From 8280facaa0ce71199011ebd3101de02c89601798 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 25 May 2023 15:23:03 +0200
Subject: Updated affiliation and started referee response.

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+\documentclass[a4paper]{article}
+
+\usepackage{fullpage}
+\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
+\usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{fullpage,amsmath,amssymb,latexsym,graphicx}
+\usepackage{newtxtext,newtxmath} % Times for PR
+
+\begin{document}
+
+Response to referees
+
+\begin{verbatim}
+----------------------------------------------------------------------
+Report of Referee A -- LK15589/Kent-Dobias
+----------------------------------------------------------------------
+
+New expressions of the scaling function of free energy, magnetization,
+and magnetic susceptibility of the Ising model in a magnetic field are
+proposed. These expressions are obtained by combining:
+
+- an essential singularity at zero magnetic field (as predicted by the
+critical droplet theory), obtained by applying the Kramers-Kronig
+relation to a scaling ansatz of the 'metastable free energy',
+
+- a parameterization (in the spirit of Schofield parameterization) in
+terms of new scaling fields of the analytical part of the scaling
+function.
+
+Even though both approaches have been introduced in the late 1960s, I
+am not aware of any other attempt to combine them. This is the great
+originality of this paper. The agreement of the proposed scaling
+functions with the Monte Carlo data presented on figure 1 is
+impressive. The improvement compared to the series expansion (8th
+order plotted on figure 1) is undeniable. It seems to me that this
+work constitutes a real progress in the field of critical phenomena.
+In the presentation, the focus is put on the 2D Ising model but the
+ideas could be applied to a broad class of systems where a continuous
+transition lies at the end of first-order transition line. For these
+reasons, I recommend the publication in Physical Review Letters.
+Questions and comments follow.
+
+1. I did not find in Ref [3] the statement that the essential
+singularity is not observable, as written by the authors. Could the
+authors tell me at which page they found this statement?
+
+2. Before equation (1), some factors are missing in the expression of
+the critical droplet size that should read $R_c={(d-1)\over d}{\Sigma
+S_d\over M|H|V_d}$.
+
+3. The steps leading to the scaling functions (7) and (8) does not
+seem to depend on any particular model but only on the dimension $d$
+and on the exponent $b$ describing the fluctuations of the spherical
+critical droplet. I am therefore wondering if the same scaling
+functions would also hold for models in different universality
+classes, the 3-state Potts model for example. Could the authors
+comment on this?
+
+4. In the particular case of the Ising model, $d=4$ is the upper
+critical dimension. Could this affect the scaling function (8), for
+example by the presence of logarithmic corrections?
+
+5. After equation (12), in the expression of $F(t,h)$, the term
+$t^2\ln t^2$ cannot come from the integration of (10). Its presence
+should be motivated.
+
+6. Did the authors try to produce the same comparison as in figure 1
+in the case of the 3D and 4D Ising model?
+
+7. There is no function $f$ in equation (13) as mentioned in the
+sentence that follows.
+
+8. The presentation of the Schofield-like parameterization (page 3) is
+really minimalist compared to the rest of the paper. I think that the
+presentation of this part could (should?) be improved. What does
+$\theta_c$ correspond to? Is it a free parameter? Why is (15) analytic
+in the range $-\theta_c<\theta <\theta_c$? What is the interest? Why
+this parameterization is more useful than the original scaling
+variable? I understand that details will be given in a forthcoming
+publication but more details would help the non-expert reader to
+appreciate the interest of the approach.
+
+9. In the conclusion, the authors wrote ``We have developed a Wolff
+algorithm for the Ising model in a field''. The idea of introducing a
+ghost spin is not new. It is mentioned in R.H. Swendsen and J.S. Wang
+(1987) \textit{Phys. Rev. Lett.} \textbf{58} 86 where it is attributed
+to the original Fortuin-Kastelyn work from 1969.
+
+10. There is a minor typo in the acknowledgment: I guess that you want
+to thank Jacques Perk.
+
+----------------------------------------------------------------------
+Report of Referee B -- LK15589/Kent-Dobias
+----------------------------------------------------------------------
+
+There are a variety of problems with this paper and it should not be
+published. Since the authors will not agree with this I will attempt
+to detail my objections:
+
+This paper appears to combine the droplet model picture from the 60's
+with some renormalization group language and a computer computation
+which is not explained and it is not clear what the authors are
+willing to call an actual result.
+
+The two dimensional Ising model in a magnetic field has been studied
+for decades and any further study must relate to these extensive
+computations. This paper fails completely to do this.
+
+1. Several references are missing:
+
+S. N. Isakov, Comm. Math. Phys. (1984) 427-443 where the essential
+singularity are the phase boundary is demonstrated.
+
+P. Fonseca and A. Zamolodchikov, J. Stat. Phys. 110 (2002) 527-590
+which gives a comprehensive scenario for the scaled free energy in the
+critical region.
+
+A. Zamolodchikov and I Ziyaldinov, Nuclear Physics B849 (2011) 654-674
+where scattering in the Ising field theory is extensively discussed.
+
+2. Several references are clearly not understood. The authors state
+the references 15-20 deal with an essential singularity in the
+magnetic susceptibility whereas papers 15-20 are concerned with a
+natural boundary in the susceptibility. Essential singularities are
+isolated singularities, natural boundaries are not. The authors say
+nothing about this natural boundary which is a major feature of the
+analyticity of the model that must be explained.
+
+3. There are completely unsubstantiated claims made at the end of the
+paper. It is said that "Our methods should allow improved
+high-precision forms for the free energy." The results of references
+15 and 16 have generated, used and analyzed series of hundreds and
+thousands of terms. There is no reason to believe that anything in
+this present paper will improve on this monumental work or on the work
+of ref. 43. Statements such as "Our methods might be generalized to
+predict similar singularities..." have no place in a scientific paper.
+
+4. The statement "Our forms both exhibit incorrect low-order
+coefficients at the transition (Fig. 2) and incorrect asymptotics as
+h|t|^{-\beta delta} becomes very large" does not inspire confidence in
+the paper.
+
+In short, I cannot find anything in this paper which makes an advance
+over the previous literature of 50 years.
+
+The paper should be rejected.
+\end{verbatim}
+\end{document}
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