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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
|As Contributors:||Klaas Landsman|
|Submitted by:||Landsman, Klaas|
|Submitted to:||SciPost Physics|
|Domain(s):||Theor. & Comp.|
|Subject area:||Quantum Physics|
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.
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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).