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Quantum spin systems versus Schroedinger operators: A case study in spontaneous symmetry breaking

by C. J. F. van de Ven, G. C. Groenenboom, R. Reuvers, N. P. Landsman

This is not the current version.

Submission summary

As Contributors: Klaas Landsman
Arxiv Link: https://arxiv.org/abs/1811.12109v2
Date submitted: 2019-06-05
Submitted by: Landsman, Klaas
Submitted to: SciPost Physics
Domain(s): Theor. & Comp.
Subject area: Quantum Physics

Abstract

Spontaneous symmetry breaking (SSB) is mathematically tied to some limit, but must physically occur, approximately, before the limit. Approximate SSB has been independently understood for Schroedinger operators with double well potential in the classical limit (Jona-Lasinio et al, 1981; Simon, 1985) and for quantum spin systems in the thermodynamic limit (Anderson, 1952; Tasaki, 2019). We relate these to each other in the context of the Curie-Weiss model, establishing a remarkable relationship between this model (for finite N) and a discretized Schroedinger operator with double well potential.

Current status:
Has been resubmitted

Ontology / Topics

See full Ontology or Topics database.

Spontaneous symmetry breaking

Author comments upon resubmission

We are very grateful for the time and energy spent by the referee on this paper, both through SciPost and through personal correspondence. The paper is much more balanced now and has a considerably improved bibliography. As to the novelty of the paper as opposed to its review character, we have now made it clearer where we feel out contributions lie. Mapping the Curie-Weiss model onto a discretized 1d Schroedinger operator with double well potential is one of those, based on a reduction of the model due to permutation symmetry as explained in section 2 (while such mappings in other models were known, and are referred to, this case seems to be new). The other, we continue to maintain, is the specific mathematical form of the symmetry-breaking perturbations of the CW model. It should be clear from this revision that we do not claim any physical novelty, but the mathematics is, as far as we know, even given the many new references added thanks to the referee, still new, though, as we wrote from the beginning, directly inspired by work in the 1980s on the classical limit of Schroedinger operators. In order to make this clearer than in v1, we have added a lengthy discussion of our perturbation in section 4, which was lacking in v1, and we have also slightly modified the perturbation in order to make the analogy with the "flea" from the 1980s clearer, including redoing the numerical simulations, which gave almost exactly the same results as before. We have also made a comparison with the condensed matter physics literature on SSB (in so far as we know it) in section 5, which emphasises our different starting point for SSB (namely a state decomposition perspective as opposed to a non-commuting limits perspective). The Introduction has also been modified in order to give the right historical perspective (as we see it). In this way all major and minor points raised by the referee have been taken into account or answered.

List of changes

- slight rephrasing of abstract
- section 1: new footnote 3 including many new references (see below) and embedding of footnote in main text. New paragraph at the end added.
- section 2: small corrections only
- section 3: streamlined, all details on discretization procedures that we eventually decided not to use (but felt it worth explaining why not) now deleted
- section 4: 2 x 2 matrix illustration added, change in "flea" family of perturbations (physics the same as before, math now more congruent with the Schroedinger operator case, numerics redone with similar results).
- section 5: Text after first paragraph (which largely comes from v1) added.
- References: nos. 2, 4, 5, 6, 7, 8, 9, 10, 15, 19, 22, 23, 24, 31, 32, 33, 35, 36, 37, 40, 42, 48, 49 added (about half suggested by the referee and half found by ourselves).


Reports on this Submission

Report 1 by Jasper van Wezel on 2019-6-26 Invited Report

Strengths

1 - The submission has a clear and pedagogical style of presentation.
2 - The presented mathematical mapping between models is rigorous.

Weaknesses

1 - Besides introducing a mapping between the Curie-Weiss model and a Schrödinger operator, the submitted manuscript does not contain any new physics.
2 - Two of the main claims of novelty are simply false (see below).

Report

Dear authors,

thank you very much for seriously considering the suggestions and comments made in my previous report.
I believe the resubmitted manuscript has a much improved discussion of how the present work relates to previous literature, and a much more balanced presentation of the current results.

