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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
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Submission summary
As Contributors:  Klaas Landsman 
Arxiv Link:  https://arxiv.org/abs/1811.12109v1 (pdf) 
Date submitted:  20181130 01:00 
Submitted by:  Landsman, Klaas 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Spontaneous symmetry breaking (SSB) is mathematically tied to either the thermodynamic or the classical limit, but physically, some approximate form of SSB must occur before the limit. For a Schroedinger operator with double well potential in the classical limit, this may indeed be accomplished by the "flea" mechanism discovered in the 1980s by JonaLasinio et al. We adapt this mechanism to the CurieWeiss model (as a paradigmatic meanfield quantum spin system), and also establish an unexpected relationship between this model (for finite N) and a discretized Schroedinger operator with double well potential.
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Reports on this Submission
Report 1 by Jasper van Wezel on 2019122 (Invited Report)
 Cite as: Jasper van Wezel, Report on arXiv:1811.12109v1, delivered 20190122, doi: 10.21468/SciPost.Report.800
Strengths
1  The submission has a clear and pedagogical style of presentation.
2  All presented work is based on mathematically rigorous arguments.
Weaknesses
1  The submitted manuscript contains only alreadyknown physics
2  It is not clear how the presented numerical results can be taken to the thermodynamic limit. This causes at least one serious issue in the manuscript (see report).
Report
This is a very nicely written paper discussing the emergence of spontaneous symmetry breaking in a version of the CurieWeiss model. It includes a thorough treatment of many aspects of spontaneous symmetry breaking, in a mathematically rigorous fashion, which will be useful to any readers interested in the details of spontaneous symmetry breaking in general, and in the CurieWeiss model in particular.
Beyond being a nice, clear, and precise description of the effects of a symmetry breaking field for finite N and hbar in a setting where it has to the best of my knowledge not been discussed in these terms before, the paper unfortunately does not offer much new insight. The idea that a finite symmetrybreaking field is required to break the symmetry of a finite system is completely standard. Similarly, the realisation that the strength of this field may be taken to infinitesimal values in the thermodynamic limit has been at heart of the way we teach spontaneous symmetry breaking for a long time.
I appreciate the present submission as an explicit and pedagogical example of spontaneous symmetry breaking. Examples that can be treated in an accessible, yet mathematically rigorous, way are rare, and it is valuable to have more such examples. Beyond that, I don't see any new physics in this manscript.
I would therefore urge the authors to rewrite their introduction in such a way that they make clear that this manuscript presents a new and pedagogical example of known physics, rather than a novel research result.
Besides this general remarks, I also noticed some other issues in the manuscript:
Major issues:
1 In the introduction, and in various places throughout the manuscript, the author fail to acknowledge, and even misrepresent, past results in the field. The realisation that finite symmetry breaking fields are needed to break the symmetry of finite systems has been around at least since the works of PW Anderson. The effect of finite perturbations and the need for suitable convergence in the thermodynamic limit has been discussed in detail for magnetic systems by various authors throughout the 1980's, and has been generalised to many other types of systems since.
The authors fail to cite much of the previous work in this direction, and instead present their own papers from the past few years as if these are the first results in the field.
1b  The failure to acknowledge previous work is made worse by the fact that the authors use their own personal terminology for concepts that have been discussed in the literature for many years under other names. For example, the "flea in Schrodinger's cat" is nothing other than the "symmetry breaking field" in any other discussion of spontaneous symmetry breaking.
2 Although I agree with the idea of the discussion in section 4.1, I am not convinced the perturbation shown in figure 9 can be regarded as a proper choice for a symmetry breaking field. As the authors show in their manuscript, the two nearly degenerate ground states become arbitrarily welllocalised in the limit of large N. Taking the limit of N to infinity while keeping the perturbation finite thus means that the ground state wavefunction will have zero overlap with any alterations of the potential away from the minima of the double well. Only perturbations that make the two minima in the potential nondegenerate can then serve as proper symmetrybreaking fields in the limit of large N.
In other words: I suspect that for the perturbation shown in figure 9, the energy difference between the nearly degenerate ground states scales as lambda/N, implying that N can always be chosen large enough for the perturbation chosen by the authors to be practically irrelevant. This is precisely opposite to the effect of a proper symmetry breaking field, which causes an energy splitting proportional to lambda*N, and which becomes more relevant with increasing N.
