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diff --git a/essential-ising.tex b/essential-ising.tex index 4bab30f..b1c17e6 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -29,6 +29,11 @@ \frac{\partial\tmp#2}{\partial#3\tmp} } +\makeatletter +\newif\ifreprint +\@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse} +\makeatother + \begin{document} \title{Essential Singularities in the Ising Universal Scaling Functions} @@ -163,6 +168,7 @@ 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} @@ -177,6 +183,20 @@ explicitly given our scaling ansatz, yielding -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] \notag \end{align} +\else +\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}) + -4BX\Gamma(\tfrac23)E_{5/3}((BX)^{-2})\bigg] +\\ + \mathcal F^{\text{\textsc{4d}}}(X)&= + \frac{A}{9\pi X^2}e^{1/(BX)^3} + \Big[3\Gamma(0,(BX)^{-3}) + -3\Gamma(\tfrac23)\Gamma(\tfrac13,(BX)^{-3}) + -\Gamma(\tfrac13)\Gamma(-\tfrac13,(BX)^{-3})\Big] +\end{align} +\fi for \textsc{4d}. At the level of truncation we are working at, the Kramers--Kronig relation does not converge in \textsc{2d}. However, the higher moments can still be |