(<\) \(0\) between the iterates, where \[\label{eq:DistanceBetweenUnitaries} \xi(A,B) \mathrel{\vcenter{:}}= \log\left(\tfrac{1}{2}\abs{1-\tfrac{1}{d}\tr(A^\dagger B)} \right),\tag{6}\] is a key quantity to consider: it governs the accuracy of the emergent time-translation symmetries in exponential precision (we will find that it is natural to measure the distance in logarithmic scale). Indeed, \(\xi_n\) for a qubit system is plotted in Fig. 2 (b)-(d), where two salient features are observed: When \(\xi_n\) is close to \(0\) (vertical orange lines), the trace distance \(\Delta_{2^n}^{(k)}\) appears to drop; while when \(\xi_n\) is large, the trace distance plateaus, as expected, confirming its important role in dynamics. However, \(\xi_n\) is also seen to behave rather erratically, as if driven by a random process: Over time there are multiple large excursions from the origin but also multiple recurrences back to it. Understanding which process dominates (if any), is paramount to understanding whether CHSE is achieved.

Quasirandom walk of \(\xi_n\) and CHSE.— We find that for a system consisting of a single qubit, the behavior of the distance \(\xi_n\), which we interpret as the position of a ‘particle’ living on the semi-infinite real line \((-\infty\) \(,\) \(0]\), can be exactly captured by a dynamical map in which the position and an angular variable \(\theta_n\) \(\in\) \([0\) \(,\) \(2\pi)\) are coupled: \[\begin{align} \label{eq:rhothetamap} \xi_{n+1}&=\xi_n+\log(1-e^{\xi_n}) + f(\theta_n) \\ \theta_{n+1}&=\arg(e^{i 2\theta_n} + (1-e^{i2\theta_n})e^{\xi_n}),\nonumber \end{align}\tag{7}\] where \(f(\theta)\) \(=\) \(\log(4\sin(\theta)^2)\). This map is derived from the map on unitaries \(\Phi\) [35]. The initial points \(\xi_1\) \(,\) \(\theta_1\) are related to the starting choice of unitaries \(A\) \(,\) \(B\) whose precise expressions and values are not important in determining the generic long-time behavior of the map. Alternatively, the dynamics can equivalently be represented as a nonlinear map \(c_{n+1}\) \(=\) \(c_{n}^2+1-\abs{c_n}^2\) of a single complex variable in the unit disk \(|c_n|\leq 1\), which accords a complementary geometrical viewpoint of the TMD in terms of motion within the disk (see [35] for details).

One may gain more physical insight into Eq. 7 when the particle is far from the origin, so that the factor \(e^{\xi_n}\) is small and can be ignored; then the angular part decouples: \[\begin{align} \xi_{n+1}&\approx\xi_n + f(\theta_n) \tag{8} \\ \theta_{n+1}&\approx 2\theta_n \mod 2\pi. \tag{9} \end{align}\] Equation 9 is the angle doubling map, which is well known to be chaotic, in the sense that the angles \(\theta_n\) behave like random numbers for a generic initial condition. These random numbers, in turn, drive the position of the particle with the length of each step given by \(f(\theta_n)\); thus the iteration in Eq. 8 describes a quasirandom walk, explaining qualitatively the behavior seen in Fig. 2 (b)-(d). Of course, the exact equations of motion in Eq. 7 feature an additional ‘soft-wall’ constraint \(\log(1-e^{\xi_n})\) preventing the particle from crossing the origin.

The equations of motion 7 also provide a quantitative understanding of the steplike structure seen in the trace distance in Fig. 2 (b)-(d). Key is the function \(f(\theta)\) [Fig. 2 (f)]: It is upper bounded by \(\log(4)\), implying a finite maximum increase in \(\xi_n\) per step; but at the same time is unbounded from below, diverging to \(-\infty\) at \(\theta\) \(=\) \(0\) \(,\) \(\pi\), implying a potentially unbounded decrease in \(\xi_n\) per step. Thus upon incurring a single large negative jump in \(\xi_n\), the system becomes almost ‘Floquet-like’ so that \(\Delta^{(k)}_{2^n}\) plateaus, and requires many small positive steps to conspire together to increase the value of \(\xi_n\) again so that \(\Delta^{(k)}_{2^n}\) drops. One can estimate the distribution of the lengths of these plateaus \(\tau\) by analyzing the distribution of the first-return step of \(\xi_n\) under Eq. 7 (i.e., the minimum \(n>1\) such that \(\xi_n\geq\xi_1\)). We numerically find a power-law distribution \(P(n)\) \(\sim\) \(n^{-3/2}\) [Fig. 2 (e)], same as for a random walk in 1D [36]. Taking \(\tau=2^n\), the distribution of plateau lengths is, thus, \(P(\tau)\) \(\sim\) \(\tau^{-1} \log(\tau)^{-3/2}\), a heavy-tailed distribution with unbounded mean.

