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Exact mean-field solution of a spin chain with short-range and long-range interactions
by Etienne Granet
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Etienne Granet |
| Submission information | |
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| Preprint Link: | scipost_202210_00055v2 (pdf) |
| Date accepted: | 2023-03-30 |
| Date submitted: | 2023-01-17 17:31 |
| Submitted by: | Granet, Etienne |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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Abstract
We consider the transverse field Ising model with additional all-to-all interactions between the spins. We show that a mean-field treatment of this model becomes exact in the thermodynamic limit, despite the presence of 1D short-range interactions. Namely, we show that the eigenstates of the model are coherent states with an amplitude that varies through the Hilbert space, within which expectation values of local observables can be computed with mean-field theory. We study then the thermodynamics of the model and identify the different phases. Among its peculiar features, this 1D model possesses a second-order phase transition at finite temperature and exhibits inverse melting.
Author comments upon resubmission
1 - I included a discussion in section 2.3.2 as requested by the referee.
2 - As the referee points out, the present method indeed works only when the short-range Hamiltonian is exactly solvable. Treating generic cases is beyond the scope of this work, but I added in the last paragraph of the conclusion a sentence to say that the exact solvability is required in this paper.
3 - I understand the comment of the referee that adding some plots would make the second part of the manuscript more accessible to a large audience. As detailed below, I added new plots to illustrate the different results. However, I also have to remark that the analytic study in Section 4 of the different phases obtained from the mean field solution of Section 3 is already very detailed, precise and complete.
4 - I added a more precise plot in Section 3.9 as requested by the referee. The obtained relative precision is of order at most $10^{-4}$ using a simple linear fit on the data. The precision and relevance of the linear fit is clearly visible in the plot.
5 - I clarified the meaning of disordered and ordered at the end of 4.1.3 as requested by the referee.
6 - I added numerical values in the left panel of Fig 4, and quoted the temperature used. In the other panels numerical values were already present through the location of the critical points, but I added some additional labels to make it clearer.
7 - I thank the referee for this good suggestion. I added a plot in Fig 3 for e.g. the first order transition in $\sigma^x$ at zero temperature, with comparison between the theory and the numerics.
Best regards
Etienne
List of changes
See comments.
Published as SciPost Phys. 14, 133 (2023)