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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 Submission thread is now published as

Submission summary

Authors (as Contributors): Klaas Landsman
Submission information
Arxiv Link: (pdf)
Date accepted: 2020-01-15
Date submitted: 2019-07-31 02:00
Submitted by: Landsman, Klaas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
Approaches: Theoretical, Computational


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.

Published as SciPost Phys. 8, 022 (2020)

Author comments upon resubmission

Since only a minor revision was asked, we tried to clarify remaining differences with the referee in a revised Discussion session, see List of Changes below. This fully answers point 1a) in the last referee report, which seems to largely concern a way of phrasing things rather than a fundamental difference of opinion, and in this new version we hope to do justice to both our own views and the referee's; see especially the new lines 9-19 on page 29. Points 1b) and 2) are more difficult to deal with, since the criticisms seem contradictory to us: on the one hand (1b) our perturbations are claimed to be the same as those in the theoretical condensed matter physics literature, while on the other hand (2) they can't work because they are localized off the bottoms of the wells, or, equivalently, away from the double peaks of the unperturbed ground state wave function. Point 2) applies verbatim to our key mathematical references (Jona-Lasionio et al, 1981; Graffi et al, 1984; Simon, 1985) which are well-attested and uncontroversial papers with a large follow-up in mathematical physics, so here we suggest that physical intuition should bow to mathematical rigor - in fact the referee's reaction (which if correct would invalidate this entire body of work, on which, as we explain in the unrevised Introduction, our entire approach is based) confirms how completely unexpected these papers were at the time, and in our view still are; our mission with this paper is partly to advertise this work, which so far has not landed at all in theoretical physics. We answer the particular technical point made by the referee in our new footnote 22 on page 28. In our view, this also dispels 1b), but we do not make a big song and dance about the novelty of the flea perturbations because, as we explain throughout the paper, we merely transfer them from the double well potential (in which context they were introduced in the literature just cited) to the Curie-Weiss model.
The minor points made by the referee are uncontroversial and have been incorporated. On the whole, our dialogue with the referee and the journal has greatly improved and clarified our paper so we repeat our gratitude, simultaneously hoping that the end has now been reached! In particular, we hope that the referee and the editor accept that there is not a single valid perspective on SSB; there is a genuine difference between the standard theoretical physics approach (which the referee follows) and the standard mathematical physics approach (which we follow), each with its own advantages, and this paper is partly meant as a bridge between the two. See also the list of changes below.

List of changes

page 17, last line added (answers minor point by referee)

page 20, sentence after (3.36) added, following referee almost verbatim

page 28, discussion after (5.26) rephrased to clarify that there are two different points of view on SSB, where we explain why we favour ours (which is the standard one in mathematical physics), upon which the entirely new paragraph in lines 9-19 on page 29 relates the two approaches again, in a way which seems to clarify the situation, leading to peaceful coexistence, or so we hope.

Reference 17 added (Note that a huge addition to the bibliography has already been made in version 2; this one was added to support our new footnote 22, see above)

Reports on this Submission

Report 2 by Bruno Nachtergaele on 2019-12-14 (Invited Report)

  • Cite as: Bruno Nachtergaele, Report on arXiv:1811.12109v3, delivered 2019-12-14, doi: 10.21468/SciPost.Report.1391


Spontaneous symmetry breaking plays a central role in many physical theories and our observed physical world exhibits symmetry breaking in many situations. The purpose of this work is to help clarify a subtlety in the way spontaneous symmetry breaking manifests itself in statistical mechanics.

A quantum system with a finite number of degrees of freedom has a unique equilibrium state and, more often than not, the ground state is also unique. These states then necessarily share all the symmetries of the Hamiltonian. Spontaneous symmetry breaking manifests itself in the thermodynamic limit but, of course, the state obtained as a limit of the unique and symmetric finite-system states, will also be symmetric. These symmetric states are a uniformly weighted mixture of the distinct pure symmetry broken phases. In the physical world, under most conditions, only pure phases are observed. Therefore, it is generally accepted that the extremal states found in an ergodic decomposition (at positive temperatures) or into pure states (at zero temperature), are the ones that describe physical reality.

The standard procedures in statistical mechanics to construct the pure phases involve adding symmetry breaking terms to the Hamiltonian, such as a uniform magnetic field that favors a particular value of the magnetization in a ferromagnet, and letting the field strength vanish in the thermodynamic limit. Another approach is to use symmetry breaking boundary conditions that recede to infinity in the thermodynamic limit. These two commonly used approaches work well. For those who want to understand what happens in an experiment on a finite sample, however, these procedures are not satisfactory. Pure phases are observed without using a magnetic field or a particular boundary condition. The accepted explanation for this is that any realistic preparation process will inevitably exhibit an instability that steers the end result to one of the pure phases.

In this paper, the authors study aspects of this instability in a particular model and take inspiration from an analogy between the thermodynamic limit and the classical limit, which I now briefly explain.

