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.

---
 referee_response.tex | 148 +++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 148 insertions(+)
 create mode 100644 referee_response.tex

(limited to 'referee_response.tex')

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 v1.2.3-70-g09d2


From fd0b72927daa9df68e0f0c83437b1392a18830a5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 25 May 2023 15:48:48 +0200
Subject: Draft of referee response complete.

---
 ising_scaling.bib    | 16 ++++++++++-
 ising_scaling.tex    |  2 +-
 referee_response.tex | 80 +++++++++++++++++++++++++++++++++++++++++++++++++++-
 3 files changed, 95 insertions(+), 3 deletions(-)

(limited to 'referee_response.tex')

diff --git a/ising_scaling.bib b/ising_scaling.bib
index ffc714c..0d0dc51 100644
--- a/ising_scaling.bib
+++ b/ising_scaling.bib
@@ -352,7 +352,7 @@
 }
 
 @article{BarmaFisherPRB,
-  title={Two-dimensional Ising-like systems: Corrections to scaling in the Klauder and double-Gaussian models},
+  title={Two-dimensional {Ising}-like systems: Corrections to scaling in the {Klauder} and double-{Gaussian} models},
   author={Barma, Mustansir and Fisher, Michael E},
   journal={Physical Review B},
   volume={31},
@@ -363,3 +363,17 @@
 }
 
 
+@article{Isakov_1984_Nonanalytic,
+ author = {Isakov, S. N.},
+ title = {Nonanalytic features of the first order phase transition in the Ising model},
+ journal = {Communications in Mathematical Physics},
+ publisher = {Springer Science and Business Media LLC},
+ year = {1984},
+ month = {12},
+ number = {4},
+ volume = {95},
+ pages = {427--443},
+ url = {https://doi.org/10.1007%2Fbf01210832},
+ doi = {10.1007/bf01210832}
+}
+
diff --git a/ising_scaling.tex b/ising_scaling.tex
index d7d2474..00de5e3 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -207,7 +207,7 @@ literature \cite{Mangazeev_2010_Scaling, Clement_2019_Respect}.
 
 In the low temperature phase, the free energy has an essential singularity at
 zero field, which becomes a branch cut along the negative-$h$ axis when
-analytically continued to negative $h$ \cite{Langer_1967_Theory}. The origin
+analytically continued to negative $h$ \cite{Langer_1967_Theory, Isakov_1984_Nonanalytic}. The origin
 can be schematically understood to arise from a singularity that exists in the
 imaginary free energy of the metastable phase of the model. When the
 equilibrium Ising model with positive magnetization is subjected to a small
diff --git a/referee_response.tex b/referee_response.tex
index 7990a4f..01ece7a 100644
--- a/referee_response.tex
+++ b/referee_response.tex
@@ -8,7 +8,12 @@
 
 \begin{document}
 
-Response to referees
+\section*{Response to referees for \texttt{LK15589/Kent-Dobias}}
+
+First, we would like to apologize for the large delay in resubmission. As is
+evident, the manuscript has undergone a significant transformation as a result
+of the reviews we received. We would like to thank the reviewers for their
+helpful notes on the original manuscript.
 
 \begin{verbatim}
 ----------------------------------------------------------------------
@@ -43,11 +48,19 @@ 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?
+\end{verbatim}
+
+The comment has been removed.
 
+\begin{verbatim}
 2. Before equation (1), some factors are missing in the expression of
 the critical droplet size that should read $R_c={(d-1)\over d}{\Sigma
 S_d\over M|H|V_d}$.
+\end{verbatim}
 
+These equations are completely changed in the new manuscript.
+
+\begin{verbatim}
 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
@@ -55,7 +68,12 @@ 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?
+\end{verbatim}
 
+The observation of the referee is true, and these models could be studied with
+a similar technique if sufficient data on their scaling functions is measured.
+
+\begin{verbatim}
 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?
@@ -63,13 +81,31 @@ 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.
+\end{verbatim}
+
+We have now clarified both of these questions in part II, where the
+relationship between flow equations and singularities in the free energy is
+discussed. For the 4D model, the presence of a marginal variable dramatically
+changes the analytic structure of the scaling function.
 
