SciPost Submission Page
Spectral anomalies and broken symmetries in maximally chaotic quantum maps
by Laura Shou, Amit Vikram, Victor Galitski
Submission summary
| Authors (as registered SciPost users): | Laura Shou · Amit Vikram |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202403_00032v3 (pdf) |
| Date submitted: | 2025-03-23 02:24 |
| Submitted by: | Shou, Laura |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Spectral statistics such as the level spacing statistics and spectral form factor (SFF) are widely expected to accurately identify "ergodicity", including the presence of underlying macroscopic symmetries, in generic quantum systems ranging from quantized chaotic maps to interacting many-body systems. 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. Notably, the same classical map can exhibit signatures of different random matrix symmetry classes in short-range spectral statistics depending on the quantization. While the long-range spectral statistics encoded in the early time ramp of the SFF are more robust and correctly identify macroscopic symmetries in several common quantizations, we also demonstrate analytically and numerically that the presence of Berry-like phases in the quantization leads to spectral anomalies, which break this correspondence. Finally, we provide numerical evidence that long-range spectral rigidity remains directly correlated with ergodicity in the quantum dynamical sense of visiting a complete orthonormal basis.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Here, we would like to address the two concerns posed by Referee 1 in the second round.
Regarding their question about mean gap ratios for unequal blocks: indeed the Referee is correct in saying that the mean gap ratios would be different from the predictions of Ref. [48] if the blocks are of unequal size. Given *just* figure 2, one could think of interpolating between the gap ratios for 1 and 2 blocks to "fit" the data. But we emphasize that just because a model can fit the data, it should not be regarded as capturing the underlying physics of the problem.
Moreover, one of the key takeaways from our work is that such a simple fit is not even possible in our case. The spectral form factor in Fig. (4) clearly shows that the early time behavior is often very different from the late time behavior (e.g., consistent with 2 full blocks in the top row at early times, but with differing behaviors at late times), and there is no consistent way to assign a number or relative size of blocks at all times. This is to be contrasted with an average between blocks of unequal size, which would just be a weighted average of the corresponding spectral form factors. Also, in Fig. (6), we show that the behavior of different spectral statistics measures is close to an interpolation between a 1 block and 2 block spectrum (for example), which again is a fit to the data but demonstrates that the fit has to be more complicated than blocks of unequal size. On the whole, we show that we can not interpret the "anomalous" gap ratios as emerging merely from differently sized "symmetry blocks" --- they are instead indicative of genuine dynamical anomalies in the system.
We thank the Referee for their feedback regarding readability of the figures. We have enlarged Figures 1, 2, 6, 11, and 15 to be easier to read. However, most of the other figures (such as Figures 3 and 4) are already at the maximum width for the text. We have considered splitting up several of the figures, but feel that this would run the risk of making it more difficult for the reader to understand the organization of figures, since we already have 16 figures. We have made sure the figures are in high resolution pdf, so that they will be easily viewable electronically.