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Entanglement evolution and generalised hydrodynamics: interacting integrable systems
by Vincenzo Alba, Bruno Bertini, Maurizio Fagotti
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Vincenzo Alba · Bruno Bertini · Maurizio Fagotti |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1903.00467v3 (pdf) |
| Date accepted: | 2019-06-19 |
| Date submitted: | 2019-06-13 02:00 |
| Submitted by: | Bertini, Bruno |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable systems, where the two rates coincide.
Author comments upon resubmission
Dear Editor,
We thank the referees for their careful reading of our manuscript and for their positive assessment. Here are our responses to their questions/comments.
Reply to Referee A:
(1)"In Fig. 3(c) the label should be J_{α,−λ}(t) (sign missing)."
Thank You. The figure was actually correct, there was a typo in the caption. We fixed it.
(2)"In Fig. 5(b) for the solid line there is only one trajectory to be seen. Do both particles indeed follow exactly the same trajectory? If yes is there a simple explanation why this happens?"
The particles corresponding to the solid lines in Fig.5(b) don’t follow exactly the same trajectory, but very close ones. Since lambda~pi/2, they are initially almost still, and their velocities remain very close to one another also when they enter the light cone. This can be understood using Eq. 14 and noting that T_{alpha,beta,-pi/2-mu}=T_{alpha,beta,pi/2-mu}, a_{alpha,-\pi/2}=a_{alpha,-pi/2}, v_{alpha,\pm pi/2}=0.
We added a sentence in the caption of Fig.5 to clarify this point.
(3)"What would change if the group velocity did not have only a single maximum?"
If the group velocity in one of the two leads does not have a single maximum we are unable to prove the non-crossing condition for particles with opposite rapidities. In particular, it is not generically possible to write \Phi_{\alpha,\lambda}(\zeta)-\Phi_{\alpha,-\lambda}(\zeta) in terms of an integral of a positive function, as done in Eq. 113.
Reply to Referee B:
"The only small improvement I would suggest is in the explanations around eq. 12. For appropriate context, I would suggest to mention that this is the hydrodynamic approximation. It is also important to mention that the result holds for x scaling with t as well, as such a scaling is what is used throughout the paper (and maybe mention that on the right-hand side, the only x,t dependence is in the density matrix; the x in the operator is not relevant)."
We agree with the referee: Eq. 12 holds in the hydrodynamic regime. Under very mild assumptions on the initial state this regime is reached for large times after the quench, when the system reached local equilibrium. This is the meaning of the large time limit in Eq. 12. In particular, this is expected to happen for very general inhomogeneous quench protocols, not only the bipartite quench considered in our work. For these reasons we decided to keep the discussion in Sec.3 at the general level, specifying the results to the case under exam in Sec.3.1.
We have added a sentence below Eq. 12 to clarify this point.
Regarding the x dependence in the operator, this can be removed when the state \rho_s(x,t) is homogeneous for each x and t. After Eq. 12 we argue that this is the case for integrable models, but to make the point clearer we decided to split the reasoning in two steps: first we argue that \rho_s(x,t) exists and then that is homogeneous at every fixed x and t.
We thank the referees for their careful reading of our manuscript and for their positive assessment. Here are our responses to their questions/comments.
Reply to Referee A:
(1)"In Fig. 3(c) the label should be J_{α,−λ}(t) (sign missing)."
Thank You. The figure was actually correct, there was a typo in the caption. We fixed it.
(2)"In Fig. 5(b) for the solid line there is only one trajectory to be seen. Do both particles indeed follow exactly the same trajectory? If yes is there a simple explanation why this happens?"
The particles corresponding to the solid lines in Fig.5(b) don’t follow exactly the same trajectory, but very close ones. Since lambda~pi/2, they are initially almost still, and their velocities remain very close to one another also when they enter the light cone. This can be understood using Eq. 14 and noting that T_{alpha,beta,-pi/2-mu}=T_{alpha,beta,pi/2-mu}, a_{alpha,-\pi/2}=a_{alpha,-pi/2}, v_{alpha,\pm pi/2}=0.
We added a sentence in the caption of Fig.5 to clarify this point.
(3)"What would change if the group velocity did not have only a single maximum?"
If the group velocity in one of the two leads does not have a single maximum we are unable to prove the non-crossing condition for particles with opposite rapidities. In particular, it is not generically possible to write \Phi_{\alpha,\lambda}(\zeta)-\Phi_{\alpha,-\lambda}(\zeta) in terms of an integral of a positive function, as done in Eq. 113.
Reply to Referee B:
"The only small improvement I would suggest is in the explanations around eq. 12. For appropriate context, I would suggest to mention that this is the hydrodynamic approximation. It is also important to mention that the result holds for x scaling with t as well, as such a scaling is what is used throughout the paper (and maybe mention that on the right-hand side, the only x,t dependence is in the density matrix; the x in the operator is not relevant)."
We agree with the referee: Eq. 12 holds in the hydrodynamic regime. Under very mild assumptions on the initial state this regime is reached for large times after the quench, when the system reached local equilibrium. This is the meaning of the large time limit in Eq. 12. In particular, this is expected to happen for very general inhomogeneous quench protocols, not only the bipartite quench considered in our work. For these reasons we decided to keep the discussion in Sec.3 at the general level, specifying the results to the case under exam in Sec.3.1.
We have added a sentence below Eq. 12 to clarify this point.
Regarding the x dependence in the operator, this can be removed when the state \rho_s(x,t) is homogeneous for each x and t. After Eq. 12 we argue that this is the case for integrable models, but to make the point clearer we decided to split the reasoning in two steps: first we argue that \rho_s(x,t) exists and then that is homogeneous at every fixed x and t.
List of changes
- Typo corrected in caption of Fig. 3
- Sentence added after Eq. 12
- Sentence added in caption of Fig. 5
- Other minor typos corrected throughout the manuscript
Published as SciPost Phys. 7, 005 (2019)