path: root/ising_scaling.tex

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\begin{document}

\title{Smooth Ising universal scaling functions}

\author{Jaron Kent-Dobias}
\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France}

\author{James P.~Sethna}
\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA}

\date\today

\begin{abstract}
\end{abstract}

\maketitle

At continuous phase transitions the thermodynamic properties of physical
systems have singularities. Celebrated renormalization group analyses imply
that not only the principal divergence but also entire additive functions are
\emph{universal}, meaning that they will appear at any critical points that
connect phases of the same symmetries in the same spatial dimension. The study
of these universal functions is therefore doubly fruitful: it provides both a
description of the physical or model system at hand, and \emph{every other
system} whose symmetries, interaction range, and dimension puts it in the same
universality class.

The continuous phase transition in the two-dimensional Ising model is perhaps
the most well studied, and its universal thermodynamic functions have likewise
received the most attention. Precision numeric work both on the lattice
critical theory and on the ``Ising'' critical field theory (related by
universality) have yielded high-order polynomial expansions of those functions
in various limits, along with a comprehensive understanding of their analytic
properties and even their full form \cite{Fonseca_2003_Ising, Mangazeev_2008_Variational, Mangazeev_2010_Scaling}. In parallel, smooth approximations of the
Ising ``equation of state'' have produced convenient, evaluable, differentiable
empirical functions \cite{Guida_1997_3D, Campostrini_2000_Critical, Caselle_2001_The}. Despite being differentiable, these approximations become
increasingly poor when derivatives are taken due to the presence of a subtle
essential singularity [refs] that is previously unaccounted for.

This paper attempts to find the best of both worlds: a smooth approximate
universal thermodynamic function that respects the global analyticity of the
Ising free energy, for both the two-dimensional Ising model (where much is
known) and the three-dimensional Ising model (where comparatively less is
known). First, parametric coordinates are introduced that remove unnecessary
nonanalyticities from the scaling function. {\bf [The universal scaling function has the nonanalyticities. You are writing it as a function with the right singularity, modulated somehow with an analytic function.]} Then the arbitrary analytic
functions that compose those coordinates are approximated by truncated
polynomials whose coefficients are fixed by matching the series expansions of
the universal function in three critical regimes: at no field and low
temperature, no field and high temperature, and along the critical isotherm.

This paper is divided into four parts. First, general aspects of the problem
will be reviewed that are relevant in all dimensions. Then, the process
described above will be applied to the two- and three-dimensional Ising models.

\section{General aspects}

\subsection{Universal scaling functions}

A renormalization group analysis predicts that certain thermodynamic functions
will be universal in the vicinity of any critical point in the Ising
universality class. Here we will explain precisely what is meant by universal.

