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Measurement induced transitions in non-Markovian free fermion ladders
by Mikheil Tsitsishvili, Dario Poletti, Marcello Dalmonte, Giuliano Chiriacò
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Submission summary
| Authors (as registered SciPost users): | Giuliano Chiriacò · Marcello Dalmonte · Mikheil Tsitsishvili |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202312_00020v1 (pdf) |
| Date submitted: | 2023-12-13 14:13 |
| Submitted by: | Chiriacò, Giuliano |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Recently there has been an intense effort to understand measurement induced transitions, but we still lack a good understanding of non-Markovian effects on these phenomena. To that end, we consider two coupled chains of free fermions, one acting as the system of interest, and one as a bath. The bath chain is subject to Markovian measurements, resulting in an effective non-Markovian dissipative dynamics acting on the system chain which is still amenable to numerical studies in terms of quantum trajectories. Within this setting, we study the entanglement within the system chain, and use it to characterize the phase diagram depending on the ladder hopping parameters and on the measurement probability. For the case of pure state evolution, the system is in an area law phase when the internal hopping of the bath chain is small, while a non-area law phase appears when the dynamics of the bath is fast. The non-area law exhibits a logarithmic scaling of the entropy compatible with a conformal phase, but also displays linear corrections for the finite system sizes we can study. For the case of mixed state evolution, we instead observe regions with both area, and non-area scaling of the entanglement negativity. We quantify the non-Markovianity of the system chain dynamics and find that for the regimes of parameters we study, a stronger non-Markovianity is associated to a larger entanglement within the system.
Author comments upon resubmission
Thank you for handling our manuscript, and for forwarding the reports provided by the Refer-
ees. We hereby resubmit our manuscript for publication in SciPost Physics Core.
Both Referees had a positive view of the manuscript and think that the investigated system
is interesting and exhibits potential new physics. They had some comments on the technical
side, in particular on the small system sizes of the numerical simulations. We have performed
additional simulations and improved the precision of our numerics.
We hope that with these changes our work is now suitable for publication in SciPost Physics
Core.
Yours sincerely,
M. Tsitsishvili
D. Poletti
M. Dalmonte
G. Chiriacò
List of changes
1. Modified Fig.(8), now showing the finite size correction to the scaling of the fermionic negativity (page 14).
2. Also modified the related discussion in the main text (end of page 13 and beginning of page 14).
3. Added Fig.(11) and corresponding paragraphs to the 4th section ”Measures of non-Markovianity”, where we use the Gaussianity of the state and measure the degree of non-Markovianity using the square trace distance measure (page 17).
4. Added an appendix D ”Residual Analysis of the data fit for Negativity” (page 25).
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-2-20 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202312_00020v1, delivered 2024-02-20, doi: 10.21468/SciPost.Report.8588
Report
Thank authors for responding to my questions and comments in great details. I am generally OK with the reply and would like to recommend for publication.
However, I realize that the authors might misunderstood my comments on the numerical calculation of $N(\phi)$. Let me clarify myself again here. I agree that sum of Gaussian density matrices, $\rho = \sum_\alpha \rho_\alpha/N_{tr}$, is generally no longer Gaussian. However, since each individual $\rho_\alpha$ is still a Gaussian state, it is not difficult to write $\rho_\alpha$ as a matrix in the Fock space (For example, if $\rho_\alpha = e^{-c^\dagger M c}$, we can first write down the matrix representation of $\rho_\alpha$ in the basis spanned by the single-particle eigenstates of $M$, and then unitarily rotate it to a reference basis). Then one just compute $\rho$ by summing these matrices, and then compute $N(\phi)$ etc. For a chain of length $L$, the dimension of the Fock space is $2^L$, which can be handled easily when $L\leq 12$.
I might miss additional technical difficulties and will be curious of authors' response. If it is indeed a feasible approach, I would strongly recommend the authors to try to extend the numerics to $L=10$ or even $L=12$.
Report #1 by Anonymous (Referee 2) on 2024-2-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202312_00020v1, delivered 2024-02-10, doi: 10.21468/SciPost.Report.8537
Report
I thank the authors for their reply. I had not appreciated some of the complications in their numerical simulations of this model (e.g. the fact that they are sometimes looking at a statistical mixture of Gaussian states which is not itself Gaussian). Nonetheless the additions and improvements are nice, including the new method based on the Frobenius distance that allows them to push the number of sites from L=4 to L=8 in the analysis of Markovianity. I am in favor of publishing the revised paper.
Author: Giuliano Chiriacò on 2024-02-26 [id 4322]
(in reply to Report 1 on 2024-02-10)We thank the Referee for the positive recommendation on the resubmitted version of our manuscript.
Author: Giuliano Chiriacò on 2024-02-26 [id 4323]
(in reply to Report 2 on 2024-02-20)We thank the Referee for the positive comments on the revised version of our manuscript.
We had actually not understood exactly what the Referee meant in the previous report. We are grateful to the Referee for clarifying it. However, the suggested method of using Gaussian states for the time evolution and then calculating the full density matrix along each trajectory, still contains an exponential complexity like the ED method. This makes it more costly than the method purely based on correlations of Gaussian states, and also more costly than the ED method, which does not require a parallel evolution and then an average over different trajectories. We would also note that the proposed method is feasible for $L\leq12$ in a single chain, which becomes $L\leq6$ in the ladder model we consider, i.e. close to the systems sizes we reach with ED and smaller than the system sizes studied with the purely Gaussian-based method. We have added a brief comment on this in the text, along with a table summarizing the computational complexity of the different methods.