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Boundary Chaos: Spectral Form Factor
by Felix Fritzsch, Tomaž Prosen
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Submission summary
Authors (as registered SciPost users):  Felix Fritzsch 
Submission information  

Preprint Link:  https://arxiv.org/abs/2312.12452v1 (pdf) 
Date submitted:  20231227 16:20 
Submitted by:  Fritzsch, Felix 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Random matrix spectral correlations is a defining feature of quantum chaos. Here, we study such correlations in a minimal model of chaotic manybody quantum dynamics where interactions are confined to the system's boundary, dubbed \textit{boundary chaos}, in terms of the spectral form factor and its fluctuations. We exactly calculate the latter in the limit of large local Hilbert space dimension $q$ for different classes of random boundary interactions and find it to coincide with random matrix theory, possibly after a nonzero Thouless time. The latter effect is due to a drastic enhancement of the spectral form factor, when integer time and system size fulfill a resonance condition. We compare our semiclassical (large $q$) results with numerics at small local Hilbert space dimension ($q=2,3$) and observe qualitatively similar features as in the semiclassical regime.
Current status:
Reports on this Submission
Strengths
1. Calculates the Spectral Form Factor analytically for a class of manybody models, also gives numerical results.
2. Clearly written
Weaknesses
1. Connections to physical models is not discussed sufficiently.
Report
The paper deals with a devilishly clever model of a manybody system that has elements of nonintegrability while keeping the simplicity in the bulk via swap operators. Pushing all the complexity to the border, it is inspired by billiards: popular models of single particle chaos. The authors mention this at places and give insights into how a certain procedure is like the Poincare surface of section.
The primary quantity that is calculated is the spectral form factor (SFF) that forms an important tool to study chaos. The boundary gates that provide the single "impurity" are taken from (1) product gates, (2) Tdual gates: random unitaries times products and (3) Haar random. ary quantity that is calculated is the spectral form factor (SFF) that forms an important tool to study chaos.
As the bulk has swap operators, the authors are able to reduce the SFF calculation to a twobody problem each of local dimension q^t. For the product gate the SFF is found to be a higher power of t than for single CUE or two independent CUE. While for the random gate case the CUE result is reached for large q. The Tdual case is also like the random but is controlled by the coupling parameter and leads to a "thouless time".
The calculations seem fairly technical, but enough details are provided in the appendix for a determined reader.
Requested changes
1. In Ref. 18, https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.121.060601
the authors have found a phase transition (maybe MBL) as a function of what in the current paper is probably $J$. Maybe the authors can comment on it in the context of their model (esp. the Tdual case).
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
 Analytical results derived at large local Hilbert space dimension $q$
Extensive numerical analysis
Weaknesses
Main results are specific to boundary chaos
Report
In this work, the authors investigate the presence of quantum chaos in noninteracting circuits locally perturbed by an interacting gate (Tdual or Haar random) acting at the system’s boundary (boundary chaos). This is done through the analysis of the spectral form factor (SFF) and of its moments. In particular, results in the limit of large local Hilbert space dimension $q$ are obtained analytically by reducing the manybody problem to an effective twobody calculation. Their analytical findings at large $q$ agree with the expected results from random matrix theory (RMT) . The authors then compare these exact results to numerical simulations performed for small $q=2,3,4$. They show that the main features of SFF and its moments are captured by RMT also at small $q$ for times < Heisenberg time and > of a Thouless time, before which the SFF displays a nonuniversal behavior. Furthermore, their extensive numerical analysis gives an estimate of the Thouless time, depending on system size and the type of interacting gates perturbing the circuit.
The paper is well written, and the derived results are of interest for the community. I believe that the results contained in this manuscript deserve publication, and easily meet the acceptance criteria of SciPost Physics Core. However, my main concerns regarding the publication in SciPost Physics is that the indepth analysis of SFF for boundary chaos does not provide general hints for other settings. I understand that this model serves as a minimal model for studying quantum manybody chaos, but I would appreciate if the authors can comment more on generic features that this study can highlight for other setting.
Some other minor comments are listed below.
The authors may include the curve obtained with numerics for q=4 L=6 in Figure 1b and qualitatively comment on the observed deviations from RMT during the initial nonuniversal regime already at the beginning of Sec. 5.
Similarly, it would be nice to add the results for q=4 L=6 in the plot of Figure 2b and show theconvergence towards a WignerDyson distribution.
For the Haar impurity, the authors extract a powerlaw dependence in t/L of the initial nonuniversal regime, with exponent $\nu\approx 4$ for the scaled SFF and for its moments. Is the value of this exponent obtained as a fitting parameter? Do the authors have an understanding for this value?
Still on the data collapse for the Tdual impurity (Figure 6c). Is there an understanding of the factor $1/L^2$ in the scaled SFF needed to observe data collapse (and consequently of the Thouless time scaling exponent $\mu=(2\nu)$)? Also, do the authors have an understanding —beyond the numerical evidence— on why the exponent $\mu$ for m=2 (Figure 8c) is different from that of the SFF? Do the authors have an estimate of the exponent $\mu$ for higher moments?
 As the authors pointed out at the end of page 10, the Thouless time (for both types of impurities) it is expected to decrease when q is increased at fixed L. It would be nice to see how much the exponent $\nu$ changes for q=2,3 in the Haar case, and possibly $\mu$ for m=1,2 and q=3,4 in the Tdual.