Suppose one controls a temperature-like parameter $T$ and a magnetic field-like
parameter $H$, which in the proximity of a critical point at $T=T_c$ and $H=0$
have normalized reduced forms $t=(T-T_c)/T_c$ and $h=H/T$. Thermodynamic
functions are derived from the free energy per site $f$, which depends on $t$
and $h$. Renormalization group analysis can be used calculated the flow of
these parameters under continuous changes of scale, yielding flow equations of
the form
\begin{align} \label{eq:raw.flow}
  \frac{dt}{d\ell}=\frac1\nu t+\cdots
  &&
  \frac{dh}{d\ell}=\frac{\beta\delta}\nu h+\cdots
  &&
  \frac{df}{d\ell}=Df+\cdots
\end{align}
where $D$ is the dimension of space and $\nu$, $\beta$, and $\delta$ are
dimensionless constants. The flow equations are truncated here, but in general
all terms allowed by symmetry are present on their righthand side. By making a
near-identity transformation to the coordinates and the free energy of the form
$u_t(t, h)=t+\cdots$, $u_h(t, h)=h+\cdots$, and $u_f(f,t,h)=f+\cdots$, one can
bring the flow equations into an agreed upon simplest normal form
\begin{align} \label{eq:flow}
  \frac{du_t}{d\ell}=\frac1\nu u_t
  &&
  \frac{du_h}{d\ell}=\frac{\beta\delta}\nu u_h
  &&
  \frac{du_f}{d\ell}=Du_f+g(u_t),
\end{align}
which are exact as written. The flow of the parameters is made exactly linear,
while that of the free energy is linearized as nearly as possible. Solving these equations for $u_f$ yields
\begin{equation}
  \begin{aligned}
    u_f(u_t, u_h)
    &=|u_t|^{D\nu}\mathcal F_\pm(u_h|u_t|^{-\beta\delta})+|u_t|^{D\nu}\int_1^{u_t}dx\,\frac{g(x)}{x^{1+D\nu}} \\
    &=|u_h|^{D\nu/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta})+|u_t|^{D\nu}\int_1^{u_t}dx\,\frac{g(x)}{x^{1+D\nu}} \\
  \end{aligned}
\end{equation}
where $\mathcal F_\pm$ and $\mathcal F_0$ are undetermined scaling functions.  The scaling functions are universal in the sense that if
another system whose critical point belongs to the same universality class has
its parameters brought to the form \eqref{eq:flow}, one will see the same
functional form (up to constant rescaling of $u_t$ and $u_h$ and choice of
$g$).

The analyticity of the free energy at finite size implies that the functions
$\mathcal F_\pm$ have power-law expansions of their arguments about zero. This
is not the case at infinity, and in fact $\mathcal
F_0(\eta)=\eta^{2/\beta\delta}\mathcal F_\pm(\eta^{-1/\beta\delta})$ itself has a power-law
expansion about zero, implying that $\mathcal F_\pm(\xi)\sim \xi^{2\beta\delta}$ for large $x$.

The free energy flow equation of the 3D Ising model can be completely linearised, giving $g(x)=0$. This is not the case for the 2D Ising model, where a term proportional to $u_t^2$ cannot be removed by a smooth change of coordinates. The scale of this term sets the relative size of $u_f$ and $u_t$.
For the constant scale of $u_t$ and $u_h$, we adopt the same convention as used by
\cite{Fonseca_2003_Ising}. This gives $g(u_t)=-\frac1{4\pi}u_t^2$. The dependence of the nonlinear scaling variables on
the parameters $t$ and $h$ is system-dependent, and their form can be
found for common model systems (the square- and triangular-lattice Ising
models) in the literature \cite{Clement_2019_Respect}.



\subsection{Essential singularities and droplets}


In the low temperature phase, the free energy as a function of field 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 can be schematically understood to arise from a
singularity that exists in the complex free energy of the metastable phase of
the model, suitably continued into the equilibrium phase. When the equilibrium
Ising model with positive magnetization is subjected to a small negative
magnetic field, its equilibrium state instantly becomes one with a negative
magnetization. However, under physical dynamics it takes time to arrive at this
state, which happens after a fluctuation containing a sufficiently large
equilibrium `bubble' occurs.

The bulk of such a bubble of radius $R$ lowers the free energy by
$2M|H|V_dR^d$, where $d$ is the dimension of space, $M$ is the magnetization,
$H$ is the external field, and $V_d$ is the volume of a $d$-ball, but its
surface raises the free energy by $\sigma S_dR^{d-1}$, where $\sigma$ is the
surface tension between the stable--metastable interface and $S_d$ is the
volume of a $(d-1)$-sphere. The bubble is sufficiently large to decay
metastable state when the differential bulk savings outweigh the surface costs.

