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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 | |
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| Preprint Link: | scipost_202210_00055v1 (pdf) |
| Date submitted: | 2022-10-08 23:03 |
| Submitted by: | Granet, Etienne |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022-12-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202210_00055v1, delivered 2022-12-13, doi: 10.21468/SciPost.Report.6303
Strengths
1. The manuscript shows that the TIFM model with an additional fully-connected interacting term can be solved exactly in the thermodynamic limit, which corresponds to a MF solution. This, in my opinion, is very interesting as contrary to popular believe the local terms do not spoil the MF result. Moreover, there is a possibility that similar argument can also be applied to a broad class of mixed-ranged Hamiltonians.
2. The analytical treatment provided in the manuscript is very innovative and presented in a detailed and readable manner that can be followed by broad audience of SciPost Physics.
Weaknesses
See report.
Report
In the manuscript, the Author has considered standard 1D TIFM with additional fully-connected interacting term. The manuscript is divided into two parts — (a) in the first part, they have solved the system in the thermodynamic limit by means of ‘density-resolved coherent states’ that correspond to a mean-field Hamiltonian (MFH), essentially proving that MF treatment of the system becomes exact in the thermodynamic limit; (b) in the second part, the Author has then analyzed the system both at zero and finite temperatures using previously found analytical results.
Overall, I believe that the present work should deserve a publication in SciPost Physics in some form. But the Author should first modify the manuscript accordingly to make it suitable for the same. My criticisms/comments are as follows:
1. The TIFM with all-to-all interaction, although looks artificial, has immense practical importance as similar systems can be realized with ultra-cold atoms on optical lattices interacting with high-finesse cavities. The Author should include a discussion about many-body cavity systems in Sec. 2.3.
2. The present analytical treatment, although very innovative and interesting, can only work if the short-range part of the Hamiltonian is exactly solvable, making similar analysis in other model with mixed-ranged interaction intractable. The Author should mention this clearly in the manuscript. It would be great, but not mandatory, if the Author can comment whether the MF treatment also becomes exact in the thermodynamic limit for such ‘unsolvable’ systems.
3. While the first part of the manuscript is almost complete, the second part must be improved to a large extent. To make the manuscript readable for broad audience, the Author should include several illustrative and quantitive plots. Please follow following points for that.
4. In the table (wrongly denoted as Figure 1) the Author has compared ED energies with the thermodynamic MF result. In my opinion, such comparison should be accompanied by a plot, where they should plot energies of exact and MF Hamiltonian for different $L$, along with their extrapolations (with error-bars), and the exact thermodynamic MF result. Also, the $1/L$ extrapolation with only $L=18, 20$ is very crude and must be improved with proper error analysis. With present-day ED methods, it is possible go upto $L=24 - 28$ using standard computers for this Hamiltonian to do such extrapolation. They may also consider other observables for such comparisons.
5. The Author should clarify what do they mean by ‘ordered’, ‘disordered’, and ‘paramagnetic’ just before the Sec. 4.2 in terms of proper order parameters.
6. The left panel of Figure 2 and both the panels in Figure 3 must include proper numerical values of the axis parameters. The temperature is also not quoted in the caption of the left panel of Figure 3.
7. The Author should show the variations of different order parameters across different phase transitions, both from the numerics (for exact and MF Hamiltonians) for different system sizes, and in the thermodynamic limit using MF results. Since, average magnetizations and average two-site correlators are diagonal in momentum basis, they should be calculable within the present framework.
Requested changes
See report.
Anonymous Report 1 on 2022-10-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202210_00055v1, delivered 2022-10-13, doi: 10.21468/SciPost.Report.5891
Strengths
The Hamiltonian (1) is solved by a mean field in the sense that local observables are obtained from a generalized paired state ansatz which, in the thermodynamic limit, makes the problem exactly equivalent to a mean-field Hamiltonian. This is a very interesting mathematical construction with possible applications beyond model (1).
Weaknesses
At first site I found the Hamiltonian (1) somewhat artificial. The coefficient of the non-local term has to be fine tuned for the presented approach to work. However, section 2.3 dispels these doubts to large extent. Solvable models have to be special but 2.3 shows that (1) is close to some generic models.
Report
This is a very imaginative mathematical physics that, however, should be readable to broad audience. I mean that the formalism is relatively straightforward and, therefore, may be applicable in other models as well. The work is certainly worth being published in Scipost Physics.