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Diffstat (limited to 'essential-ising.tex')
| -rw-r--r-- | essential-ising.tex | 79 |
1 files changed, 52 insertions, 27 deletions
diff --git a/essential-ising.tex b/essential-ising.tex index b1c17e6..312becf 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -3,12 +3,12 @@ % Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. % Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. % -\documentclass[aps,prl,reprint]{revtex4-1} +\documentclass[aps,prl,preprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} -\mathtoolsset{showonlyrefs=true} +%\mathtoolsset{showonlyrefs=true} \def\[{\begin{equation}} \def\]{\end{equation}} @@ -124,12 +124,26 @@ F_\c\sim\Sigma^d(M|H|)^{-(d-1)}$. Assuming the singular scaling forms $\Sigma=|g_t|^\mu\mathcal S(g_h|g_t|^{-\Delta})$ and $M=|g_t|^\beta\mathcal M(g_h|g_t|^{-\Delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form -\begin{align} - \Delta F_c& +\def\eqcritformone{ \sim\mathcal S^d(g_h|g_t|^{-\Delta})(-g_h|g_t|^{-\Delta}\mathcal - M(g_h|g_t|^{-\Delta}))^{-(d-1)}\notag\\ - &\sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}). -\end{align} + M(g_h|g_t|^{-\Delta}))^{-(d-1)} +} +\def\eqcritformtwo{ + \sim\mathcal G^{-(d-1)}(g_h|g_t|^{-\Delta}). +} +\ifreprint +\[ + \begin{aligned} + \Delta F_c&\eqcritformone + \\ + &\eqcritformtwo + \end{aligned} +\] +\else +\[ + \Delta F_c\eqcritformone\eqcritformtwo +\] +\fi Since both surface tension and magnetization are finite and nonzero for $H=0$ at $T<T_c$, $\mathcal G(X)=\O(X)$ for small $X$. The decay rate of the metastable state will be roughly given by the Boltzmann factor for the @@ -168,33 +182,43 @@ energy in $H$ in good agreement with transfer matrix expansions \cite{lowe.1980.instantons}. Here, we compute the integral to come to explicit functional forms. In \textsc{3d} and \textsc{4d} this can be computed explicitly given our scaling ansatz, yielding -\ifreprint -\begin{align} - \mathcal F^{\text{\textsc{3d}}}(X)&= - \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} - \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2})\\ - &\hspace{10em}-4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] - \notag -\\ - \mathcal F^{\text{\textsc{4d}}}(X)&= - \frac{A}{9\pi X^2}e^{1/(BX)^3} - \Big[3\Gamma(0,(BX)^{-3})\\ - &\hspace{2em}-3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) - -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] - \notag -\end{align} -\else -\begin{align} +\def\eqthreedeeone{ \mathcal F^{\text{\textsc{3d}}}(X)&= \frac{AB^{1/3}}{12\pi X^2}e^{-1/(BX)^2} \bigg[\Gamma(\tfrac16)E_{7/6}((BX)^{-2}) +} +\def\eqthreedeetwo{ -4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] -\\ +} +\def\eqfourdeeone{ \mathcal F^{\text{\textsc{4d}}}(X)&= \frac{A}{9\pi X^2}e^{1/(BX)^3} \Big[3\Gamma(0,(BX)^{-3}) +} +\def\eqfourdeetwo{ -3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] +} +\ifreprint +\begin{align} + &\begin{aligned} + \eqthreedeeone\\ + &\hspace{6em} + \eqthreedeetwo + \end{aligned} + \\ + &\begin{aligned} + \eqfourdeeone + \\ + &\hspace{2em} + \eqfourdeetwo + \end{aligned} +\end{align} +\else +\begin{align} + \eqthreedeeone\eqthreedeetwo + \\ + \eqfourdeeone\eqfourdeetwo \end{align} \fi for \textsc{4d}. @@ -275,8 +299,9 @@ better express the equation of state of the Ising model in the whole of its parameter space. \begin{acknowledgments} - The authors would like to thank Tom Lubensky for a reason that Jim should - really flesh out. + The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien + for helpful conversations. This work was partially supported by NSF grant + DMR-1312160. \end{acknowledgments} \bibliography{essential-ising} |