This critical bubble occurs with free energy cost
\begin{equation}
  \begin{aligned}
    \Delta F_c
      &\simeq\left(\frac{S_d\sigma}d\right)^d\left(\frac{d-1}{2V_dM|H|}\right)^{d-1} \\
      &\simeq T\left(\frac{S_d\mathcal S(0)}d\right)^d\left[\frac{2V_d\mathcal M(0)}{d-1}ht^{-\beta\delta}\right]^{-(d-1)}
  \end{aligned}
\end{equation}
where $\mathcal S(0)$ and $\mathcal M(0)$ are the critical amplitudes for the
surface tension and magnetization, respectively \textbf{[find more standard
notation]} \cite{Kent-Dobias_2020_Novel}.
In the context of statistical mechanics, Langer demonstrated that the decay rate is asymptotically proportional to the imaginary part of the free energy in the metastable phase, with (assuming Arrhenius behavior)
\begin{equation}
  \operatorname{Im}f\propto\Gamma\sim e^{-\beta\Delta F_c}=e^{-1/(B|h||t|^{-\beta\delta})^{d-1}}
\end{equation}
which can be more rigorously related in the context of quantum field theory [ref?].

\begin{figure}
  \includegraphics{figs/F_lower_singularities.pdf}
  \caption{
    Analytic structure of the low-temperature scaling function $\mathcal F_-$
    in the complex $\xi=u_h|u_t|^{-\beta\delta}\propto H$ plane. The circle
    depicts the essential singularity at the first order transition, while the
    solid line depicts Langer's branch cut.
  } \label{fig:lower.singularities}
\end{figure}
  
This is a singular contribution that depends principally on the scaling
invariant $ht^{-\beta\delta}\simeq u_h|u_t|^{-\beta\delta}$. It is therefore
suggestive that this should be considered a part of the singular free energy
$f_s$, and moreover part of the scaling function that composes it. We will therefore make the ansatz that
\begin{equation}
  \operatorname{Im}\mathcal F_-(\xi)=A\Theta(-\xi)|\xi|^{-b}e^{-1/(B|\xi|)^{d-1}}\left(1+O(\xi)\right)
\end{equation}
\cite{Houghton_1980_The}
The exponent $b$ depends on dimension and can be found through a more careful
accounting of the entropy of long-wavelength fluctuations in the droplet
surface \cite{Gunther_1980_Goldstone}.

\subsection{Yang--Lee edge singularities}

At finite size, the Ising model free energy is an analytic function of
temperature and field because it is the logarithm of a sum of positive analytic
functions. However, it can and does have singularities in the complex plane due
to zeros of the partition function at complex argument, and in particular at
imaginary values of field, $h$. Yang and Lee showed that in the thermodynamic
limit of the high temperature phase of the model, these zeros form a branch cut
along the imaginary $h$ axis that extends to $\pm i\infty$ starting at the
point $\pm ih_{\mathrm{YL}}$ \cite{Yang_1952_Statistical, Lee_1952_Statistical}.
The singularity of the phase transition occurs because these branch cuts
descend and touch the real axis as $T$ approaches $T_c$, with
$h_{\mathrm{YL}}\propto t^{\beta\delta}$. This implies that the
high-temperature scaling function for the Ising model should have complex
branch cuts beginning at $\pm i\xi_{\mathrm{YL}}$ for a universal constant
$\xi_{\mathrm{YL}}$.

\begin{figure}
  \includegraphics{figs/F_higher_singularities.pdf}
  \caption{
    Analytic structure of the high-temperature scaling function $\mathcal F_+$
    in the complex $\xi=u_h|u_t|^{-\beta\delta}\propto H$ plane. The squares
    depict the Yang--Lee edge singularities, while the solid lines depict
    branch cuts.
  } \label{fig:higher.singularities}
\end{figure}

The Yang--Lee singularities are critical points in their own right, with their own universality class different from that of the Ising model \cite{Fisher_1978_Yang-Lee}.