Minor issues:
1  On the bottom of page 9, and again in various places further on, it is confusing to talk about "new ground states". Two states simply become degenerate up to numerical precision, and any superposition of them is a (numerically) allowed ground state. If you always find a particular "new" ground state in the numerics, such as a localised state, that must mean there is a symmetry breaking field in your numerical implementation of the calculation. Without such a field, you ought to find random superpositions of the two degenerate states.
This should really be explained already on page 9.
On a related note, the number 80 in N=80 is arbitrary, and must depend on the machine precision of the computer used to do the calculations. The authors really ought to mention that.
2  in the first line of the introduction, the authors choose ferromagnets as their example for the archetype of spontaneous symmetry breaking. I suspect the authors are aware that this is actually a very bad example. Ferromagnets are very special in the sense that their symmetrybroken state is an exact eigenstate of their symmetric Hamiltonian for any N. This is not at all generic, and in fact severely affects the physics associated with symmetry breaking in ferromagnets (see for example 10.1016/j.aop.2015.07.008 and references therein).
3  It seems like the matrix elements discussed in theorem 2.1 are included in those of appendix 2 in 10.1103/PhysRevB.74.094430 ?
Requested changes
1  I suggest the authors rewrite their introduction in such a way that they make clear that this manuscript presents a new and pedagogical example of known physics, rather than a novel research result.
2  Please add references to and discussion of previous results in the literature.
I would also suggest using, or at least mentioning, standard terminology.
3  The authors should discuss the behaviour of their symmetry broken states with increasing N, in order to establish a connection between their lowN numerics, and the largeN limit that is relevant for actual physical systems.
4  Please explain clearly that finding particular "numerical ground states" is the effect of choosing a particular numerical implementation for the calculation, which is equivalent to including a symmetry breaking field.
Author: Klaas Landsman on 20190201 [id 419]
(in reply to Report 1 by Jasper van Wezel on 20190122)Dear referee (Jasper) and editor, We are very grateful for this detailed and elaborate report and enter this correspondence trusting we will reach an agreement. The correct literature attributions, though important, are secondary and will be discussed below. Our primary point, in respectful disagreement with the referee, concerns the novelty of the paper. The specific mechanism we use (and sometimes may even appear to propose) for SSB was never meant to be new; anything in this direction was meant to be expository and we are pleased that the referee has appreciated this expository aspect.
Though we acknowledge, with hindsight, that both the abstract and the order of presentation in the Introduction are evidently misleading in this respect, the main novel point of the paper (in section 3) is the discovery (as we believe it to be) that the ground state of the quantum CurieWeiss model (in a small magnetic field), compressed onto the N+1 dimensional subspace in which (due to permutation symmetry) the ground state must lie (as explained in section 2), is the same as the ground state of a discretized double well potential on the line. For this we provide both extensive numerical and analytic evidence. This direct link between a quantum spin model and a discretized Schroedinger operator came to us unexpectedly, and we were unable to find anything like it in the literature: what has been remarked so far were (largely spectral) analogies between say the quantum Ising model and continuum double well potentials; what we provide is a direct mapping to the discretized double well potential, with splendid convergence properties for large N. So we ask the referee to either acknowledge this novelty or provide literature where this discovery is predated or at least anticipated (of course, we have done our best to find such papers ourselves but were unable to  we have obviously been unable to familiarize ourselves all the thousands of papers on these topics).
A second novel point is the specific perturbation we propose in section 4 for the analogue of what we call the "flea" perturbation for the continuum double well potential (see below for terminology and priority issues etc.) for the CurieWeiss model. This was actually quite hard to find, and our analysis shows that and how it does the job. In response to point 2 by the referee, despite its decreasing behaviour for large N (which is needed since the energy gap between the original ground state and the perturbed ground state must of course go to zero for N to infinity), according to which it vanishes as an inverse power of N, it is still big enough to tilt the ground state into a symmetrybreaking one, and this is, just like the double well case, because, small as it is, the perturbation still dominates the energy gap in the unperturbed model. This brings us to the third point:
Further, smaller details a matters of slight disagreement can be settled in the next stage, without any doubt. Let us again express our gratitude to the referee and the editor for their efforts spent on our paper so far and in the future.