I still think the discussion in the present paper is a worthwhile pedagogical example of spontaneous symmetry breaking in an accessible, yet mathematically rigorous, setting. I also appreciate the point of the authors that the presented mapping from the Curie-Weiss model onto a discretized 1d Schroedinger operator with double well potential is new.

However, I also continue to be of the opinion that the other two points presented by the authors as their main results, are not new:

1) In the final page of the conclusions, the authors emphasize they believe one of the main differences between their work and the standard approach to SSB in the cond-mat literature, is that their approach involves only a single limit, rather than two non-commuting ones.

In fact, this is simply untrue.
The authors do use two limits, and they do not commute.

The authors even write this themselves, in the conclusions, where they state:
"all we need is that ∆E → 0 more rapidly than δV → 0 as hbar → 0".
The two limits taken by the authors are ∆E → 0 (which corresponds to N → infinity), and δV → 0. These are precisely the same two limits as the ones in equation (5.26), used in the standard SSB literature. In both cases, they do not commute.
The trick played by the authors, of parameterising both ∆E and δV as functions of hbar, and then requiring that ∆E vanishes more rapidly than δV, is just precisely equivalent to taking the limit N-> infinity before taking the limit δV → 0.

1b) The same point comes up in a second guise on page 5, where the authors write:
"on a first analysis (to be corrected in what follows!) there is no ssb for any finite N"

The authors seem to suggest here that there is SSB for finite N.
If so, then for which N?
The only possible answer is some scale set by the ratio of the two energy scales in the model: that of the double well potential, and that of the "flea". In other words, for symmetry breaking in finite N, a non-zero "flea" is required. Spontaneous symmetry breaking (with an infinitely weak "flea") is possible only in the infinite N limit.
So again, we see that two non-commuting, singular limits are in fact present in the author's work, and they appear in precisely the same way as in the standard approach.

2) In section 4, the authors argue there is a fundamental difference between their "flea" potential, and the symmetry-breaking perturbation introduced in the standard SSB literature.

Again, this is simply not true.

The requirements on the "flea" potential formulated between equations 4.2 and 4.3 are precisely such that the support of the "flea" coincides with the tails of the gaussian (localised) ground states. Of course this must be the case, for if the support of the flea would not coincide with the support of either ground state, it could have no effect.

As the authors argue in several places in the article, the width of the Gaussian ground state in the scaled coordinates is proportional to 1/sqrt(N), so that the ground states become exponentially localised in the middle of the wells in the limit N-> infinity.
In that limit then, the "flea" must have support in the middle of the well, and corresponds simply to a lifting of the degeneracy between the two wells, which would be the standard symmetry-breaking field proposed by the standard cond-mat SSB literature.

It should be noted here that the authors seem to suggest on page 21 that the "flea" is strictly zero in the centre of the well. This cannot work in the limit of infinite N. Again, the argument is simple, and was already made in my first report: in the infinite N limit, the ground state is a Dirac delta distribution at the well centre. If the perturbation is zero there, the two states localised in either of the two wells are strictly degenerate, and a single symmetric superposition of them will be the unique ground state.

The authors can probably not see this in their numerical approach, since it cannot go beyond N=60, but the fact that there cannot be an effect of any "flea" that is zero in the centre of the wells in the infinite N limit is obvious.

Besides these major points, I also noticed some minor things:

- on page 20, the authors mention several energy scales that do not have the units of energy. I suppose they mean these energies are for certain specific values of B and J?
Even so, they really ought to write them in terms of B and J, or introduce some units.

- The result of equation 3.36 simply shows that the double well potential is locally quadratic and therefore has the spectrum of a harmonic oscillator. It would be good to point this out, and also to point out that the approximation will break down at higher excitation energies.

In conclusion,
I remain of the opinion that the discussion provided by the authors is worthy of publication, as a mainly education piece with perhaps the slight novelty of formulating a new mapping between the Curie Weiss model and a Schroedinger operator.

I do not believe, however, that the specific limit and the specific form of the perturbation considered by the authors are different from those used in the standard SSB literature, and I therefore do not judge this paper to be a significant advancement of the field.

  • validity: high
  • significance: ok
  • originality: ok
  • clarity: high
  • formatting: good
  • grammar: excellent

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