\cite{Cardy_1985_Conformal}
\cite{Connelly_2020_Universal}
\cite{An_2016_Functional}
\cite{Zambelli_2017_Lee-Yang}
\cite{Gliozzi_2014_Critical}

\subsection{Schofield coordinates}

The invariant combinations $u_h|u_t|^{-\beta\delta}$ or
$u_t|u_h|^{-1/\beta\delta}$ are natural variables to describe the scaling
functions, but prove unwieldy when attempting to make smooth approximations.
This is because, when defined in terms of these variables, scaling functions
that have polynomial expansions at small argument have nonpolynomial expansions
at large argument. Rather than deal with the creative challenge of dreaming up
functions with different asymptotic expansions in different limits, we adopt
different coordinates, in terms of which a scaling function can be defined that
has polynomial expansions in \emph{all} limits.

In all dimensions, the Schofield coordinates $R$ and $\theta$ will be implicitly defined by
\begin{align} \label{eq:schofield}
  u_t(R, \theta) = Rt(\theta)
  &&
  u_h(R, \theta) = R^{\beta\delta}h(\theta)
\end{align}
where $t$ and $h$ are polynomial functions selected so as to associate different scaling limits with different values of $\theta$. We will adopt standard forms for these functions, given by
\begin{align} \label{eq:schofield.funcs}
  t(\theta)=1-\theta^2
  &&
  h(\theta)=\left(1-\frac{\theta^2}{\theta_c^2}\right)\sum_{i=0}^\infty h_i\theta^{2i+1}
\end{align}
This means that $\theta=0$ corresponds to the high-temperature zero-field line,
$\theta=1$ to the critical isotherm at nonzero field, and $\theta=\theta_c$ to
the low-temperature zero-field (phase coexistence) line.

In practice the infinite series in \eqref{eq:schofield.funcs} cannot be
entirely fixed, and it will be truncated at finite order. We will notate the
truncation an upper bound of $n$ by $h^{(n)}$. The convergence of the
coefficients as $n$ is increased will be part of our assessment of the success
of the convergence of the scaling form.

One can now see the convenience of these coordinates. Both scaling variables depend only on $\theta$, as
\begin{align}
  \xi&=u_h|u_t|^{-\beta\delta}=h(\theta)|t(\theta)|^{-\beta\delta} \\
  \eta&=u_t|u_h|^{-1/\beta\delta}=t(\theta)|h(\theta)|^{-1/\beta\delta}.
\end{align}
Moreover, both scaling variables have polynomial expansions in $\theta$ near zero, with
\begin{align}
  &\xi= h'(0)|t(0)|^{-\beta\delta}\theta+\cdots  && \text{for $\theta\simeq0$}\\
  &\xi=h'(\theta_c)|t(\theta_c)|^{-\beta\delta}(\theta-\theta_c)+\cdots && \text{for $\theta\simeq\theta_c$}
  \\
  &\eta=-2(\theta-1)h(1)^{-1/\beta\delta}+\cdots && \text{for $\theta\simeq1$}.
\end{align}
Since the scaling functions $\mathcal F_\pm(\xi)$ and $\mathcal F_0(\eta)$ have
polynomial expansions about small $\xi$ and $\eta$, respectively, this implies
both will have polynomial expansions in $\theta$ at all three places above.

Therefore, in Schofield coordinates one expects to be able to define a global
scaling function $\mathcal F(\theta)$ which has a polynomial expansion in its
argument for all real $\theta$. For small $\theta$ $\mathcal F(\theta)$ will
resemble $\mathcal F_+$, for $\theta$ near one it will resemble $\mathcal F_0$,
and for $\theta$ near $\theta_c$ it will resemble $\mathcal F_-$. This leads us
to expect that the singularities present in these functions will likewise be
present in $\mathcal F(\theta)$. This is shown in Figure
\ref{fig:schofield.singularities}. Two copies of the Langer branch cut stretch
out from $\pm\theta_c$, where the equilibrium phase ends, and the Yang--Lee
edge singularities are present on the imaginary-$\theta$ line, where they must be since $\mathcal F$ has the same symmetry in $\theta$ as $\mathcal F_+$ has in $\xi$.

