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Hilbert Space Diffusion in Systems with Approximate Symmetries
by Rahel L. Baumgartner, Luca V. Delacrétaz, Pranjal Nayak, Julian Sonner
Submission summary
Authors (as registered SciPost users):  Pranjal Nayak 
Submission information  

Preprint Link:  scipost_202407_00050v1 (pdf) 
Date submitted:  20240729 09:47 
Submitted by:  Nayak, Pranjal 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this paper, we describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. At early times, the symmetries allow us to organize the Hilbert space into approximately decoupled sectors, each of which contributes independently to the SFF. At late times, the SFF transitions into the final ramp of the fully mixed chaotic Hamiltonian. For approximate continuous symmetries, the transitional behaviour is governed by a universal process that we call Hilbert space diffusion. The diffusion constant corresponding to this process is related to the relaxation rate of the associated nearly conserved charge. By implementing a chaotic sigma model for Hilbertspace diffusion, we formulate an analytic theory of this process which agrees quantitatively with our numerical results for different examples.
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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, nonlocal 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 manybody 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 nonlocal, the Thouless time is not explicitly stated. Is it $t_{Th} \sim \frac{1}{Q \Gamma}$? Or is it harder to estimate?
Recommendation
Publish (meets expectations and criteria for this Journal)
Author: Pranjal Nayak on 20240902 [id 4732]
(in reply to Report 1 on 20240826)We thank the referee for their time and feedback. In response to the question raised by the referee, please note that from the diffusion behaviour for the nonlocal exploration that follows from (2.21), the Thouless time is $\ln(Q)/(Q \Gamma)$. At this timescale, the effective number of sectors differs from 1 by an amount 1/Q. We will include a comment stating the same in the revised version of the manuscript.