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Entanglement dynamics in Rule 54: exact results and quasiparticle picture
by Katja Klobas, Bruno Bertini
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Bruno Bertini · Katja Klobas |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202104_00017v2 (pdf) |
| Date accepted: | 2021-09-15 |
| Date submitted: | 2021-08-26 13:55 |
| Submitted by: | Klobas, Katja |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the entanglement dynamics generated by quantum quenches in the quantum cellular automaton Rule 54. We consider the evolution from a recently introduced class of solvable initial states. States in this class relax (locally) to a one-parameter family of Gibbs states and the thermalisation dynamics of local observables can be characterised exactly by means of an evolution in space. Here we show that the latter approach also gives access to the entanglement dynamics and derive exact formulas describing the asymptotic linear growth of all Rényi entropies in the thermodynamic limit and their eventual saturation for finite subsystems. While in the case of von Neumann entropy we recover exactly the predictions of the quasiparticle picture, we find no physically meaningful quasiparticle description for other Rényi entropies. Our results apply to both homogeneous and inhomogeneous quenches.
Author comments upon resubmission
We sincerely thank both referees for their careful reading of our papers, for their relevant comments and for their very positive assessment. We have made a number of modifications to the manuscript to accomodate the comments of the referees, and we expanded the discussion about the quasiparticle picture. To help identifying the changes we highlighted them in red in the new version.
Response to Referee 1
Q: My question is the following. It is known that Rule 54 can be solved using the Bethe Ansatz method (as shown for instance in Ref. [67]). Therefore, in principle one could expect that the folded transfer matrix could also be diagonalized using Bethe Ansatz. Have the authors explored this direction? How does this relate to the tensor-network approach presented by the authors? In fact, a similar approach could presumably be applied to more general (Floquet) integrable evolutions, such as the Heisenberg XXZ chain, where transfer matrices can be also diagonalized via Bethe Ansatz. Has this approach been explored or do the authors believe there is some fundamental obstacle?
R: We thank the referee for the question, which is indeed very natural and interesting. Approaches based on the Bethe Ansatz diagonalisation of a space transfer matrices have a relative long history in the literature of integrable models. Indeed, this is the main idea of the so called quantum transfer matrix approach
[1] Suzuki, M., and M. Inoue (1987), Prog. Theor. Phys. 78, 787.
[2] Kluemper, A. (1992), Ann. Phys. 504, 540.
[3] Kluemper, A. (1993), Z. Phys. B 91, 507.
Although this approach has mostly been applied to the determination of thermodynamic properties, a program to use it to determine the non-equilibrium dynamics from a class of “solvable” initial states has been proposed in
[4] B. Pozsgay, J. Stat. Mech. (2013) P10028.
[5] L. Piroli, B. Pozsgay, and E. Vernier, J. Stat. Mech. (2017) 023106; Nucl. Phys. B 933, 454 (2018),
where it has been used to determine the time (real and imaginary) evolution Loschmidt echo in the XXZ chain. Note that the latter quantity depends on a non-folded transfer matrix.
The problem of this approach is that, apart from being technically very complicated, it can concretely be applied to determine only a few eigenvalues of the space transfer matrix [5]. In the real-time case these eigenvalues are the leading ones for short times but become sub-leading for large enough times [5]. We believe that this is the main reason why this approach has not yet successfully been applied to study the dynamics of local observables and entanglement, which involve more complicated folded transfer matrices.
We believe that a similar problem would arise for Rule 54: even though in principle one can determine the spectrum of W using this approach (in fact, to apply the standard quantum transfer matrix approach one would need a ABA treatment of Rule 54 which is currently not fully developed) this is probably going to be more complicated than the treatment employed here (another advantage of the technique used here is that it is in principle independent of integrability). However, as the leading eigenvectors are ultimately uniquely defined and do not depend on the method used to obtain them, it would be very interesting to find them using integrability methods. First it would be interesting to see whether the “solvable” initial states which we find here correspond to the Bethe Ansatz ones (we believe that this is the case because our solvable states produce pairs of quasiparticles as the solvable states in Bethe Ansatz). Second it would be interesting to understand whether a Bethe Ansatz approach provides an explanation
for the remarkably simple form of the fixed points in Rule 54.
