path: root/essential-ising.tex

%  Ising model abrupt transition.
%
%  Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017.
%  Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved.
%
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\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,xifthen,mfpic}

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\title{Essential Singularity in the Ising Abrupt Transition}
\author{Jaron Kent-Dobias}

\date{April 20, 2017}

\begin{document}

\def\[{\begin{equation}}
\def\]{\end{equation}}

\def\im{\mathop{\mathrm{Im}}\nolimits}
\def\dd{\mathrm d}
\def\O{\mathcal O}
\def\ei{\mathop{\mathrm{Ei}}\nolimits}
\def\b{\mathrm b}

\newcommand\pd[3][]{
  \ifthenelse{\isempty{#1}}
    {\def\tmp{}}
    {\def\tmp{^#1}}
  \frac{\partial\tmp#2}{\partial#3\tmp}
}

\maketitle

It's long been known that the decay rate $\Gamma$ of metastable states in
statistical mechanics is often related to the metastable free energy $F$ by
\cite{langer.1967.condensation,langer.1969.metastable,gaveau.1989.analytic}
\[
  \Gamma\propto\im F
\]
What exactly is meant by `metastable free energy' is important to establish,
since formally the free energy relies on the existence of an equilibrium
state. Here one can imagine either analytic continuation of the free energy
through an abrupt phase transition, or restriction of the partition function
trace to states in the vicinity of the local free energy minimum that
characterizes the metastable state. In any case, the free energy develops a
nonzero imaginary part in the metastable region. Heuristically, this can be
thought of as similar to what happens in quantum mechanics with a non-unitary
Hamiltonian: the imaginary part describes loss of probability in the system
that corresponds to decay. 

One can estimate the scaling of the decay rate of the {\sc 2d} Ising model
using ideas from nucleation theory. In this framework, the metastable state
decays when a sufficiently large domain in the stable state forms to grow
stably to fill out the whole system. The free energy of a domain of $N$ spins
causes a free energy change
\[
  \Delta F=\Sigma N^\sigma-MHN
\]
where $\Sigma$ is the surface tension and $1-\frac1d\leq\sigma<1$. This is
maximized by
\[
  N_c=\bigg(\frac{MH}{\sigma\Sigma}\bigg)^{-1/(\sigma-1)}
\]
which corresponds to a free energy change
\[
  \Delta F_c\sim\bigg(\frac\Sigma{(MH)^\sigma}\bigg)^{1/(1-\sigma)}
\]
The rate of formation is proportional to the Boltzmann factor,
\[
  \Gamma\sim e^{-\beta \Delta
  F_c}=e^{-\beta(\Sigma/(MH)^\sigma)^{1/(1-\sigma)}}
\]
For domains whose boundary is minimal, $\sigma=1-\frac1d$ and this becomes
\[
  \Gamma\sim e^{-\beta(\Sigma/(MH)^\sigma)^{d-1}}
\]
There are known scaling forms for the surface tension and magnetization, $\Sigma\sim t^\mu\mathcal S(ht^{-\beta\delta})$ with $\mu=-\nu+\gamma+2\beta$
\cite{widom.1981.interface} and $M\sim t^\beta\mathcal M(ht^{-\beta\delta})$.
Since both the surface tension and magnetization have nonzero finite values at
the first-order transition $h=0$, $\mathcal S(0)=\O(1)$ and $\mathcal
M(0)=\O(1)$. It follows that
\[
  \Gamma\sim e^{-1/\mathcal G(ht^{-\beta\delta})^{d-1}}
\]
with $\mathcal G(X)=\O(X)$. This establishes the form of $\im F$
besides the prefactor. Results from field theory predict that, for small $h$,
\[
  \im F\simeq
  \begin{cases}
    \big(\frac
    h{t^\Delta}\big)^{-(d-3)d/2}(g^*)^{-d(d-1)/4}\exp\big[-B\big(\frac
    h{|t|^\Delta}\big)^{-(d-1)}(g^*)^{-(d+1)/2}\big] & d=2,4\\
    \big(\frac
    h{t^\Delta}\big)^{-7/3}(g^*)^{-8/3}\exp\big[-B\big(\frac
    h{t^\Delta}\big)^{-2}(g^*)^{-2}\big]
    & d=3
  \end{cases}
\]
with $\Delta=3-\frac\epsilon2$, $g^*=2\pi^2\frac\epsilon{n+8}$
\cite{houghton.1980.metastable,gunther.1980.goldstone}. This is consistent
with our form above. We therefore predict that
\[
  \im F=t^{2-\alpha}\mathcal F(ht^{-\beta\delta})^{-(d-3)d/2}e^{-1/\mathcal
    G(ht^{-\beta\delta})^{d-1}}
\]
In {\sc 2d} we have
\[
  \im F=t^2\mathcal F(ht^{-\Delta})e^{-1/\mathcal G(ht^{-\Delta})}
\]
with $\Delta=\beta\delta=\frac{15}8$. In terms of $X=ht^{-\Delta}$, this is
\[
  \im F=t^2\mathcal F(X)e^{-1/\mathcal G(X)}\simeq At^2|X|e^{-1/B|X|}
\]

