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Thermalization of long range Ising model in different dynamical regimes: a full counting statistics approach
by Nishan Ranabhat, Mario Collura
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Mario Collura · Nishan Ranabhat |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.00533v3 (pdf) |
| Date accepted: | 2024-02-06 |
| Date submitted: | 2023-12-07 12:57 |
| Submitted by: | Ranabhat, Nishan |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study the thermalization of the transverse field Ising chain with a power law decaying interaction $\sim 1/r^{\alpha}$ following a global quantum quench of the transverse field in two different dynamical regimes. The thermalization behavior is quantified by comparing the full probability distribution function (PDF) of the evolving states with the corresponding thermal state given by the canonical Gibbs ensemble (CGE). To this end, we used the matrix product state (MPS)-based Time Dependent Variational Principle (TDVP) algorithm to simulate both real time evolution following a global quantum quench and the finite temperature density operator. We observe that thermalization is strongly suppressed in the region with strong confinement for all interaction strengths $\alpha$, whereas thermalization occurs in the region with weak confinement.
Author comments upon resubmission
Dear Editor,
We express our sincere thanks to the esteemed referees for dedicating their valuable time to thoroughly reviewing our manuscript and for offering constructive comments. We also extend our apologies for the relative delay in resubmission. In response to the insightful assessments and suggestions provided by the reviewers, we implemented substantial additions and modifications to our previous manuscript. We believe that these enhancements have improved the comprehensiveness of our work, rendering it worthy of publication in SciPost Physics core. Below, we systematically address the questions raised by the referees and the additions we made to the manuscript,
1) Detail on calculating PDF in MPS formalism (Report 2)
We have added Appendix B2 to outline the procedure to calculate full probability distribution function in MPS formalism along with a tensor network diagram in Figure 12.
2) Benchmark TDVP for with exact diagonalization(Report 2)
We have introduced Appendix B3 to address concerns regarding the accuracy of the numerical data presented in the manuscript. In Figure 13, we have bench-marked the TDVP data for 14-site system with exact diagonalization. The details of time evolution with exact diagonalization procedure is presented in Appendix A. In Panel (a) of Figure 13, the absolute error in the energy density is plotted for different system parameters as a function of the inverse temperature. The absolute error remains consistently on the order of 10^(-5) or smaller for all cases. In Panels (b), (c), and (d), we illustrate the absolute error in kink density for various system parameters during the real-time evolution. The errors converge and consistently remain on the order of 10^(-6) for all the cases considered.
3) Evidence on data convergence (Report 1 and 2)
To assess the convergence of TDVP data for larger system sizes, we plot the relative error (the absolute difference between TDVP results with different bond dimensions) in distance to thermalization (DT). This is performed for three values of bond dimensions chi = 60, 90, 128, as shown in Figure 14 in Appendix B3. The errors converge and consistently remain on the order of 10^(-3) or smaller across all cases considered.
4) Brief summary on dynamical regimes (Report 1)
In the first paragraph of section 4, we have added a brief overview of dynamical quantum phase transitions (DQPT) and introduced various dynamical regimes within the long-range Ising model. We have supplemented this information with several references to better understand the background and previous contributions to the field.
5) Choice of post-quench parameters (Report 1)
The choice of two values of transverse fields is based on insights from our previous study (reference number [54] in the References section), which indicated that quenches near the dynamical critical points do not exhibit conclusive equilibration within the time frame feasible with our numerical resources. Consequently, we focused our investigation on two points: h_f = 0.3 and h_f = 0.6. These points reside in the dynamical ferromagnetic and paramagnetic regimes, respectively, and demonstrate a conclusive equilibration.
6) Thermalization and weak confinement (Report 1)
We have introduced Figure 5 to the manuscript, which depicts the post-quench evolution of domain-wall kinks. Because confinement bounds the dynamics of domain-wall kinks during temporal evolution, this parameter serves as a relevant and robust indicator of thermalization. The inclusion of this result further supports the findings shown in Figures 3 and 4.
7) Language and structure (Report 1)
We have edited the manuscript to improve the grammar and language. We relocated the details of the finite-temperature simulations to the appendix. The primary figures have been integrated into the main text. Additionally, for enhanced visualization, we switched panels (a), (b), and (c) of Figures 3 and 4 to the log scale while presenting panels (d), (e), and (f) in 2D for clarity. As a further enhancement, we have introduced Figure 1, encapsulating the central idea of the manuscript.
List of changes
The list of changes made are as follows;
1) Added appendices A, B2, and B3
2) Added figure 1 in main text
3) Expanded section 4 by adding figure 5
4) re-plotted figure 3 and 4 for better visualization
5) added a paragraph on dynamical regimes on section 4
6) re-located the discussion on simulation of finite temperature states to appendix B1
Published as SciPost Phys. Core 7, 017 (2024)
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2023-12-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.00533v3, delivered 2023-12-22, doi: 10.21468/SciPost.Report.8332
Report
The main idea and results of this work are relevant and of interest and I believe the revision did sufficiently address the concerns regarding the novel contribution from this work. Indeed, I have to commend the Authors for the revision of this manuscript. I find that the current version of the manuscript presents the work very clearly with sufficient background and citation to place the work within the broader field. Furthermore, the other points raised in the previous round of Referee reports were also addressed. Nevertheless, a few minor typos and grammatical/stylistic errors remain, but as they are not drastic I leave it to the Authors to decide whether or not they wish to address those before publication (see e.g. multiple definition of LRIM and other typos I will refrain from listing here).
Perhaps I can also take this opportunity to bring up one final suggestion, which would at least to me make things easier to understand quickly. Namely looking at the definition of P(O,t) in equation (5), I feel it would be easier to understand the definition if you simply formulate it in a manner similar to equation (7)
\[P(O,t)=\sum_{\{\sigma_i\}}|C_{\{\sigma_i\}}(t)|^2\delta\left(O-\langle\{\sigma_i\}|\hat O|\{\sigma_i\}\rangle\right)\]
which allows $\{\sigma_i\}$ to run over the complete computational basis.
To keep this short, I think apart from perhaps reformulating the presentation of equation (5) and clearing out the remaining typos and grammatical/stylistic errors, as pointed out above, the manuscript meets the criteria for publication in SciPost Physics.