+\begin{verbatim}
 6. Did the authors try to produce the same comparison as in figure 1
 in the case of the 3D and 4D Ising model?
+\end{verbatim}
+
+We do not, though it would not be difficult to apply these techniques to the 3D
+model. For the 4D, as mentioned above, some substantial changes would need to
+be made to the parametric form. In addition, less data on the scaling functions
+are available in 3D and especially 4D.
 
+\begin{verbatim}
 7. There is no function $f$ in equation (13) as mentioned in the
 sentence that follows.
+\end{verbatim}
+
+This is no longer relevant to the modified manuscript.
 
+\begin{verbatim}
 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
@@ -79,16 +115,29 @@ 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.
+\end{verbatim}
 
+In the new manuscript, the treatment of the Schofield parameterization has now
+been made central.
+
+\begin{verbatim}
 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.
+\end{verbatim}
 
+Indeed true, numeric references have since been removed.
+
+\begin{verbatim}
 10. There is a minor typo in the acknowledgment: I guess that you want
 to thank Jacques Perk.
+\end{verbatim}
+
+The name has been corrected.
 
+\begin{verbatim}
 ----------------------------------------------------------------------
 Report of Referee B -- LK15589/Kent-Dobias
 ----------------------------------------------------------------------
@@ -117,7 +166,14 @@ critical region.
 
 A. Zamolodchikov and I Ziyaldinov, Nuclear Physics B849 (2011) 654-674
 where scattering in the Ising field theory is extensively discussed.
+\end{verbatim}
+
+We thank the referee for their helpful references, and we have cited the first
+two. The second one was especially relevant to our study. We don't see the
+relevance of the scattering to the scaling functions we study here, but perhaps
+future work may examine it as well.
 
+\begin{verbatim}
 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
@@ -125,7 +181,14 @@ 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.
+\end{verbatim}
+
+The natural boundary mentioned is purported to exist in the complex temperature
+dependence susceptibility of the lattice Ising model. It is not clear to us
+why it should be present in the scaling function of the free energy for the
+Ising universality class. We have removed the inaccurate comments.
 
+\begin{verbatim}
 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
@@ -134,12 +197,27 @@ 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.
+\end{verbatim}
 
+We believe that our transformed technique and manuscript can substantiate this
+claim, in a specific sense. Though the free energy computed point by point in
+our references by Mangazeev et al. and Fonseca et al. are more accurate, they
+are not functional forms: they are tables of data. We now show in the
+manuscript that our functional form exponentially approaches the numeric values
+of the scaling function and its derivatives measured in the aforementioned
+works.
+
+\begin{verbatim}
 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.
+\end{verbatim}
 
+The asymptotic problems of the old manuscript have been repaired by treating
+more carefully the parametric coordinates.
+
+\begin{verbatim}
 In short, I cannot find anything in this paper which makes an advance
 over the previous literature of 50 years.
 
-- 
cgit v1.2.3-70-g09d2


From 90e8393b4069ace54c73975e3159aaf0db782ab7 Mon Sep 17 00:00:00 2001
From: jps6 <jps6@cornell.edu>
Date: Thu, 25 May 2023 15:35:15 +0000
Subject: Update on Overleaf.