\begin{align}
  F(X)
  &=\frac1\pi\int_{-\infty}^\infty\frac{\im F(X')}{X'-X}\,\dd X'
  =\frac{At^2}\pi\int_{-\infty}^0\frac{|X'|e^{-1/B|X'|}}{X'-X}\,\dd X'\\
  &=-\frac{At^2}\pi\int_0^\infty\frac{X'e^{-1/BX'}}{X'+X}\,\dd X'
\end{align}
since $\im F=0$ for $X>0$. $\pd{}h=\pd Xh\pd{}X=t^{-\Delta}\pd{}X$.
Unfortunately this integral doesn't converge, and it seems we cannot evaluate
this result at the level of truncation we've chosen. However, 

\[
  F(H)=At^{2-\alpha}\sum_{n=0}^\infty f_nX^n
\]
\[
  f_n=\frac1\pi\int_{-\infty}^0\frac{\im F(X)}{X^{n+1}}\,\dd X
  =\frac{(-1)^{n+1}}\pi\int_0^{\infty}\frac{Xe^{-1/BX}}{X^{n+1}}\,\dd X
  =\frac1\pi(-1)^{n+1}B^{n-1}\Gamma(n-1)
\]
for $n>1$. 

\begin{align}
  \chi
  &=\pd[2]Fh
  =t^{-2\Delta}\pd[2]FX
  =-\frac{2}\pi At^{2-2\Delta}\int_0^\infty\frac{X'e^{-1/BX'}}{(X+X')^3}\,\dd
  X'\\
  &=\frac2\pi
  \frac{ABt^{-\gamma}}{(BX)^3}\big[BX(1-BX)+e^{1/BX}\ei(-1/BX)\big]
\end{align}

\[
  \lim_{X\to0}\chi=-\frac4\pi ABt^{-\gamma}
\]

\[
  \beta^{-1}\chi=C_{0\pm}|t|^{-7/4}+C_{1\pm}|t|^{-3/4}+\O(1)
\]
$C_{0-}=0.025\,536\,971\,9$ $C_{1-}=-0.001\,989\,410\,7$
\cite{barouch.1973.susceptibility}

CORRECTIONS TO SCALING, $u_t$ and $u_h$ instead of $t$ and $h$.

\begin{align}
  u_h
  &=h[1+c_ht+dht^2+e_hh^2+f_ht^3+\O(t^4,th^2)]\\
  u_t
  &=t+b_th^2+c_t^2+d_t^3+e_tth^2+f_tt^4+\O(t^5,t^2h^2,h^4)
\end{align}
\begin{align}
  c_h=\frac{\beta_c}{\sqrt2}
  &&
  d_h=\frac{23\beta_c^2}{16}
  &&
  f_h=\frac{191\beta_c^3}{48\sqrt2}\\
  c_t=\frac{\beta_c}{\sqrt2}
  &&
  d_t=\frac{7\beta_c^2}6
  &&
  f_t=\frac{17\beta_c^3}{6\sqrt2}\\
  e_t=b_t\beta_c\sqrt2
  &&
  b_t=-\frac{E_0\pi}{16\beta_c^2}
\end{align}
$E_0=0.040\,325\,5003$ $e_h=-0.007\,27(15)$
\[
  F(t,h)-F(t,0)=\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2n}(t)h^{2n}
\]
\[
  \chi(t,h)=\pd[2]Fh=\chi_2(t)+\sum_{n=1}^\infty\frac1{(2n)!}\chi_{2(n+1)}h^{2n}
\]

\begin{align}
  \chi
  &=\pd[2]Fh
  =\pd[2]{F_\b}h
  +\frac d{y_t}\bigg(\frac d{y_t}-1\bigg)|u_t|^{d/y_t-2}\bigg(\pd{u_t}h\bigg)^2
\end{align}

\input{figs/scaling_func.tex}

\bibliographystyle{plain}
\bibliography{essential-ising}

\end{document}