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Boundary Chaos: Exact Entanglement Dynamics
by Felix Fritzsch, Roopayan Ghosh, Tomaž Prosen
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Submission summary
| Authors (as registered SciPost users): | Felix Fritzsch |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2301.08168v2 (pdf) |
| Date submitted: | 2023-02-01 16:32 |
| Submitted by: | Fritzsch, Felix |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We compute the dynamics of entanglement in the minimal setup producing ergodic and mixing quantum many-body dynamics, which we previously dubbed {\em boundary chaos}. This consists of a free, non-interacting brickwork quantum circuit, in which chaos and ergodicity is induced by an impurity interaction, i.e., an entangling two-qudit gate, placed at the system's boundary. We compute both the conventional bipartite entanglement entropy with respect to a connected subsystem including the impurity interaction for initial product states as well as the so-called operator entanglement entropy of initial local operators. Thereby we provide exact results in a particular scaling limit of both time and system size going to infinity for either very small or very large subsystems. We show that different classes of impurity interactions lead to very distinct entanglement dynamics. For impurity gates preserving a local product state forming the bulk of the initial state, entanglement entropies of states show persistent spikes with period set by the system size and suppressed entanglement in between, contrary to the expected linear growth in ergodic systems. We observe similar dynamics of operator entanglement for generic impurities. In contrast, for T-dual impurities, which remain unitary under partial transposition, we find entanglement entropies of both states and operators to grow linearly in time with the maximum possible speed allowed by the geometry of the system. The intensive nature of interactions in all cases cause entanglement to grow on extensive time scales proportional to system size.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-5-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2301.08168v2, delivered 2023-05-03, doi: 10.21468/SciPost.Report.7137
Strengths
1. Computation of exact results for dynamics of entanglement in a many body system
2. Extremely clear exposition of the diagrammatic methods upon which the calculations are based
Report
This paper presents interesting results on the dynamics of entanglement in a many qubit circuit in 1+1 dimensions, where the dynamics is only nontrivial at the boundary. Earlier work has established the apparent limited complexity of such a system, it has the spectral properties typical of quantum chaos. This work shows that these properties coexist with unusual entanglement dynamics, including revivals at regular intervals due to the simple dynamics of the bulk.
The work makes extensive use of tensor network diagrams, with most of the formal development begin carried out with some very intricate graphical manipulations. Fortunately these are explained very clearly, with the diagrams forming an integral part of the text. The developed formalism is able to deal with both state and operator entanglement measures on an almost identical footing.
I am happy to recommend acceptance, with only a few changes to request.
Requested changes
1. A few minor typos "intergrable" (p3) "vecotrization" (p7)
2. In the description of the diagrams Eq. (24) and (25) it might be useful to explicitly state the values of $L$, $l$, and $t$
3. In Eq. (40) and after it might be clearer to state that the eigenvalue $\lambda_0$ depends on $\tau$.
4. In Figure 2 a specific choice of $J=\pi/4-0.05$ is made. Is there any particular reason for this? Perhaps a clarifying sentence.
Report #1 by Pieter W. Claeys (Referee 1) on 2023-3-17 (Invited Report)
- Cite as: Pieter W. Claeys, Report on arXiv:2301.08168v2, delivered 2023-03-17, doi: 10.21468/SciPost.Report.6916
Strengths
1- Efficient tensor network representation of entanglement entropy in a minimal model of boundary chaos.
2- Exact results for (nontrivial) entanglement dynamics of both initial product states and initial local operators.
Weaknesses
1- Results are restricted to the very specific setup of boundary chaos.
2- Relatively short discussion of entanglement dynamics for initial product states with generic impurity.
Report
In this work the authors consider a minimal model for ergodic and mixing quantum many-body dynamics termed 'boundary chaos', where a brickwork quantum circuit consisting of non-interacting swap gates is perturbed by a single impurity interaction placed at the boundary. In a previous work by two of the authors it was shown that such a model is indeed chaotic and ergodic, in terms of both the spectral statistics and the dynamical correlations. Here the methods developed for this model are extended to the calculation of entanglement dynamics for initial product states and an initial local operator. The geometry of the setup already allows for an efficient tensor network representation of the entanglement entropy, such that numerical results can be obtained for large system sizes and long times. The authors then focus on various scenarios where exact analytics results can be obtained.
1) Initial product states: The authors consider two specific choices of impurity, leading to qualitatively different dynamics. If the impurity supports a 'vacuum' state, the entanglement dynamics exhibits periodic revivals at times proportional to the system size, and if the impurity is chosen to be T-dual the entanglement dynamics exhibit a staircase structure. For generic impurities no analytical treatment is possible, but numerical results suggest a mixture of these two scenarios.
2) Initial local operator: For generic impurities the operator entanglement dynamics is shown to behave in the same way as the entanglement dynamics of an initial product state with an impurity supporting a vacuum, i.e. exhibiting periodic revivals, due to a direct analogy in the calculation when moving to a 'vectorized' representation. For a T-dual impurity a staircase structure is again obtained.
These calculations are underlied by identifying a transfer matrix in the tensor network representation of the entanglement entropy and restricting to cases where the leading eigenspectrum of this transfer matrix can be analytically obtained, similar to various calculations in the literature on dual-unitary circuits.
The analytical calculations are clearly presented and convincing, and well supported by numerics. The paper is overall very well written and the results are sure to be of interest to the community working on entanglement dynamics and chaotic many-body systems. I am happy to recommend this paper for publication in SciPost Physics, provided some minor clarifying comments are addressed. The calculation of entanglement entropies in chaotic quantum many-body dynamics is a notoriously difficult problem, and the authors here present an exact solution for the entanglement dynamics in a minimal but nontrivial model. For this reason I believe that this paper satisfies the SciPost Physics acceptance criteria.
Requested changes
1- In Eq. (21) the definition of the permutation operator is somewhat ambiguous, and does not correspond to the previous definition (C5) from Ref. [65], which is referred to for more details. When trying to reproduce Eq. (22) for $L=5$ I find e.g. the inconsistent result that $\sigma_0(2) = 5 \textrm{ mod 5 }=\sigma_0(5) = 0 \textrm{ mod } 5$, but everything seems to be consistent when using the definition from Ref. [65]. Can the authors clarify?
2- When discussing the dynamics starting from an initial product state the authors consider the cases where the impurity either supports a vacuum state or is T-dual, leading to qualitatively different dynamics. Can the authors comment on what happens if the impurity supports both a vacuum state and is T-dual?
3- Section 3.2.3, presenting numerical results on the entanglement dynamics starting from an initial product state and generic impurity, is somewhat short and does not allow the reader to reproduce the conclusions. A single numerical result is given for a randomly chosen impurity and it is mentioned that the dynamics is a mix of the previously discussed cases, but it is not clear how generic/reproducable this behavior is. It would be useful to give the numerical parametrization of the impurity in Appendix to make the results reproducible. A brief discussion on how the observed dynamics depend on 'how close the impurity is to T-dual' could also be useful, but I do not insist on this.
4- On page 12, I assume that the transfer matrices correspond to the rows of the tensor network (25) rather than the columns, as is written. Is this correct?
5- Analytical results are obtained for Rényi entropies $R_n$ of arbitrary order $n$, but numerical comparisons are restricted to the second Rényi entropy $n=2$. Could the authors comment on how they would expect the numerics and the effect of the subleading terms in the analytical calculatoin to change for higher Rényi index?
6- There are some typos in the manuscript ('intergrable', 'impuirity','vecotrization', 'wit', 'dynamcis', 'sencond').