The location of the Yang--Lee edge singularities can be calculated directly from the coordinate transformation \eqref{eq:schofield}. Since $h(\theta)$ is an odd real polynomial for real $\theta$, it is imaginary for imaginary $\theta$. Therefore, one requires that
\begin{equation}
  i\xi_{\mathrm{YL}}=\frac{h(i\theta_{\mathrm{YL}})}{(1+\theta_{\mathrm{YL}}^2)^{-\beta\delta}}
\end{equation}
The location $\theta_c$ is not fixed by any principle and will be left a floating parameter.

\begin{figure}
  \includegraphics{figs/F_theta_singularities.pdf}
  \caption{
    Analytic structure of the global scaling function $\mathcal F$ in the
    complex $\theta$ plane. The circles depict essential singularities of the
    first order transitions, the squares the Yang--Lee singularities, and the
    solid lines depict branch cuts.
  } \label{fig:schofield.singularities}
\end{figure}

\subsection{Singular free energy}

As we have seen in the previous sections, the unavoidable singularities in the
scaling functions are readily expressed as singular functions in the imaginary
part of the free energy.

Our strategy follows. First, we take the known singular expansions of the imaginary parts of the scaling functions $\mathcal F_{\pm}(\xi)$ and produce simplest form accessible under polynomial coordinate changes of $\xi$. Second, we assert that the imaginary part of $\mathcal F(\theta)$ must have this simplest form. Third, we perform a Kramers--Kronig type transformation to establish an explicit form for the real part of $\mathcal F(\theta)$. Finally, we make good on the assertion posited in the second step by fixing the Schofield coordinate transformation to produce the correct coefficients known for the real part of $\mathcal F_{\pm}$.

This success of this stems from the commutative diagram below. So long as the
application of Schofield coordinates and the dispersion relation can be said to
commute, we may assume we have found correct coordinates for the simplest form
of the imaginary part to be fixed in reality by the real part.
\[
  \begin{tikzcd}[row sep=large, column sep = 9em]
  \operatorname{Im}\mathcal F_\pm(\xi) \arrow{r}{\text{Kramers--Kronig in $\xi$}} \arrow[]{d}{\text{Schofield}} & \operatorname{Re}\mathcal F_{\pm}(\xi) \arrow{d}{\text{Schofield}} \\%
  \operatorname{Im}\mathcal F(\theta) \arrow{r}{\text{Kramers--Kronig in $\theta$}}& \operatorname{Re}\mathcal F(\theta)
\end{tikzcd}
\]

\begin{figure}
  \includegraphics{figs/contour_path.pdf}
  \caption{
    Integration contour over the global scaling function $\mathcal F$ in the
    complex $\theta$ plane used to produce the dispersion relation. The
    circular arc is taken to infinity, while the circles around the
    singularities are taken to zero.
  } \label{fig:contour}
\end{figure}

