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Dual symplectic classical circuits: An exactly solvable model of many-body chaos
by Alexios Christopoulos, Andrea De Luca, D L Kovrizhin, Tomaž Prosen
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Alexios Christopoulos |
| Submission information | |
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| Preprint Link: | scipost_202307_00022v1 (pdf) |
| Date submitted: | July 17, 2023, 11:33 a.m. |
| Submitted by: | Alexios Christopoulos |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, for these models, the rotational symmetry leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Montecarlo simulations, displaying excellent agreement for different choices of observables.
Current status:
Reports on this Submission
Report #3 by Pieter W. Claeys (Referee 3) on 2023-9-15 (Invited Report)
- Cite as: Pieter W. Claeys, Report on arXiv:scipost_202307_00022v1, delivered 2023-09-15, doi: 10.21468/SciPost.Report.7828
Strengths
1- Exactly solvable models of classical dynamics 2- Natural generalization of dual-unitarity to classical dynamics 3- Detailed analysis and exact reuslts for a representative example
Weaknesses
1- No explicit discussion of notions of chaos/integrability
Report
Specifically, it is shown that in these models two-point correlation functions vanish everywhere except on the edge of the causal light cone, where they can be calculated using a transfer operator formalism. In the specific case of an Ising Swap model with additional single spin rotations, the authors explicitly analyse this transfer operator. It is shown that the transfer operator conserves total angular momentum and various exact results on the eigenspectrum are presented, including autocorrelation functions for the $S_z$ spin component.
The results are interesting, convincing, and well presented. Furthermore, there are various results in the growing literature on dual-unitarity that can subsequently be studied in these classical circuits, such that this work opens up new pathways in this research direction. As such, I am happy to recommend this work for publication in SciPost Physics provided some questions/comments are addressed.
Requested changes
1- One of the most remarkable properties of dual-unitary circuits is that it could be explicitly shown that these circuits satisfy the usual 'definition' of quantum chaos, with major contributions from one of the authors. However, even though the title of this work is "An exactly solvable model of many-body chaos", there is no discussion about whether or not these circuits are chaotic (although the ergodicity is discussed). Since classical chaos is well defined, it would be interesting if the authors could explicitly comment on the chaotic properties of their analysed Ising swap model. Lyapunov exponents are mentioned in the introduction, but then not discussed in the main text. In a related comment, classical and quantum notions of integrability are defined differently, and it would be interesting if the authors could comment on how these notions are satisfied in the presented model when mentioning the integrable points of the Ising swap gate.
Some minor comments 2- On page 3, it is mentioned that the spectrum of the Jacobian needs to include pairs of eigenvalues and that the Lyapunov exponents appear in pairs. Could the authors provide either an argument or a reference as to why that is the case? 3- In the same section, when introducing the model, the authors write "symplectic maps always involve $d$-pairs of conjugate variables,the configuration $q$ and the momentum $p$, which can be seen as the coordinates of a $2d$ dimensional manifold", before discussing such properties as those in my previous point. However, in the specific example considered in this manuscript the authors consider the dynamics of spin variables. It might be worthwhile to also discuss/mention spin variables in that section and the connection with typical pairs of conjugate variables. 4- Small typo below Eq. (25): "The transfer operator is just the Perron–Frobenius of ..." 5- Conservation of total angular momentum typically follows from rotational symmetry, and it might be useful to discuss this symmetry in the main text. Right now it is mentioned below Eq. (B.6) that "This is not surprising, since as we can see from (B.3),(B.6) the transfer operator is just a composition of rotations, which preserve the total angular momentum." It would be useful to include such a discussion in the main text.
Report #2 by Anonymous (Referee 2) on 2023-9-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202307_00022v1, delivered 2023-09-13, doi: 10.21468/SciPost.Report.7823
Report
This set of models, which are natural classical versions of so-called dual-unitary circuits previously studied, were not introduced in this paper. However, the authors develop a general formalism to compute analytically dynamical correlation functions, similar to the quantum case. The general formalism is nicely applied to one specific model (the Ising swap model) and the predictions are tested quantitatively against numerical Montecarlo data.
Overall, I think this is a strong paper. Although perhaps not extremely innovative, it fills a gap in the literature, by extending to the classical case the formalism previously developed in the context of dual-unitary quantum circuits. The draft is well written and all the predictions are supported by convincing numerics. I don't have any particular comment on how to improve the readability of the draft, so I recommend publication of the manuscript as is.
I have, however, just two questions/curiosities, and two comments about references.
First, I believe that, compared to the quantum case, this model could be useful to study more easily the periodicity of the classical orbits and how this depends on integrability/non-integrability of the model. It would be interesting to study in particular the scaling of the orbit sizes with the system size. Do the authors have some intuition about this aspect?
Second, would it be possible to also generalize, say, tri-unitary quantum circuits (as introduced in https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.043046) to the classical case?
Regarding the literature, when stating
"Interestingly, dual unitary quantum circuits can exhibit strongly chaotic quantum dynamics, whose classical simulation is in general expected to be exponentially hard in system size"
I believe it would be important to cite
https://quantum-journal.org/papers/q-2022-01-24-631/
where a rigorous result along these lines was proven.
Finally, two very trivial comments. It appears T. Prosen is misspelled (I believe?) as "T. c. v. Prosen" in multiple entries in the literature. Also, journal names appear sometimes in short-hand notation (e.g. Phys. Rev. Lett.), sometimes with full names (e.g. Physics Letters A). The authors might want to make the notation uniform.
See the attached document for the changes to the manuscript and answers to the reviewers.
Attachment:
Report #1 by Anonymous (Referee 1) on 2023-8-28 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202307_00022v1, delivered 2023-08-28, doi: 10.21468/SciPost.Report.7736
Strengths
- Generalizes ideas from dual unitary circuits to classical deterministic ones
- Proves several exact results, in particular for two-point correlators
- Brings methods more commonly used in quantum dynamics (circuits, tensor networks) to classical many body
- Shows explicitly (and in full detail for a class of examples) how deterministic overall dynamics contracts to stochastic dynamics for a subset of degrees of freedom (2-pt correlators in this case)
Weaknesses
Report
Requested changes
-
Montecarlo -> Monte Carlo in the abstract.
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Please clarify what the "hooks" are meant to be at the right and left edges of Fig.2.
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Present the Appendices in the same order as they are mentioned in the main text.
See the attached document for the changes to the manuscript and answers to the reviewers.
Attachment:
Thank you very much for your comments and requested changes. I am going to take them into consideration. The "hooks" or else the curly edges of the graph in Fig.2 indicate the periodic boundary conditions in space for our system. Following your suggestion, I am going to clarify this in the caption of the respective figure.
Author: Alexios Christopoulos on 2023-11-27 [id 4150]
(in reply to Report 3 by Pieter W. Claeys on 2023-09-15)See the attached document for the changes to the manuscript and answers to the reviewers.
Attachment:
resubmission_letter_Ug6bl8v.pdf