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Adiabatic gauge potential and integrability breaking with free fermions
by Balázs Pozsgay, Rustem Sharipov, Anastasiia Tiutiakina, István Vona
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Authors (as registered SciPost users):  Balázs Pozsgay 
Submission information  

Preprint Link:  https://arxiv.org/abs/2402.12979v2 (pdf) 
Date submitted:  20240306 13:37 
Submitted by:  Pozsgay, Balázs 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We revisit the problem of integrability breaking in free fermionic quantum spin chains. We investigate the socalled adiabatic gauge potential (AGP), which was recently proposed as an accurate probe of quantum chaos. We also study the socalled weak integrability breaking, which occurs if the dynamical effects of the perturbation do not appear at leading order in the perturbing parameter. A recent statement in the literature claimed that integrability breaking should generally lead to an exponential growth of the AGP norm with respect to the volume. However, afterwards it was found that weak integrability breaking is a counterexample, leading to a crossover between polynomial and exponential growth. Here we show that in free fermionic systems the AGP norm always grows polynomially, if the perturbation is local with respect to the fermions, even if the perturbation strongly breaks integrability. As a byproduct of our computations we also find, that in free fermionic spin chains there are operators which weakly break integrability, but which are not associated with known long range deformations.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2024628 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2402.12979v2, delivered 20240628, doi: 10.21468/SciPost.Report.9312
Report
Report on the manuscript entitled Adiabatic gauge potential and integrability breaking with free fermions by Balázs Pozsgay, Rustem Sharipov, Anastasiia Tiutiakina and István Vona.
The Authors say to incorporate a new scenario to the already known exponential or polynomial scaling with system size of the norm of socalled adiabatic gauge potential (AGP) for chaotic or integrable manybody quantum systems, respectively. Namely, for free fermionic models subjected to local (with respect to fermions) weak or strong integrability breaking perturbations, the AGP norm scales polynomially when $\lambda$ is set to zero. The weak and strong local perturbations are represented by four and six fermions terms respectively. For the former kind of perturbation, the Authors explicitely proved that indeed they are “always weakly breaking integrability”. Their staments are supported by numerical results on the integrable XX model. However, for finite $lambda$ departing from zero, the Authors also find a crossover to the exponential scaling for large enough $\lambda$.
In general, I do find the manuscript well written, technically sound and timely. I consider this work as a good contribution to the theory of manybody quantum chaos. I feel comfortable suggesting publication in SciPost, however, before this happens I would like to ask the Authors to clarify something that could seem trivial but it is confusing me.
The picture they draw departs from considering a $\lambda$dependent Hamiltonian $H(\lambda) =H_0 + \lambda V$, where in particular $H_0 =$ XX. To my understanding, for $\lambda = 0$ the AGP norm scales polynomially with system size, just as it should be expected for an integrable model, the XX model. Once \lambda increases from zero the exponential scaling with system size is eventually recovered, just as it should be for chaotic systems. The crossover from polynomial to exponential scaling of the AGP norm for intermediate values of $\lambda$ should be also expected, since this kind of behavior is well known in the manybody quantum chaos literature, say, for instance, from a level spacing statistics perspective. What does mean $\lambda=0$ in the context of the AGP not of the Hamiltonian? Please, could you clarify if I missunderstood?
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Publish (meets expectations and criteria for this Journal)
Report #1 by Anonymous (Referee 1) on 2024530 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2402.12979v2, delivered 20240530, doi: 10.21468/SciPost.Report.9157
Report
Recently the scaling behavior of the adiabatic gauge potential has been introduced as a new measure for quantifying and defining quantum chaos. Specifically, according to current literature results one expects exponential growth of the AGP norm with volume for strong integrability breaking. The present manuscript gives analytical bounds for strongly integrability breaking local perturbations in a fermion system which show that the growth of the AGP norm is only polynomial, contradicting these claims in the literature. The analysis is backed up by compelling numerical data.
In summary the manuscript provides an interesting results on a very current much debated topic. I support publication in its current form.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)