Free energy and metastable states in the square-lattice J1-J2 Ising model

Calculations and analysis were carefully performed.

Poorly organized and difficult to understand.

The author investigated the free energy of the square-lattice J1-J2 Ising model using the random local field approximation (RLFA). The author compared the prediction by the RLFA with the exact solution and Monte Carlo results for small systems. This paper continued with another article by the same author [25]. This paper aimed to study the metastable states identified in the previous paper in terms of free energy. The free energy calculated by the RLFA indicated the existence of the metastable states, and they were confirmed in the exact calculations and MC simulations. The geometric slab-droplet transition was also investigated.

The subject matter addressed in this paper is somewhat specialized. Still, the calculations and analysis have been carefully performed, and we are willing to publish it with minor modifications.

While the subject matter of this paper is somewhat very specialized, the calculations and analyses were performed carefully. Therefore, I agree to publish the manuscript with minor modifications.

Here are my comments.

(1) Organization of the manuscript

The paper is poorly organized and requires much effort from the reader. It also took a lot of work to understand the correspondence between the figures.

For example, Fig. 1 and Fig. 4 seem to correspond, but it isn't easy to see how they should be compared from the figures.
The symbols T_0, T_1, and T_2 appear without explanation in the caption of Fig. 1, and their definitions appear discretely in the text.

Also, Fig. 2 (a), Fig. 6 and Fig. 2 (b), and Fig. 8 seem to correspond to each other, but it isn't easy to read because it was necessary to refer to several places simultaneously to understand the context.

It is also difficult to extract useful information from Fig. 8(b).

The author should structure the manuscript more clearly for the reader.

(2) The definitions of the order parameters

The polarization m was adopted as the order parameter in the analysis with RLFA, while the total spin M was used as the order parameter for the exact solutions and MC simulations, which makes it difficult to compare the figures.

Also, since this paper deals only with homogeneous polarization, m seems identical to magnetization. Is it correct? For example, the caption of Fig. 3 refers to m as magnetization. Then, why not call all of them magnetization?

It would be easier to see if we use m as the order variable instead of the total spin M in Figs. 5-9. Is there any particular reason to use M?

(3) Comparison with MFA

The author claimed that the RLFA predicted the slab-droplet transition that MFA does not. Also, the author claimed that such geometric phase transition is present in the finite system with periodic boundary conditions.

If I understand correctly, RLFA is a kind of MFA, and the boundary conditions cannot be considered explicitly.

The author should discuss why RLFA could predict the slab-droplet transition, which MFA could not.

1- Modify figures to make it easier for readers to compare.
2- Add the detailed comparison with MFA

very detailed analysis
the self-consistent representation

The big difference in the range of J2 for the first order phase transition between RLFA and previous numerical calculation, especially the recent results, needs more convincing discussions. Perhaps, the RLFA itself , the analysis of the entropy , or something else, is accountable.

Compared to the relevant paper [20] in the references, the author made a further investigation by RLFA and restricted free energy. As demonstrated, the free energy barrier explained the formation of the metastable states as the temperature and J2 varied.

By a detailed calculation of size 6*6 in fig.5, the author illuminated the slab-droplet geometric phase transition. By RLFA, the non-analytic behaviors signals the phase transition. It is a perspective angle to enrich the understanding for J1-J2 model.

I feel the main context is too long, it would be better to move some figures and relevant discussions to the supplemental materials for the compactness in the article, if possible.

The author have made a self-consistent analysis for the results from RLFA, MC, and restricted free energy. I still wonder the big difference of the range of J2 determining the first order phase transition between previous numerical calculation and the results in this paper. More convincing discussions, e.g. the characters of RLFA , the effect from the entropy, or something else, is to be discussed for the completeness.

In addition, a typo in line #94, less then-> less than.

More convincing discussions about the big difference of J2 regime between RLFA and previous numerical results for the first order phase transition the are needed.

1- Use of a sophisticated mean field theory based ansatz to understand a known problem of meta-stability in a toy model.

2- Comparison with Monte Carlo simulations provided.

1- Results do not appear to add obvious value over a recent paper by the same author (https://journals.aps.org/pre/abstract/10.1103/PhysRevE.107.034124), where the same problem has been tackled using the same method.

2- The indicators for meta-stability are indirect and the results are confirmed using differences in free energy based on expectations of slab versus droplet pictures. This is not a smoking gun signature. I would suggest looking at correlation functions (if possible) in both RLFA and MC simulations to confirm the presence of these spatial structures.

3- The discussion about the ability of RLFA to capture the meta-stable states is not sufficient, and requires added details which can convince the reader as to the added expression ability provided by RLFA.

4- Monte Carlo simulations are not performed at a sufficient size. If they are scaled up, direct snapshots can be shown to indicate the expected spatial structures.

The author has considered the problem of meta-stability created by competing interactions in an Ising model on a square lattice. The method used here is a mean field ansatz which is capable of treating nearest neighbor correlations. Stable spatial patterns are argued for using differences in free energy at fixed magnetization. A comparison to Monte Carlo results is provided. As differences in free energy are employed, the inference of particular spatial patterns is indirect and it would be beneficial is something more direct (such as correlation functions) could be provided. The central claim is the ability of the particular flavor of mean field ansatz to represent various meta-stable spatial patterns, and this is not adequately supported by the data presented

1- A more detailed explanation of the RLFA procedure and its ability to encode spatial patterns is required.

2- Direct evidence (such as correlation functions) should be provided using both RLFA and MC simulations. This is definitely possible for the latter. For RLFA, this may be a challenging task, but it would definitely strengthen the claim the author wishes to make.