This question of the referee made us realise that it is appropriate to mention such integrability-based approaches to study the space transfer matrix. However, we decided to include a brief discussion about them in Paper I, where we survey a number of different approaches based on the “space-like” evolution.
Response to Referee 2
Q: In the introduction the authors discuss the applicability of the quasiparticle picture to describe the entanglement dynamics. They mention that it has not been applied in systems where the dynamics is not unitary. This is not entirely correct, as it has been extended
recently for free fermions and free bosons models in
https://doi.org/10.1103/PhysRevB.103.L020302
and
https://arxiv.org/abs/2106.11997
and in a related setting in
https://scipost.org/10.21468/SciPostPhys.7.2.024
R: We thank the referee for pointing out these relevant references, which we added to our bibliography.
We decided, however, not to quote them in the introduction, where the discussion is about the physical behaviour, i.e. whether or not the entanglement entropies grow linearly, rather than the precise technique used to describe it. Indeed, in the scenarios discussed in the above references the entropies either saturate or grow logarithmically in time as opposed to the standard linear growth that one observes in the unitary case.
In our view it is more appropriate to mention the above references in Sec. 5, when discussing the quasiparticle picture.
Q: The authors observe that the dynamics of the Rényi entropies is seemingly not compatible with the quasiparticle picture. As they stress this is in accord with previous observations in the literature (Ref. 73). However, they should mention that it is still possible to derive the steady state Rényi entropies (as done in Ref. 73) for interacting integrable systems.
R: We thank the referee for pointing out this. We are now mentioning this point both in Sec. 2 (after equation 2.6) and in Sec. 4 (after equation 4.9).
Q: The most important results of the paper is the calculation of the Renyi entropies. However, in the introduction they are barely discussed. I would suggest to move part of the discussion on the Renyi entropies in the introduction, for instance mentioning why they are important and what has been done already in the literature (i.e., the calculation of their steady-state value in integrable systems).
R: The referee is indeed right in saying that the calculation of the time evolution of Rényi entropies is our main result. However, we prefer to leave the discussion in the introduction on a slightly more general level (talking about entanglement rather than its measures). This is for two main reasons: (i) we believe that talking about entanglement is more significant from the physical point of view; (ii) in many cases the same techniques can be applied to the calculation of several entanglement measures, not only the Renyi entropies.
We nevertheless agree with the referee that it is useful to slightly expand the discussion about Rényi entropies, accounting for the previous literature on the matter. Especially in the case of interacting integrable models. Therefore we expanded the discussion after Eq. 2.6.
Response to Referee 1
Q: My question is the following. It is known that Rule 54 can be solved using the Bethe Ansatz method (as shown for instance in Ref. [67]). Therefore, in principle one could expect that the folded transfer matrix could also be diagonalized using Bethe Ansatz. Have the authors explored this direction? How does this relate to the tensor-network approach presented by the authors? In fact, a similar approach could presumably be applied to more general (Floquet) integrable evolutions, such as the Heisenberg XXZ chain, where transfer matrices can be also diagonalized via Bethe Ansatz. Has this approach been explored or do the authors believe there is some fundamental obstacle?
R: We thank the referee for the question, which is indeed very natural and interesting. Approaches based on the Bethe Ansatz diagonalisation of a space transfer matrices have a relative long history in the literature of integrable models. Indeed, this is the main idea of the so called quantum transfer matrix approach
[1] Suzuki, M., and M. Inoue (1987), Prog. Theor. Phys. 78, 787.
[2] Kluemper, A. (1992), Ann. Phys. 504, 540.
[3] Kluemper, A. (1993), Z. Phys. B 91, 507.
Although this approach has mostly been applied to the determination of thermodynamic properties, a program to use it to determine the non-equilibrium dynamics from a class of “solvable” initial states has been proposed in
[4] B. Pozsgay, J. Stat. Mech. (2013) P10028.
