From 8280facaa0ce71199011ebd3101de02c89601798 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 25 May 2023 15:23:03 +0200 Subject: Updated affiliation and started referee response. --- ising_scaling.tex | 2 +- referee_response.tex | 148 +++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 149 insertions(+), 1 deletion(-) create mode 100644 referee_response.tex diff --git a/ising_scaling.tex b/ising_scaling.tex index 53f76ad..d7d2474 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -29,7 +29,7 @@ linkcolor=purple \title{Precision approximation of the universal scaling functions for the 2D Ising model in an external field} \author{Jaron Kent-Dobias} -\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\affiliation{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma} \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} diff --git a/referee_response.tex b/referee_response.tex new file mode 100644 index 0000000..7990a4f --- /dev/null +++ b/referee_response.tex @@ -0,0 +1,148 @@ +\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} -- cgit v1.2.3-54-g00ecf