As $\theta\to\infty$, $\mathcal F(\theta)\sim\theta^{2/\beta\delta}$. In order that the contribution from the arc of the contour vanish, we must have the integrand vanish sufficiently fast at infinity. Since $2/\beta\delta<2$ in all dimensions, we will simply use 2.
\begin{equation}
  0=\oint_{\mathcal C}d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)}
\end{equation}
where $\mathcal C$ is the contour in Figure \ref{fig:contour}. The only
nonvanishing contributions from this contour are along the real line and along
the branch cut in the upper half plane. For the latter contributions, the real
parts of the integration up and down cancel out, while the imaginary part
doubles. This gives
\begin{equation}
  \begin{aligned}
    0&=\left[\int_{-\infty}^\infty+\lim_{\epsilon\to0}\left(\int_{i\infty-\epsilon}^{i\theta_{\mathrm{YL}}-\epsilon}+\int^{i\infty+\epsilon}_{i\theta_{\mathrm{YL}}+\epsilon}\right)\right]
      d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\
     &=\int_{-\infty}^\infty d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)}
     +2i\int_{i\theta_{\mathrm{YL}}}^{i\infty}d\theta'\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\
     &=-i\pi\frac{\mathcal F(\theta)}{\theta^2}+\mathcal P\int_{-\infty}^\infty d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)}
     +2i\int_{i\theta_{\mathrm{YL}}}^{i\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)}
  \end{aligned}
\end{equation}
In principle one would need to account for the residue of the pole at zero, but since its order is less than two and $\mathcal F(0)=\mathcal F'(0)=0$, this evaluates to zero.
\begin{equation}
  \mathcal F(\theta)
  =\frac{\theta^2}{i\pi}\mathcal P\int_{-\infty}^\infty d\vartheta\,\frac{\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)}
  +\frac{2\theta^2}\pi\int_{i\theta_{\mathrm{YL}}}^{i\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(\theta')}{\vartheta^2(\vartheta-\theta)}
\end{equation}
\begin{equation}
  \operatorname{Re}\mathcal F(\theta)
  =\frac{\theta^2}{\pi}\mathcal P\int_{-\infty}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)}
  -\frac{2\theta^2}\pi\int_{\theta_{\mathrm{YL}}}^{\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(i\vartheta)}{\vartheta(\vartheta^2+\theta^2)}
\end{equation}
Because the real part of $\mathcal F$ is even, the imaginary part must be odd. Therefore
\begin{equation}
  \operatorname{Re}\mathcal F(\theta)
  =\frac{\theta^2}{\pi}
  \int_{\theta_c}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2}\left(\frac1{\vartheta-\theta}+\frac1{\vartheta+\theta}\right)
  -\frac{2\theta^2}\pi\int_{\theta_{\mathrm{YL}}}^{\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(i\vartheta)}{\vartheta(\vartheta^2+\theta^2)}
\end{equation}

Now we must make our assertion of the form of the imaginary part of $\operatorname{Im}\mathcal F(\theta)$. Since both of the limits we are interested in---\eqref{eq:langer.sing} along the real axis and \eqref{eq:yang.lee.sing} along the imaginary axis---have symmetries which make their imaginary contribution vanish in the domain of the other limit, we do not need to construct a sophisticated combination to have the correct asymptotics: a simple sum will do!



\section{The 2D Ising model}

\subsection{Definition of functions}

The scaling function for the two-dimensional Ising model is the most
exhaustively studied universal forms in statistical physics and quantum field
theory.
\begin{equation} \label{eq:free.energy.2d.low}
  f_s(u_t, u_h)
  = |u_t|^2\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta})
    +\frac{u_t^2}{8\pi}\log u_t^2
\end{equation}
where the functions $\mathcal F_\pm$ have expansions in nonnegative integer powers of their arguments.
\begin{equation} \label{eq:free.energy.2d.mid}
  f_s(u_t, u_h)
  = |u_h|^{2/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta})
    +\frac{u_t^2}{8\pi}\log u_h^{2/\beta\delta}
\end{equation}
where the function $\mathcal F_0$ has a convergent expansion in nonnegative integer powers of its argument.
To connect with Mangazeev and Fonseca, $\mathcal F_0(x)=\tilde\Phi(-x)=\Phi(-x)+(x^2/8\pi) \log x^2$ and $\mathcal F_\pm(x)=G_{\mathrm{high}/\mathrm{low}}(x)$.

Schofield coordinates allow us to define a global scaling function $\mathcal F$ by
\begin{equation} \label{eq:schofield.2d.free.energy}
  f_s(R, \theta) = R^2\mathcal F(\theta) + t(\theta)^2\frac{R^2}{8\pi}\log R^2
\end{equation}
The scaling function $\mathcal F$ can be defined in terms of the more
conventional ones above by substituting \eqref{eq:schofield} into \eqref{eq:free.energy.2d.low} and
\eqref{eq:free.energy.2d.mid}, yielding
\begin{equation} \label{eq:scaling.function.equivalences.2d}
  \begin{aligned}
    &\mathcal F(\theta)
    =t(\theta)^2\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right]
      +\frac{t(\theta)^2}{8\pi}\log t(\theta)^2 \\
    &=|h(\theta)|^{2/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right]
      +\frac{t(\theta)^2}{8\pi}\log h(\theta)^{2/\beta\delta}
  \end{aligned}
\end{equation}
Examination of \eqref{eq:scaling.function.equivalences.2d} finds that $\mathcal F$ has expansions in integer powers in the entire domain $-\theta_c\leq0\leq\theta_c$.


