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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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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
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)
Submission & Refereeing History
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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.
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.