The thermodynamic limit turns the order parameter characterizing a phase transition with spontaneous symmetry breaking into a classical observable, with a deterministic value labeling the particular pure phase the system is in. The authors note the following analogy with the emergence of spontaneous symmetry breaking in the classical limit of a quantum particle in a symmetric double-well potential. In that system, there are two energy minimizing states of the classical particle and only states where the particle is sitting in one of the minima are physical. At positive values of Planck's constant, the system has a unique ground state with the symmetry of the potential and this symmetry is preserved in the classical limit. In other words, the classical limit ground state is a mixed state with equal probability of finding the particle in either minimum.

In the 1980's, mathematical works appeared that showed how a vanishingly small perturbation of a double-well potential, the proverbial flea on an elephant (which does not need to break the degeneracy of the classical energy minima) leads to a classical limit where the particle occupies just one of the minima. In the present work the authors show an analogous behavior in the thermodynamic limit of the mean-field Ising model (aka the Curie-Weiss model) in a transverse external field. For small transverse field, the spin flip symmetry is spontaneously broken in the ground state. A specific class of tiny perturbations is introduced and it is shown with the aid of numerical calculations that they steer the ground state in the thermodynamic limit to one of the pure ground states. In the process, the analogy between the thermodynamic limit of the Curie-Weiss and the classical limit of a double well system is made concrete by a mathematical reduction of the first to an instance of the second.

The value of this work is primarily of an illustrative and pedagogical nature. The apparent contradiction between the standard story and the physically observations is often a stumbling block for teachers and students alike. This paper, as its title promises, presents a case study that validates the explanation based on a generic instability. I found the paper to be very well-written. The cited references certainly do not exhaust the literature on the topic but they do provide sufficient background, historic context, and pointers to interesting further reading.

I believe this work will be a useful addition to the literature that may help people think more clearly and in mathematically precise terms about the important physical phenomenon of spontaneous symmetry breaking. It is unlikely to be the last word on the topic, but I expect to be a widely appreciated contribution of lasting value.

I recommend that this paper be accepted for SciPost in its present form.

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Report 1 by Jasper van Wezel on 2019-8-16 (Invited Report)

  • Cite as: Jasper van Wezel, Report on arXiv:1811.12109v3, delivered 2019-08-16, doi: 10.21468/SciPost.Report.1115


I thank the authors for the added discussions.

The new paragraphs do clarify the difference between the CM and MathPhys literature on this topic, and thus alert the reader to look into this further. However, they do not add any arguments to the discussion between the author and myself, and I am not convinced that any of my previous objections were invalid.

I also do not have any new arguments myself, beyond what I already wrote in my previous report. I therefore recommend that the editor decides whether or not they are happy to publish, taking into account the differences of opinion and arguments in the already available reports.


To the authors: I would like to clarify one thing - that my earlier points 1b and 2 do not contradict each other.
1b points out that the approach of the "flea" advocated by the authors is the same as that of a "symmery-breaking field" which is standard in the CM literature.
2 points out that one does have to choose an appropriate symmetry-breaking field, and that the authors have not at all proven their perturbation to remain appropriate all the way to the thermodynamic limit.

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Author:  Klaas Landsman  on 2019-08-19  [id 582]

(in reply to Report 1 by Jasper van Wezel on 2019-08-16)
answer to question
reply to objection

Hi, We agree to now leave this to the editor and emphasize that we have followed the previous editorial recommendation to (re)submit a minor revision. On the one hand, the dialogue with the referee has improved the paper almost beyond recognition, but on the other hand we regret that no complete agreement has been reached. Apparently, the subject lends itself to various interpretations, like works of modern art. We do believe that we answered all points raised by the referee and our previous replies explain in considerable detail (or at least intended to explain in detail) how we did so.

In reply to the last two points made by the referee we would like to point out that: 1) Apparently no perturbation is regarded as novel (by the referee) as long as it falls within the scope of general symmetry-breaking terms. Of course our flea perturbation falls within that scope - the novelty (or so we think) lies in its detailed mathematical form. We are not P.W. Anderson and we emphasized that this novelty is limited (but in our view still very interesting) from the start. 2) It is true that our evidence for symmetry breaking triggered by our perturbation is numerical and of course these were carried out for finite N, and hence not "all the way to the thermodynamic limit". However, the totally analogous flea perturbations for the double well potential have been rigorously proven, in the papers we cite throughout ours, viz. refs. [20], [15], and [40], to do this job for hbar -> 0, and since our perturbation is obtained from that one through the mapping explained in chapter 3, under the substitution hbar -> 1/N, we in fact collect two kinds of evidence for its validity. Hence we take issue with the claim that we "have not at all proven [our] perturbation to remain appropriate all the way to the thermodynamic limit". Since our mapping is itself only defined for finite N, we would agree that we have not strictly proved this (i.e. analytically), but inserting "at all" wrongly suggests that we have no evidence at all (sic). See especially page 26.

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