Draft of referee response complete.

3 files changed, 95 insertions, 3 deletions

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.