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\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. 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}
----------------------------------------------------------------------
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?
\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
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?

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
$\theta_c$ 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.
\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
----------------------------------------------------------------------

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.
\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.

\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
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}

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
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.
\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 approaches the numeric values
of the scaling function and its derivatives measured in the aforementioned
works exponentially with iterative fitting.

\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.

The paper should be rejected.
\end{verbatim}
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