\begin{equation} \label{eq:im.f.func.2d}
  f(x)=\Theta(-x) |x| e^{-1/|x|}
\end{equation}
\begin{equation}
  \operatorname{Im}\mathcal F(\theta)=A\left\{f\left[\tilde B(\theta_c-\theta)\right]+f\left[b(\theta_c+\theta)\right]\right\}
\end{equation}

\begin{equation}
  \begin{aligned}
    \operatorname{Re}\mathcal F(\theta)
      &=G(\theta^2)-\frac{\theta^2}\pi\int d\vartheta\, \frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\
      &=G(\theta^2)+\frac A\pi\left\{f[\tilde B(\theta_c-\theta)]+f[\tilde B(\theta_c+\theta)]\right\}
  \end{aligned}
\end{equation}
for arbitrary analytic function $G$ with
\begin{equation}
  G(x)=\sum_{i=0}^\infty G_ix^i
\end{equation}
and $f$ is
\begin{equation}
  f(x)=xe^{1/x}\operatorname{Ei}(-1/x)
\end{equation}
the Kramers--Kronig transformation of \eqref{eq:im.f.func.2d}, where $\operatorname{Ei}$ is the exponential integral.

\subsection{Iterative fitting}

\subsection{Comparison with other smooth forms}

\section{The three-dimensional Ising model}

\cite{Butera_2011_Free}

The three-dimensional Ising model is easier in some ways, since its hyperbolic critical point lacks stray logarithms.

\begin{equation} \label{eq:free.energy.3d.low}
  f_s(u_t, u_h)
  = |u_t|^{D\nu}\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta})
\end{equation}
\begin{equation} \label{eq:free.energy.3d.mid}
  f_s(u_t, u_h)
  = |u_h|^{D\nu/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta})
\end{equation}
\begin{equation} \label{eq:schofield.3d.free.energy}
  f_s(R, \theta) = R^2\mathcal F(\theta)
\end{equation}
\begin{equation} \label{eq:scaling.function.equivalences.3d}
  \begin{aligned}
    \mathcal F(\theta)
    &=t(\theta)^{2-\alpha}\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] \\
    &=|h(\theta)|^{(2-\alpha)/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right]
  \end{aligned}
\end{equation}
\begin{equation} \label{eq:im.f.func.3d}
  f(x)=\Theta(-x) |x|^{-7/3} e^{-1/|x|^2}
\end{equation}
\begin{equation}
  f(x)=\frac{e^{-1/x^2}}{12}\left[
    \frac4x\Gamma\big(\tfrac23\big)\operatorname{E}_{\frac53}(-x^{-2})
    -\frac1{x^2}\Gamma\big(\tfrac16\big)\operatorname{E}_{\frac76}(-x^{-2})
  \right]
\end{equation}

\section{Outlook}

The successful smooth description of the Ising free energy produced in part by analytically continuing the singular imaginary part of the metastable free energy inspires an extension of this work: a smooth function that captures the universal scaling \emph{through the coexistence line and into the metastable phase}. Indeed, the tools exist to produce this: by writing $t(\theta)=(1-\theta^2)(1-(\theta/\theta_m)^2)$ for some $\theta_m>\theta_c$, the invariant scaling combination

\begin{acknowledgments}
  The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien
  for helpful conversations. The authors would also like to think Jacques Perk
  for pointing us to several insightful studies. JPS thanks Jim Langer for past
  inspiration, guidance, and encouragement. This work was supported by NSF
  grants DMR-1312160 and DMR-1719490.
\end{acknowledgments}

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