Hilbert Space Diffusion in Systems with Approximate Symmetries

The authors describe universal behaviour of the spectral form factor (SFF) in presence of approximate global symmetries in a generic quantum system.

1. A particular strength of the work is the integration of the analysis into the sigma model description of chaotic systems. The pattern of the symmetry breaking dictates the nature of approach of the SFF ramp towards the random matrix theory ( RMT) predicted ramp. The sigma model effective action analysis via saddle point methods in the limit of large number of sectors ($N_q$ ) justify that the SFF factorizes into the RMT SFF times the time dependent number of sectors that decay from $N_q$ to 1.

2. The authors go on to study various scenarios and also confirm their quantitative predictions via very clear and reproducible numerics.

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The paper resolves a significant open question regarding how approximate symmetries affect the approach to random matrix universality. Prior work established how exact symmetries affect spectral statistics, but the intermediate regime of approximate symmetries remained poorly understood. This work provides a comprehensive framework for this transitional regime. The work has significant implications for understanding quantum thermalization and information scrambling in complex (even dynamical) systems which most of the time have approximate symmetries. I recommend this article for publication.

I also seek a few answers which the authors may choose to answer either as reply to this report / as additional parts to their paper, however some of these questions maybe outside the scope of the present work.

1. Is there a generalization of the $\sqrt{t}$ diffusive behaviour to other power laws when the perturbation connects not all but finite number (>1) of sectors? If there is a hierarchy of symmetry breakings, what kind of generalized diffusion is expected? Are there any such hierarchical SFF behaviour examples that the authors’ work may explain?

2. Can there be any interesting universalities also around the AA saddle that can carry any transport signatures of the approximate symmetries in the plateauing behaviour, are these supposed to always be 1/L suppressed or can they compete when $N_q$ scales as some power of L (for instance in Hilbert space fragmentation like scenarios) ?

3. Do the authors expect that the lessons of approximate symmetry breaking and Hilbert space diffusion also to hold for other scenarios [e.g. they mention U(1) breaking holographic example in 5.1] An interesting scenario is the independent spin sector chaos in a large c conformal field theory (CFT) as discussed in https://arxiv.org/abs/2302.14482. If the spin organization gets broken due to a deformation of the CFT, can one import the results from here to say anything about Hilbert space Diffusion phenomena in the perturbed CFT / can it give any holographic clues ?

4. If the authors had included $\beta$ dependence also in the SFF or discussed a microcanonical filtered SFF, then can the bottleneck mentioned at the end of section 4 be made more quantitative?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

In my previous report I had asked the authors only one question about the Thouless time of non-local interactions which they addressed and have included in the new version. I am satisfied with their reply. I stand by my previous report (reproduced below) and recommend that this paper be published in SciPost.

OLD REPORT:

In this paper, the authors study spectral form factor in chaotic system with approximate symmetries. Random matrix universality explains the ramp in the spectral form factor, however any structure in the system leads to deviation from the random matrix result. The authors quantitatively find these deviations from the low lying modes of the symmetry broken effective action. This sigma model analysis closely mirrors chiral symmetry breaking and chiral perturbation theory in QCD.

The authors consider in detail two cases for the symmetry breaking term. 1) Local interactions that connect only neighboring charge sectors which leads to a diffusive $\sqrt{t}$ behaviour for the spectral form factor at intermediate times. 2) All to all, non-local interactions of the charge sectors that leads to a fast exponential decay to the ramp. Both these cases are also verified numerically in a toy model. The first case is also studied in the charged SYK model. In all cases the authors find excellent agreement between numerics and the analytic predictions.

This paper meets all the general acceptance criteria of SciPost Physics and satisfies multiple expectations. For a general system, it is hard to get analytical handle on the spectral form factor, and often numerical analysis is the only viable tool. So the fact that the authors have a quantitative understanding is impressive. Their method is based on symmetry breaking and could be applied to a myriad of systems in the future, from many-body systems to holographic theories. Many potential applications are discussed in the final section.

Overall it is a good piece of work and I recommend it for publication.

I have only one question for the authors. In the case of local interactions, the process is diffusive and the Thouless time is found to be $t_{Th} \sim \frac{Q^2}{\Gamma}$. However when the interaction term is non-local, the Thouless time is not explicitly stated. Is it $t_{Th} \sim \frac{1}{Q \Gamma}$? Or is it harder to estimate?

Publish (meets expectations and criteria for this Journal)