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Generalised Gibbs Ensemble for spherically constrained harmonic models
by Damien Barbier, Leticia F. Cugliandolo, Gustavo S. Lozano, Nicolás Nessi
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Submission summary
| Authors (as registered SciPost users): | Nicolas Nessi |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2204.03081v2 (pdf) |
| Date submitted: | 2022-04-14 12:02 |
| Submitted by: | Nessi, Nicolas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We build and analytically calculate the Generalised Gibbs Ensemble partition function of the integrable Soft Neumann Model. This is the model of a classical particle which is constrained to move, on average over the initial conditions, on an $N$ dimensional sphere, and feels the effect of anisotropic harmonic potentials. We derive all relevant averaged static observables in the (thermodynamic) $N\rightarrow\infty$ limit. We compare them to their long-term dynamic averages finding excellent agreement in all phases of a non-trivial phase diagram determined by the characteristics of the initial conditions and the amount of energy injected or extracted in an instantaneous quench. We discuss the implications of our results for the proper Neumann model in which the spherical constraint is imposed strictly.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-6-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2204.03081v2, delivered 2022-06-08, doi: 10.21468/SciPost.Report.5209
Strengths
Impressive work on an exactly solvable model where the GGE can be worked out and studied in full detail.
This is clearly an on-going work with quite some history already but the authors make an effort to guide a reader to the main points and to recall necessary background.
Weaknesses
none
Report
This work is in a long series of studies on non-equilibrium dynamics of spin glasses. Here this framework is used to formulate and solve the construction of a GGE through the explicit handling and the conserved quantities. This works presents a veritable tour de force through non-equilibrium dynamics and does advance the subject.
To be accepted for publication.
Requested changes
Some minor remarks for consideration and eventual changes:
1. on p. 9 should one not have 2 Lagrange multipliers for the 2 constraints (6) ?
Or else do you assume from the outset such initial conditions that one of the constraints is automatic ? Please explain.
2. in eq. (11) it may be helpful to remind the reader that { , } are Poisson brackets (and not, e.g. anti-commutators)
3. in section 3.3 or Figure 3 (and elsewhere) you call the average <I(lambda)> a `spectrum'. I find this confusing, since I would naturally think of the eigenvalues of some operator, which I fail to see. Or is this long-standing jargon in this special type of study ?
4. in the phase diagram Figure 4a, all phases I, IIa, IV are white, in contrast what can be gleaned from the text, where it should only be phase I.
It is not directly clear if the phase IV goes up to T_0/J_0 =1 or not.
Please edit the colour coding of the phases.
5. in figure 4b, where is phase IV ? Or is this just meant to be the reproduction of an old figure from [10] ? Please explain.
6. similarly, at the end of p. 19 "... and a new one that we recognise here ..."
maybe the reader should be directly directed to figure 5 where phase IV appears.
Report #1 by Anonymous (Referee 2) on 2022-6-1 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2204.03081v2, delivered 2022-06-01, doi: 10.21468/SciPost.Report.5165
Strengths
1- One of the few works on GGE and equilibration in classical integrable systems.
2- Rigorous analysis and careful numerical check.
3- Extensive study of a paradigmatic model.
Weaknesses
1- No major weaknesses except for the paper length.
Report
This paper presents a detailed investigation of out-of-equilibrium dynamics in the Soft Neumann Model. This is a paradigmatic model that has been very useful to understand equilibrium properties of classical systems. Indeed, the model is solvable but it is nontrivial as it is non mean field and not free. While the model has been studied extensively at equilibrium, its out-of-equilibrium dynamics has not been explored much. The authors investigate the effect of quantum quenches in the SNM. Their main result is an analytic derivation of the GGE that describes the local properties of the steady state. The analysis performed in the paper is very careful, also as pertains the numerical checks. The paper is well written although quite long. I think that this paper provides a solid reference for future research in this field. Therefore, I recommend publication in the present form.
Requested changes
At some places some sentences are in green, perhaps the authors want to correct that.
Author: Nicolas Nessi on 2022-07-05 [id 2637]
(in reply to Report 1 on 2022-06-01)We thank the referee for the careful reading of our manuscript, we have removed the green color in some of the sentences of the paper.
Author: Nicolas Nessi on 2022-07-05 [id 2636]
(in reply to Report 2 on 2022-06-08)We thank the referee for the careful reading of the manuscript, and for her/his comments that we address below.
1- We thank the referee for bringing up this issue. In page 9, in equation (9) specifically, we define the Hamiltonian that governs the dynamics of our system. It turns out that the Lagrange multiplier corresponding to the secondary constraint vanishes identically: imposing the primary constraint on the dynamics automatically enforces the secondary constraint, be it exactly or on average. This implies that if we chose suitable initial conditions satisfying the primary and secondary constraint, the dynamics generated by (9) will preserve both. The situation is different when we are dealing with measures over phase space, as in page 19, equation (45), where we have to introduce the Lagrange multiplier corresponding to the secondary constraint in order to pick up configurations that indeed satisfy both constraints. We added an explanatory statement around this issue after Eq. (9).
2- We have explicitly stated that the brackets denote the Poisson bracket.
3- We call the $\lambda_{\mu}$’s “spectrum” because they can be considered as the eigenvalues of a random symmetric interaction matrix $J_{i,j}$, in particular, when we consider the connection with the Sherrington-Kirkpatrick model (section 2.4 and Appendix C).
4- The color code of Fig. 4a refers only to the regions of the phase diagram in which $\langle I_N \rangle$ is larger (dashed blue) or smaller (white) than zero, we do not intend to show the limits between the different phases in this particular figure, this is done in Fig. 5.
5- We thank the referee for raising this point. Yes, indeed, Fig. 4a is a reproduction of the phase diagram that we were able to determine from our previous analysis in Ref. [10], where, using the magnitudes that we studied back then, we could not differentiate phase I from phase IV. We have added an explanatory sentence about this in the caption of Fig. 4b.
6- We added a reference to Section V and figure 5 for clarity.