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Exact mean-field solution of a spin chain with short-range and long-range interactions

by Etienne Granet

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Submission summary

Authors (as registered SciPost users): Etienne Granet
Submission information
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
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics


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

I thank both referees for their careful reading and positive comments about the draft. In particular I thank the first referee for their very positive comments and recommendation for publication. As for the particular points raised by the second referee, here are my answers and list of changes:

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


List of changes

See comments.

Published as SciPost Phys. 14, 133 (2023)

Reports on this Submission

Anonymous Report 4 on 2023-2-21 (Invited Report)


The Author has considered all the points of my previous report to varying extent in the updated manuscript. Therefore, I recommend its publication in SciPost Physics. I apologize to the Author and the Editor for the delay.

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Anonymous Report 3 on 2023-1-19 (Invited Report)


I sustain my previous recommendation to publish the manuscript.

Thanks to the detailed suggestions by the second referee its readability has been further improved beyond the original version.

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