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diff --git a/essential-ising.tex b/essential-ising.tex index ba8e452..d641a57 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -119,8 +119,10 @@ methods of critical droplet theory for the decay of an Ising system in a metastable state, i.e., an equilibrium Ising state for $T<T_\c$, $H>0$ subjected to a small negative external field $H<0$. The existence of an essential singularity has also been suggested by transfer matrix -\cite{mccraw.1978.metastability,enting.1980.investigation} and \textsc{rg} -methods \cite{klein.1976.essential}. It has long been known that the decay +\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg} +methods \cite{klein.1976.essential}, and a different kind of essential +singularity is known to exist in the zero-temperature susceptibility +\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}. It has long been known that the decay rate $\Gamma$ of metastable states in statistical mechanics is often related to the metastable free energy $F$ by $\Gamma\propto\im F$ \cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. @@ -264,10 +266,10 @@ At the level of truncation of \eqref{eq:im.scaling} at which we are working the Kramers--Kronig relation does not converge in \twodee. However, higher moments can still be extracted, e.g., the susceptibility, by taking \[ - \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh - =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. + \chi=\pd MH=-\frac1{T}\pd[2]Fh + =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] -With a scaling form defined by $T_\c\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, +With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] @@ -325,7 +327,7 @@ temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$T_\c\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant +$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant $A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The resulting scaling functions $\fX$ and $\fM$ are plotted as solid lines in Fig.~\ref{fig:scaling_fits}. As can be seen, there is very good agreement @@ -405,7 +407,7 @@ energy~\cite{caselle.2001.critical}, and should have implications for the scalin of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized to predict similar singularities in systems where nucleation and metastability are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and +transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and freezing transitions in glasses, spin glasses, and other disordered systems. @@ -424,7 +426,8 @@ freezing transitions in glasses, spin glasses, and other disordered systems. \begin{acknowledgments} The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. JPS thanks Jim Langer for past inspiration, + 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} |