From 5a2365cecda885db2278ba8e9621b96899f72b21 Mon Sep 17 00:00:00 2001 From: "James P. Sethna" Date: Thu, 27 Jul 2017 12:18:50 -0400 Subject: JPS edits to paper and ackn --- essential-ising.tex | 49 ++++++++++++++++++++++++++++++++++--------------- 1 file changed, 34 insertions(+), 15 deletions(-) diff --git a/essential-ising.tex b/essential-ising.tex index 0c42b65..cdd7cd6 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -373,24 +373,43 @@ correction appears to match data quite well. \label{fig:scaling_fits} \end{figure} -We have used results from the properties of the metastable Ising ferromagnet -and the analytic nature of the free energy to derive universal scaling -functions for the free energy, and in \twodee the magnetization and -susceptibility, in the limit of small $t<0$ and $h$. Because of an essential -singularity in these functions at $h=0$---the abrupt transition line---their -form cannot be brought into that of any regular function by analytic -redefinition of control or thermodynamic variables. These predictions match -the results of simulations well. Having demonstrated that the essential -singularity in thermodynamic functions at the abrupt transition leads to -observable scaling effects, we hope that these functional forms will be used in -conjunction with traditional perturbation methods to better express the -equation of state of the Ising model in the whole of its parameter space. +Abrupt phase transitions, such as the jump in magnetization in the Ising +model below $T_c$, are known to imply essential singularities in the free +energy that are usually thought to be unobservable in practice. We have +argued that this essential singularity controls the universal scaling +behavior near continuous phase transitions, and have derived an explicit +analytical form for the singularity in the free energy, magnetization, +and susceptibility for the Ising model. We have developed a Wolff algorithm +for the Ising model in a field, and showed that incorporating our singularity +into the scaling function gives good convergence to the simulations in \twodee. + +Our results should allow improved high-precision functional forms for the free +energy~\cite{CaselleXXX}, and should have implications for the scaling +of correlation functions~\cite{YJXXX,XXX}. 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{ALHN}, the melting of 2D crystals~\cite{XXX}, and +freezing transitions in glasses, spin glasses, and other disordered systems. + + +%We have used results from the properties of the metastable Ising ferromagnet +%and the analytic nature of the free energy to derive universal scaling +%functions for the free energy, and in \twodee the magnetization and +%susceptibility, in the limit of small $t<0$ and $h$. Because of an essential +%singularity in these functions at $h=0$---the abrupt transition line---their +%form cannot be brought into that of any regular function by analytic +%redefinition of control or thermodynamic variables. These predictions match +%the results of simulations well. Having demonstrated that the essential +%singularity in thermodynamic functions at the abrupt transition leads to +%observable scaling effects, we hope that these functional forms will be used in +%conjunction with traditional perturbation methods to 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, Andrea Liu, and Randy Kamien - for helpful conversations. We would also like to thank Jim Langer for his - insightful canonical papers on this subject. This work was partially supported by NSF grant - DMR-1312160. + for helpful conversations. JPS thanks Jim Langer for past inspiration, + guidance, and encouragement. This work was supported by NSF grants + DMR-1312160 and DMR-1719490. \end{acknowledgments} \bibliography{essential-ising} -- cgit v1.2.3-54-g00ecf