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Chaos enhancement in large-spin chains

by Yael Lebel, Lea F. Santos, Yevgeny Bar Lev

Submission summary

Authors (as Contributors): Yevgeny Bar Lev
Submission information
Arxiv Link: (pdf)
Date submitted: 2022-04-06 18:30
Submitted by: Bar Lev, Yevgeny
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We study the chaotic properties of a large-spin XXZ chain with onsite disorder and a small number of excitations above the fully polarized state. We show that while the classical limit, which is reached for large spins, is chaotic, enlarging the spin suppresses quantum chaos features. We attribute this suppression to the occurrence of large and slow clusters of onsite excitations, and propose a way to facilitate their fragmentation by introducing additional decay channels. We numerically verify that the introduction of such relaxation channels restores chaoticity for large spins, so that only three excitations are required to achieve strong level repulsion and ergodic eigenstates.

Current status:
Awaiting resubmission

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2022-7-28 (Invited Report)


1- Identifies instance of model with classical chaos but with slow approach to quantum chaos especially for large spins where naively quantum chaos might be expected to be more pronounced.
2- The ergodic properties of model studied in the paper are of general interest especially in the spin one-half case. This paper partially addresses the physics of the higher spin case.


Possible weaknesses:

1- The authors could strengthen their introduction with a discussion of the broader significance of their findings.
2- The non-gaussian distributions and non-GOE r-statistics are interesting results but there appears to be little understanding of these features. The discussion of slowly relaxing clusters sounds reasonable but various aspects of this picture have not been clearly established it seems.
3- Although the paper presents results for various system sizes, spin sizes and magnetization sectors (and despite figures showing some trends) there is no systematic overview of where anomalous distributions can be observed.
4- The results are fairly clear on the finite size features of r-statistics and off-diagonal matrix element distributions. But the paper seems to miss an opportunity to explain in the text the significance for the thermalizing behavior in the thermodynamic limit (to the degree this can be accessed by finite size scaling and despite the fact that various system sizes were examined) and the connection to the spin one-half case - for example by tuning the disorder strength.


The general context for the work is how higher spin models approach the classical limit and the degree to which s>1/2 are like their spin-1/2 counterparts. The particular higher spin models studied here are of interest also from the perspective of MBL.
Perhaps the most appropriate acceptance criterion here is #3: "Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work".

At the moment the paper has some tantalizing results that could be presented in a more systematic way to help guide future work in these directions.

Requested changes

- The matrix element distributions suggest persistent non-gaussianity for the system sizes investigated. Is this consistent with the finite size scaling of the r-statistics? i.e. it would be good to see the r-statistics data re-plotted versus L.
- The data plotted in Fig 3 is the most fine-grained picture of the anomalous large spin behavior. What is the regime where the distribution is gaussian and what is the regime where it is cusped as S, N and disorder strength are varied? Can one conclude anything from these trends? Maybe a "phase diagram" would be helpful here.
- Is there some understanding of the cusps at zero in Fig 3 (c,d)?
- Relatedly, the authors conjecture that clusters of excitations can have slow dynamics but the results that are presented are r-stats and matrix element distributions. Is it clear that the fat tails are coming from such clusters in real space?
- Is it clear that the departure from GOE statistics can be accounted for by states contributing to the fat tails in the matrix element distribution?
- I do not understand the claimed "slow-relaxing clusters". This is despite the plausible statements just before section V and section V itself where additional terms are added to the Hamiltonian that assist the system towards chaos and that target highly excited local states. My confusion relates to (i) the discrepancy between static observables and dynamical statements in the paper (ii) the role of disorder. Concretely, it would be good to present some measure (e.g. the entanglement) across the spectrum to look for the distribution of anomalous states and their density. How does this picture vary with disorder strength (in the relatively low disorder regime)?
- Again on the slow-relaxing clusters. It would be good to have a concrete picture either directly tied to calculations of the dynamics or features of the spectrum. e.g. Is there any sense in which they are emergent single particle states (e.g. bound states) that localize in the disorder potential and bias the spectrum towards Poissonian r-stats? If not, how should one think of them and their effects on the spectrum and especially the fat tails and cusps in the matrix element distribution.

  • validity: high
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2022-5-12 (Invited Report)


In their work, the authors address the chaotic features of spin chains described by a quantum non-integrable Hamiltonian, the anisotropic Heisenberg model in the presence of a weak disordered field and its connection to the -expected- classical limit; in which the magnitude of the local spin is very large. In particular, the authors are interested in approximating the minimum number of excitations required to observe the signatures of chaos both in the low-magnitude and large-magnitude spin cases.
It is shown that the classical model (for which the spin magnitude is large and local spin matrices commute) is chaotic, in the sense of the presence of a positive Lyapunov exponent that signals this regime.
The authors then proceed to evaluate numerical quantities that are considered to signal chaotic features, such as level spacing statistics and probabilistic distributions of off-diagonal matrix elements of local observables in the energy eigenbasis. Their results indicate that increasing the magnitude of the local spin reduce chaotic behaviour, so they propose to introduce certain perturbations that might enhance chaos in this regime.
I find the paper easy to read and I particularly like their conclusion. I, however, would not recommend publication unless the following points are appropriately addressed:

Requested changes

1- Section IV: Random matrices fail to describe the richness of the structure of microscopic Hamiltonians, such as the Heisenberg model presented here, as posited by the eigenstate thermalisation hypothesis. Not only are treatments of random matrices oblivious to energetic considerations (effectively only partially describing infinite-temperature ensembles), but it has now been pointed out that the matrix elements of local observables in the energy eigenbasis cannot be independent and identically distributed random variables. Granted, they appear to be so in very small frequency scales.
This was suggested to be reason behind the exponential growth of out-of-time order correlators (and positive Lyapunov exponents) by Foini and Kurchan in:
The frequency-dependent presence of statistical correlations was later characterised by Richter et. al. in:
Furthermore, it was later shown that the frequency-dependent presence of statistical correlations results in an important consequence in the dynamics of high order correlation functions by Brenes et. al. in:
I believe the authors need to discuss these considerations in further detail, as I find the first paragraph in Section IV to be an oversimplification of what has now been described as intricate qualities of quantum chaos.
2- The authors employ level spacing statistics and distributions of matrix elements of local observables in the energy eigenbasis to infer chaotic behaviour. What are the caveats of considering these quantities as sole indicators? Level spacing distributions, as presented in this paper for instance, only considers local effects between the eigenvalues. Quantum chaos has been characterised from the perspective of deformations in the adiabatic gauge potential:
and spectral form factors:
How do these connect to each other? Are we allowed to assume that a generic system with a GOE or GUE distribution of spacings will in turn display all the qualities of a chaotic system, such as, for instance, the emergence of hydrodynamical behaviour of transport?
I believe a discussion on these topics is of the essence.
3- Section VI: "...In previous studies of spin- 1/2 chains, it was established that at least 3 spin excitations are required to achieve quantum chaos..." This statement needs citations. The authors are probably referring to their previous submission to SciPost.
4- Section III: The algorithms and methods described in Refs. [34] and [35] should, at least, be described in a minimal fashion in the paper.
5- The results exposed in Figure 1 need to be further described. I, for instance, do not understand why there are several points of MLE for a single phase deviation $\delta \theta$. At first I thought each point for a given $\delta \theta$ corresponded to a disorder realisation, however, in the caption it is stated that each point is an ensemble average. I believe further clarification is required.

I believe these are additions that would be appreciated by a future reader.

  • validity: good
  • significance: good
  • originality: ok
  • clarity: high
  • formatting: excellent
  • grammar: perfect

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