---
 referee_response.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

(limited to 'referee_response.tex')

diff --git a/referee_response.tex b/referee_response.tex
index 01ece7a..b33d3b1 100644
--- a/referee_response.tex
+++ b/referee_response.tex
@@ -183,10 +183,11 @@ nothing about this natural boundary which is a major feature of the
 analyticity of the model that must be explained.
 \end{verbatim}
 
-The natural boundary mentioned is purported to exist in the complex temperature
-dependence susceptibility of the lattice Ising model. It is not clear to us
-why it should be present in the scaling function of the free energy for the
-Ising universality class. We have removed the inaccurate comments.
+Our scaling function indeed does not show any evidence of a natural boundary or logarithmic corrections at complex temperatures in a field: we see only the branch cut of the dominant logarithmic singularity in the free energy. This is to be expected, because our calculation focuses on the universal scaling function as it depends upon the relevant variables $t$ and $h$, and does not incorporate singular corrections to scaling from irrelevant operators.
+
+The logarithmic corrections seen in the susceptibility are thought by these authors to come from singular corrections to scaling from these irrelevant operators. Furthermore, these logarithms are thought by Perk (private communication) to be associated with the lattice models, so they should not be seen in (say) the $\phi^4$ theory or membrane Ising phase transitions. 
+
+We expect that a natural boundary in the susceptibility in the complex plane in the lattice model is due to these corrections to scaling, and thus should not be expected to manifest itself in the universal scaling function we calculate.
 
 \begin{verbatim}
 3. There are completely unsubstantiated claims made at the end of the
-- 
cgit v1.2.3-70-g09d2


From aab3c8e2b5fcad46e41f2f76e1dd342031f8afe0 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 26 May 2023 10:49:08 +0200
Subject: Edited the referee response.

---
 referee_response.tex | 27 ++++++++++++++++++++-------
 1 file changed, 20 insertions(+), 7 deletions(-)

(limited to 'referee_response.tex')

diff --git a/referee_response.tex b/referee_response.tex
index b33d3b1..0f2ce95 100644
--- a/referee_response.tex
+++ b/referee_response.tex
@@ -13,7 +13,22 @@
 First, we would like to apologize for the large delay in resubmission. As is
 evident, the manuscript has undergone a significant transformation as a result
 of the reviews we received. We would like to thank the reviewers for their
-helpful notes on the original manuscript.
+helpful notes on the original manuscript. The first reviewer was supportive and
+asked instructive questions. The second reviewer, though critical, led us to
+some great insights.
+
+The manuscript now focuses on the approximation of the 2D Ising universal
+scaling function by a smooth functional form. Though the singularity discussed
+in the original manuscript still plays an important role, our approximation now
+encompasses the whole parameter space of the relevant scaling fields. We
+compare this form to the values of the universal scaling function and its
+derivatives previously measured in the literature, and find exponential
+convergence with the amount of data fit.
+
+We believe that the substantial changes to our manuscript merit its
+reconsideration for publication. Though the new manuscript is so different from
+the old one as to likely deserve a new reviewing cycle, we respond to the
+original reviews here, to make clear how the revised manuscript addresses them.
 
 \begin{verbatim}
 ----------------------------------------------------------------------
@@ -169,9 +184,7 @@ where scattering in the Ising field theory is extensively discussed.
 \end{verbatim}
 
 We thank the referee for their helpful references, and we have cited the first
-two. The second one was especially relevant to our study. We don't see the
-relevance of the scattering to the scaling functions we study here, but perhaps
-future work may examine it as well.
+two. The second one was especially relevant to our study.
 
 \begin{verbatim}
 2. Several references are clearly not understood. The authors state
@@ -202,11 +215,11 @@ predict similar singularities..." have no place in a scientific paper.
 
 We believe that our transformed technique and manuscript can substantiate this
 claim, in a specific sense. Though the free energy computed point by point in
-our references by Mangazeev et al. and Fonseca et al. are more accurate, they
+our references by Mangazeev et al.\ and Fonseca et al.\ are more accurate, they
 are not functional forms: they are tables of data. We now show in the
-manuscript that our functional form exponentially approaches the numeric values
+manuscript that our functional form approaches the numeric values
 of the scaling function and its derivatives measured in the aforementioned
-works.
+works exponentially with iterative fitting.
 
 \begin{verbatim}
 4. The statement "Our forms both exhibit incorrect low-order
-- 
cgit v1.2.3-70-g09d2

