Boundary Chaos: Spectral Form Factor

1. Calculates the Spectral Form Factor analytically for a class of many-body models, also gives numerical results.
2. Clearly written

1. Connections to physical models is not discussed sufficiently.

The paper deals with a devilishly clever model of a many-body system that has elements of non-integrability 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) T-dual 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 two-body 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 T-dual 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.

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 T-dual case).

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

- Analytical results derived at large local Hilbert space dimension $q$
-Extensive numerical analysis

-Main results are specific to boundary chaos

In this work, the authors investigate the presence of quantum chaos in noninteracting circuits locally perturbed by an interacting gate (T-dual 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 many-body problem to an effective two-body 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 non-universal 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 in-depth 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 many-body 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 non-universal 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 Wigner-Dyson distribution.

-For the Haar impurity, the authors extract a power-law dependence in t/L of the initial non-universal 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 T-dual 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 T-dual.