Jasper van Wezel on 20190206 [id 429]
(in reply to Klaas Landsman on 20190201 [id 419])Dear authors,
Thank you very much for your detailed reply, and for explaining the background to your presentation. I would like to answer the points made in your reply:
Please allow me to be clear: I do acknowledge that the specific example of correspondence between the two models found by the authors in the present work may not have been known before (I have not encountered it, as far as I'm aware), and may be new in that regard. So if the authors insist that the primary novelty of their work is in the mapping between models and in the choice of symmetry breaking field, and that the rest is primarily educational, I accept that. I would just like to point out that the novelty claimed by the authors seems limited if seen in the light of the many related results already in the literature, and I would ask the authors to revise their paper to acknowledge that. I would suggest that instead, they can emphasize the important educational aspect of their work.
I appreciate the point made by the authors that there may be a mismatch in the extent to which spontaneous symmetry breaking has been covered in the condensed matter versus mathematical physics literature. However, since the authors address a typical condensed matter topic (that of order in spin systems), I think they should respect the established condensed matter physics literature. In terms of standard terminology, this means they should at least mention commonly used and wellestablished names like "symmetry breaking field" in favour of "flea". In terms of giving credit to work in the literature on this subject, I would like to at least mention the work of Lieb and Mattis ( J. Math. Phys. 3, 749 (1962)) showing how energy levels are ordered in magnetic systems in general, and that the exact ground state of any magnetic system (barring the pure ferromagnet) is symmetric; the works of both Kaiser and Peschel (J. Phys. A 22, 4257 (1989)) and of Kaplan, Von der Linden, and Horsch (Phys. Rev. B 42, 4663 (1990)) which both discuss how the symmetry in the antiferromagnet is broken spontaneously, including the role of the symmetry breaking field as well as that of the spectrum of lowenergy states; and finally, the work of Bernu, Lhuillier, and Pierre (Phys. Rev. Lett. 69, 2590 (1992)) giving a numerical treatment of symmetry breaking in finite antiferromagnets (particularly relevant with regard to the present submission). I believe these works together established the modern understanding of spontaneous symmetry breaking in magnetic systems. More recently, a lot of progress has been made by various authors, including Watanabe, Brauner, and Beekman, to name just a few. There are certainly many more relevant references that could be added: this is an active field of research, building on a long history. I do agree with the authors that it is not easy to find a textbook that explains the particular aspects of symmetry breaking which are the subject of the current paper in a clear and concise manner. I can however point the authors to a set of lecture notes written in 1994 which shows that, even then, the subject was already covered in detail in (good) condensed matter lectures at the MSc level. Finally, although cryptic in some aspects, the papers of Anderson were certainly instrumental in establishing and promoting this modern understanding of spontaneous symmetry breaking in condensed matter theory.
Finally, with regard to their choice of symmetrybreaking term, I don't think the authors address my concern at all. As I wrote before: " I suspect that [..] the energy difference between the nearly degenerate ground states scales as lambda/N [..] This is precisely opposite to the effect of a proper symmetry breaking field, which causes an energy splitting proportional to lambda*N, and which becomes more relevant with increasing N." In their reply, the authors simply state : "despite its [.. vanishing ..] as an inverse power of N, it is still big enough to tilt the ground state into a symmetrybreaking one." But this is precisely the thing I question. I do not think that it is enough to tilt the ground state into a symmetrybroken one. My complaint is precisely that from the calculations shown in the current manuscript, it is impossible to tell whether or not the symmetry will be broken in the limit of large N (rather than for N<80). Based on the fact that the wavefunction localises exponentially in the centre of the well, and thus has an exponentially small overlap with the perturbation chosen by the authors, one would expect the influence of the perturbation to disappear in the limit of large N. If the authors want to argue that their perturbation is nonetheless a suitable symmetrybreaking field, they will have to show explicitly that how the symmetrybreaking effect scales with system size, or equivalently, that the limits of large N and vanishing lambda do not commute. On a related note: the authors mention they found choosing a suitable symmetrybreaking field to be "quite hard". I do not really understand this, since generically, if a symmetry is to be broken by any infinitesimal perturbing field, the perturbation should be conjugate to the broken symmetry. In this case, that simply means that any suitable symmetry breaking field raises the (minimum) energy of one of the potential wells with respect to the other. The fact that the perturbation chosen by the authors does not do this, is another reason for suspecting that their choice of symmetrybreaking field may not be adequate in the largeN limit.
The authors did not yet address any of the other points raised in my report, which therefore remain open.