Spectral anomalies and broken symmetries in maximally chaotic quantum maps

1. The anomalies in level statistics of quantized A-baker's maps are studied comprehensively, yielding a thorough understanding of the phenomenon.

2. Short-time behavior of the spectral form factor (SFF) is not only observed numerically but also analytically explained with a semiclassical periodic orbit expansion.

3. Deviations from random matrix theory are traced back to approximate symmetries of the quantized maps, enriching the understanding of the phenomenon.

4. Spectral anomalies are linked with the dynamics under the considered quantized maps and understood in terms of quantum cyclic ergodicity.

1. The manuscript is not easy to follow, it's readability could be improved (see Requested changes below).

2. The notions of the macroscopic symmetry of the classical map and the quantum symmetry of the quantized map are not clearly distinguished and sometimes mixed. The manuscript plainly demonstrates that the relation between the two types of symmetries is not direct and depends on the quantization type. The manuscript's narrative rooted in the premise that both types of symmetries are directly related unnecessarily complicates the reasoning (see Requested changes below).

The manuscript "Spectral anomalies and broken symmetries in maximally chaotic quantum maps" by L. Shou, A. Vikram, and V. Galitski investigates eigenvalue statistics of various quantizations of A-baker's maps. The A-baker's maps are classically chaotic and ergodic and possess two macroscopic discrete symmetries: time-reversal symmetry (TR) and reflection symmetry. Various quantizations of these classical maps exhibit different degrees of symmetry even though all such quantizations reproduce the respective A-baker's map in the semiclassical limit $N \to \infty$.

Properties of spectral statistics and their connection with various types of ergodic behavior are fundamental problems in the study of quantum chaos and dynamics of quantum many-body systems. The present manuscript demonstrates that the discrete macroscopic symmetries are only sometimes reflected in the level statistics of the quantized systems. While this phenomenon was noted in earlier studies referenced in the manuscript, the present manuscript addresses this problem systematically and exhaustively, painting a comprehensive picture of the phenomenon. The manuscript thus provides an important step in a previously identified and long-standing research problem, fulfilling the criteria of SciPost Physics. For this reason, I am inclined to recommend the manuscript for publication, which I will happily do when the authors respond to my remarks below concerning the presentation of the results and the associated reasonings.

1. The abstract states, "By studying various quantizations of maximally chaotic maps that break a discrete classical symmetry upon quantization, we demonstrate that this approach can be misleading and fail to detect macroscopic symmetries."
Sec. 1.2: "We aim to illustrate the unreliability of common spectral statistics in identifying discrete symmetries as may be present in quantized chaotic maps or many-body systems."
Sec. 3.1 "Due to the classical TR and reflection symmetries of the classical A-baker's map, one would expect its quantizations to exhibit spectral statistics similar to a 2-block COE matrix (a direct sum of two independent, equal sized COE matrices)."

The statements above show that the authors seem to imply that level statistics of the quantized maps *should* reflect the macroscopic symmetries of the classical map. This is clearly not the case due to the possibility of quantum mechanical breaking of the symmetry, as written explicitly already in lines 54-55, and then e.g. at the very start of Sec. 4.1. A clear separation of the notions of the macroscopic symmetry of the classical map and the quantum symmetry (associated with a commuting operator) of the quantized map is lacking. Highlighting this non-trivial correspondence would allow the authors to present the results more clearly without undermining the significance of the findigns.

2. The authors write, "These violations are striking in the context of the use of spectral statistics to identify discrete symmetries of the time evolution operator. While such diagnostics are effective in a variety of systems exhibiting block RMT behavior [10,45–48], our results show they cannot always be relied upon, even in simple systems with a well-defined classical limit." Ref. 48 concerns quantum symmetries of quantum systems. The results of the present manuscript highlight the non-triviality of the relation between the level statistics and the macroscopic symmetries of the underlying classical map but never demonstrate the incompatibility of the quantum symmetry of the quantized map and the spectral statistics of the system. Hence, the above quote from the manuscript appears to be misleading.

3. Section 3 does not look like an "Overview of results". While page 7 contains a summary of the results, the subsequent subsections provide more and more detailed accounts of the results, and the whole Section is inconsistent. For instance, Sec. 3.1 does not provide definitions of level spacing distribution and the mean gap ratio, even though these quantities are shown in Fig. 2 (the two quantities are defined only in Sec. 4.2. by Eq 22 and 23). Conversely, Sec. 3.2 provides the definition of SFF, Eq. (8), which is then repeated in Sec. 5 as Eq. 28.
While Sec. 3.1 may serve as an overview of results about short-range spectral statistics, Sec. 3.2, 3.3, and 3.4 extend over nine pages, which is definitely too much for an "Overview of results."
The consequence of this particular section structure is that the results are described in a fragmented and hard-to-follow manner. The idea of Sec. 3 as an overview of the results is not materialized properly in the present manuscript version.

4. Lines 552-553: "there are dips in the mean gap ratio at specific values of $N$ , which typically correspond to powers of the slope". The dips in the mean gap ratio value in Fig. 8 appear to be equidistant, and the horizontal scale is linear in $N$. This seems to be in contradiction with the statement of the dips being related to the powers of $A$. Also, the dips are denoted only with a single color; does this mean they are observed only for a single value of $A$?

5. Line 575: would it be possible to quantify the degree of symmetry breaking by the smallness of the Hilbert-Schmidt norm of this commutator?

6. The parameter $A$ is sometimes called the "slope" of the baker's map. Can this name be introduced somewhere in Sec. 2 (so that the distinction between SFF's slope is more apparent)?

7. The study focuses on eight types of quantizations of the baker's map, which are summarized in Table 1. Could Table 1 also contain information about the quantum symmetries of the given quantization, i.e. TR and reflection symmetry (and mention also the approximate symmetry)? This could simplify the reader's job of linking the observed properties of level statistics with the degree of symmetry of the quantum system.

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