Turning now to the important question of whether the random walker wanders off to (negative) infinity or remains near its (finite) starting position at late times: interestingly, because the function \(f(\theta)\) governing the step size has zero mean and finite variance, the random walk belongs to the Gaussian universality class, and so both generically happen infinitely often, rigorously captured by the following lemma.

The proof, taken from Ref. [37], is presented in the Supplemental Material [35].

Statement (ii) of Lemma 1 formalizes the long timescales of emergent time-translation symmetry and nonergodicity. Conversely, statement (i) guarantees that despite these long timescales, eventual uniform ergodicity is achieved: We further show that each recurrence near the initial point \((\xi_1,\theta_1)\) is accompanied by a strict decrease in the degree of dissimilarity between the ensemble of time-evolution unitaries of the TMD to Haar random ones (proof in Supplemental Material [35]):

Since Lemma 1 (i) implies these strict decreases happen infinitely often, \(\lim_{n \to \infty} M_n = 0\) and hence \(\lim_{T \to \infty} D_T = 0\) [35], yielding the expected result from the MEP that the TMD ultimately achieves maximal ergodicity.

We note that the technical reason Ref. [18] did not identify the CHSE of the TMD is their claim that \(\xi_n\to-\infty\) (\(\phi_m\to 0\) in their notation), which does not hold generically [statement (i) of Lemma 1].

Discussion and outlook.—Our study of the aperiodic TMD has confirmed the applicability of the MEP: it correctly predicts that the drive, which lacks any explicit symmetries, ultimately produces an ergodic temporal ensemble of states, uniformly distributed over the Hilbert space. At the same time, our work has also explained how this maximal ergodicity is attained dramatically slowly and in a scale-free manner (at least for a qubit), a dynamical phenomenon we dub CS-CHSE. Our work has direct implications for experiments probing CHSE (such as in a nitrogen-vacancy center setup, recently performed [9]), in correctly interpreting the apparent nonergodic behavior they would find at short times if they were to implement the TMD. We leave as an open question why the long periods of nonergodicity seem to vanish with increasing system size [Fig. 1 (a)], for which it would be useful to understand the invariant measure of the map \(\Phi\) in arbitrary dimension.

Beyond the TMD, it is natural to inquire about the generality of CS-CHSE. To that end, we perform a preliminary numerical study of aperiodic discrete quantum drives constructed by applying basic unitaries \(A\) \((B)\) according to the characters \(0\) \((1)\) appearing in a binary morphic word — an infinite word generated by iterating substitution rules \(0\) \(\to\) \(w_0\) \(,\) \(1\) \(\to\) \(w_1\) starting from \(0\), where \(w_0\) \(,\) \(w_1\) are some fixed subwords [26]. For example, the TMW studied in this work is the morphic word defined by \(0\) \(\to\) \(01\), \(1\) \(\to\) \(10\).

In Fig. 3 we plot the trace distances between the temporal and Haar ensembles for some morphic drives constructed from words with choices of \(w_0\) and \(w_1\) of length at most 3. We find three distinct behaviors: first, drives which display “regular" CHSE (green lines), in the form of a power-law decay, regardless of dimension; these include the generalized Fibonacci drives (\(0\) \(\to\) \(0^m1\), \(1\) \(\to\) \(0\)) studied in [3] and generalizations like the Octonacci drive \((0\) \(\to\) \(010\), \(1\) \(\to\) \(0)\) [38]. Second, drives which appear to display CS-CHSE (yellow-red lines) for a qubit [Fig. 3 (a)], which include the TMD, but also drives from other sequences, like the period-doubling word \((0\) \(\to\) \(01, 1\) \(\to\) \(00)\) [39]. This shows that CS-CHSE is not particular to the TMD. While in the present examples CS-CHSE is seen to vanish with increasing Hilbert-space dimension [Fig.3 (b)], understanding if there are more general morphic drives wherein CS-CHSE survives in the many-body regime would be an important direction for future work. Third, there is a small proportion of drives (black and gray lines) which exhibit a yet slower decay of the trace distance, present for any dimension. However, these have arguably a trivial origin, as the underlying words have arbitrarily long streaks of repeated characters (e.g. for \(0\) \(\to\) \(001, 1\) \(\to\) \(11\), each substitution doubles the length of the blocks of 1’s) ⁶; hence, we term them ‘asymptotically Floquet drives’. Nevertheless, given that asymptotically Floquet drives are strictly speaking aperiodic, it is an open and nontrivial question whether CHSE is ultimately achieved or not in their dynamics. We see that there is a rich interplay between the combinatorial structure and complexity of the aperiodic morphic words and the degree of ergodicity in the quantum drives they generate, which deserves to be better studied.

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