From 0d76bc2985132cdc8199fb542ee9ef6178a0c01d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 2 Aug 2017 22:47:40 -0400 Subject: many changes to incorporate new low-order info, small corrections to scaling functions --- essential-ising.tex | 34 ++++++++++++++++++---------------- 1 file changed, 18 insertions(+), 16 deletions(-) (limited to 'essential-ising.tex') diff --git a/essential-ising.tex b/essential-ising.tex index ea8e9eb..c98e938 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -202,7 +202,7 @@ same singular behavior as the real part of the equilibrium free energy, and that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, where \[ - \fiF(X)=-A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} + \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} \label{eq:im.scaling} \] and $\Theta$ is the Heaviside function. Results from combining an analysis of @@ -224,14 +224,14 @@ given our scaling ansatz, yielding \def\eqthreedeeone{ \fF^\threedee(Y/B)&= \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} - \bigg[4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2}) + \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2}) } \def\eqthreedeetwo{ - -\Gamma(\tfrac16)E_{7/6}(-Y^{-2})\bigg] + -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg] } \def\eqfourdeeone{ \fF^\fourdee(Y/B)&= - \frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} + -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} \Big[3\ei(-Y^{-3}) } \def\eqfourdeetwo{ @@ -242,7 +242,7 @@ given our scaling ansatz, yielding \begin{align} &\begin{aligned} \eqthreedeeone\\ - &\hspace{8em} + &\hspace{7em} \eqthreedeetwo \end{aligned} \\ @@ -264,13 +264,13 @@ 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 \[ - \chi=\pd[2]Fh - =\frac2\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. + \chi=\pd MH=-\frac1{T_\c}\pd[2]Fh + =-\frac2{\pi T_\c}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. \] With a scaling form defined by $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] + \fX^\twodee(Y/B)=\frac{AB^2}{\pi T_\c Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} \] Scaling forms for the free energy can then be extracted by direct integration @@ -278,9 +278,9 @@ and their constants of integration fixed by known zero field values, yielding \begin{align} \label{eq:mag_scaling} \fM^\twodee(Y/B) - &=\fM(0)+\frac{ABT_\c}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ + &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ \fF^\twodee(Y/B) - &=\fF(0)+T_\c Y\bigg(\frac{\fM(0)}B+\frac{AT_\c}\pi e^{1/Y}\ei(-1/Y)\bigg) + &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. @@ -325,8 +325,8 @@ $C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of the susceptibility scaling function, then -$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant -$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_0^-$. The +$\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi T_\c$ and the constant +$A$ is fixed to $A=\pi T_\c\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. 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 between our proposed functional forms and what is measured. However, there @@ -349,8 +349,10 @@ where $F_n'(Y)=f_n(Y)$ and The functions $f_n$ have been chosen to be pure integer power laws for small argument, but vanish appropriately at large argument. This is necessary because the susceptibility vanishes with $h|t|^{-\beta\delta}$ while bare -polynomial corrections would not. We fit these functions to our numeric data -for $N=0$ while requiring that $C_0^-/T_\c=\fX'(0)=c_0+2AB^2/\pi$. The +polynomial corrections would not. We fit these functions to known moments of +the free energy's scaling function +\cite{mangazeev.2008.variational,mangazeev.2010.scaling} our numeric data +for $N=0$. The resulting curves are also plotted as dashed lines in Fig.~\ref{fig:scaling_fits}. Our singular scaling function with one low-order correction appears to match data quite well. @@ -385,10 +387,10 @@ 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{caselle.2001.critical}, and should have implications for the scaling -of correlation functions~\cite{YJXXX,XXX}. Our methods might be generalized +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{ALHN}, the melting of 2D crystals~\cite{XXX}, and +transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{XXX}, and freezing transitions in glasses, spin glasses, and other disordered systems. -- cgit v1.2.3-70-g09d2