[5] L. Piroli, B. Pozsgay, and E. Vernier, J. Stat. Mech. (2017) 023106; Nucl. Phys. B 933, 454 (2018),
where it has been used to determine the time (real and imaginary) evolution Loschmidt echo in the XXZ chain. Note that the latter quantity depends on a non-folded transfer matrix.
The problem of this approach is that, apart from being technically very complicated, it can concretely be applied to determine only a few eigenvalues of the space transfer matrix [5]. In the real-time case these eigenvalues are the leading ones for short times but become sub-leading for large enough times [5]. We believe that this is the main reason why this approach has not yet successfully been applied to study the dynamics of local observables and entanglement, which involve more complicated folded transfer matrices.
We believe that a similar problem would arise for Rule 54: even though in principle one can determine the spectrum of W using this approach (in fact, to apply the standard quantum transfer matrix approach one would need a ABA treatment of Rule 54 which is currently not fully developed) this is probably going to be more complicated than the treatment employed here (another advantage of the technique used here is that it is in principle independent of integrability). However, as the leading eigenvectors are ultimately uniquely defined and do not depend on the method used to obtain them, it would be very interesting to find them using integrability methods. First it would be interesting to see whether the “solvable” initial states which we find here correspond to the Bethe Ansatz ones (we believe that this is the case because our solvable states produce pairs of quasiparticles as the solvable states in Bethe Ansatz). Second it would be interesting to understand whether a Bethe Ansatz approach provides an explanation
for the remarkably simple form of the fixed points in Rule 54.
This question of the referee made us realise that it is appropriate to mention such integrability-based approaches to study the space transfer matrix. However, we decided to include a brief discussion about them in Paper I, where we survey a number of different approaches based on the “space-like” evolution.
Response to Referee 2
Q: In the introduction the authors discuss the applicability of the quasiparticle picture to describe the entanglement dynamics. They mention that it has not been applied in systems where the dynamics is not unitary. This is not entirely correct, as it has been extended
recently for free fermions and free bosons models in
https://doi.org/10.1103/PhysRevB.103.L020302
and
https://arxiv.org/abs/2106.11997
and in a related setting in
https://scipost.org/10.21468/SciPostPhys.7.2.024
R: We thank the referee for pointing out these relevant references, which we added to our bibliography.
We decided, however, not to quote them in the introduction, where the discussion is about the physical behaviour, i.e. whether or not the entanglement entropies grow linearly, rather than the precise technique used to describe it. Indeed, in the scenarios discussed in the above references the entropies either saturate or grow logarithmically in time as opposed to the standard linear growth that one observes in the unitary case.
In our view it is more appropriate to mention the above references in Sec. 5, when discussing the quasiparticle picture.
Q: The authors observe that the dynamics of the Rényi entropies is seemingly not compatible with the quasiparticle picture. As they stress this is in accord with previous observations in the literature (Ref. 73). However, they should mention that it is still possible to derive the steady state Rényi entropies (as done in Ref. 73) for interacting integrable systems.
R: We thank the referee for pointing out this. We are now mentioning this point both in Sec. 2 (after equation 2.6) and in Sec. 4 (after equation 4.9).
Q: The most important results of the paper is the calculation of the Renyi entropies. However, in the introduction they are barely discussed. I would suggest to move part of the discussion on the Renyi entropies in the introduction, for instance mentioning why they are important and what has been done already in the literature (i.e., the calculation of their steady-state value in integrable systems).
R: The referee is indeed right in saying that the calculation of the time evolution of Rényi entropies is our main result. However, we prefer to leave the discussion in the introduction on a slightly more general level (talking about entanglement rather than its measures). This is for two main reasons: (i) we believe that talking about entanglement is more significant from the physical point of view; (ii) in many cases the same techniques can be applied to the calculation of several entanglement measures, not only the Renyi entropies.
We nevertheless agree with the referee that it is useful to slightly expand the discussion about Rényi entropies, accounting for the previous literature on the matter. Especially in the case of interacting integrable models. Therefore we expanded the discussion after Eq. 2.6.
List of changes
All changes are denoted in red in the manuscript.
Published as SciPost Phys. 11, 107 (2021)