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| -rw-r--r-- | essential-ising.bib | 11 | ||||
| -rw-r--r-- | essential-ising.tex | 90 | ||||
| -rw-r--r-- | figs/fig-susmag.gplot | 6 | ||||
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diff --git a/essential-ising.bib b/essential-ising.bib index 3e39487..556c4e7 100644 --- a/essential-ising.bib +++ b/essential-ising.bib @@ -406,3 +406,14 @@ publisher={APS} } +@article{zinn.1996.universal, + title={Universal surface-tension and critical-isotherm amplitude ratios in three dimensions}, + author={Zinn, Shun-yong and Fisher, Michael E}, + journal={Physica A: Statistical Mechanics and its Applications}, + volume={226}, + number={1-2}, + pages={168--180}, + year={1996}, + publisher={Elsevier} +} + diff --git a/essential-ising.tex b/essential-ising.tex index 819775b..a64f75b 100644 --- a/essential-ising.tex +++ b/essential-ising.tex @@ -64,10 +64,10 @@ \begin{document} -\title{Essential Singularities in the Ising Universal Scaling Functions} +\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} \author{Jaron Kent-Dobias} \author{James P.~Sethna} -\affiliation{Cornell University} +\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} \date\today @@ -76,7 +76,7 @@ used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine - asymptotic scaling functions for the free energy in the vicinity of the + scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at zero field. Analogous forms for the magnetization and susceptibility in two-dimensions are fit to numeric data and show good @@ -102,13 +102,7 @@ thermodynamic variables. \textsc{Rg} analysis predicts that the singular part of the free energy per site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ in the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\Delta})$ \footnote{Technically -we should write $\fF_{\pm}$ to indicate that the universal scaling -function takes a different form for $t<0$ and $t>0$, but we will restrict -ourselves entirely to $t<0$ and hence $\fF_-$ for the purposes of this -paper.}, where $\Delta=\beta\delta$ and $t$, $h$ are analytic functions of -$t$, $h$ that transform exactly linearly under \textsc{rg} -\cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of +$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase $t<0$ \cite{cardy.1996.scaling,aharony.1983.fields}. When studying the properties of the Ising critical point, it is nearly always assumed that $\fF(X)$, the universal scaling function, is an analytic function of $X$. However, it has long been known that there exists an essential singularity in $\fF$ @@ -140,22 +134,24 @@ the system that corresponds to decay. In critical droplet theory, the metastable state decays when a domain of the equilibrium state forms whose surface-energy cost for growth is outweighed by -bulk-energy gains. There is numerical evidence that, near the critical point, droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy cost of the surface of a droplet -scales with its radius $R$ like $\Sigma S_\dim R^{\dim-1}$ and that of its -bulk scales like $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and +bulk-energy gains. There is numerical evidence that, near the critical point, +droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy +cost of the surface of a droplet of radius $R$ +is $\Sigma S_\dim R^{\dim-1}$ and that of its +bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ is the surface tension of the equilibrium--metastable interface. The critical droplet size then is $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. Assuming the typical singular scaling forms -$\Sigma/T=|t|^\mu\fS(h|t|^{-\Delta})$ and $M=|t|^\beta\mathcal -M(h|t|^{-\Delta})$ and using known hyperscaling relations +$\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal +M(h|t|^{-\beta\delta})$ and using known hyperscaling relations \cite{widom.1981.interface}, this implies a scaling form \def\eqcritformone{ T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)}\frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta}\fM(h|t|^{-\beta\delta}))^{\dim-1}} } \def\eqcritformtwo{ - T\fG^{-(\dim-1)}(h|t|^{-\Delta}) + T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) } \ifreprint \[ @@ -181,7 +177,7 @@ and the critical domain radius $R_c$, with Bh|t|^{-\beta\delta}=\bigg(\frac{\Gamma(1+\dim/2)}{\pi^{\dim/2}\fS(0)(\xi_0^-)^{\dim-1}}\bigg)^{1/(\dim-1)}\frac\xi{R_\c} \] where $\xi=\xi_0^-|t|^{-\nu}$ for $t<T_c$. Since $\fS(0)(\xi_0^-)^{\dim-1}$ is a -universal amplitude ratio, $\frac{Bh|t|^{-\beta\delta}}{\xi/R_c}$ is a +universal amplitude ratio \cite{zinn.1996.universal}, $\frac{Bh|t|^{-\beta\delta}}{\xi/R_c}$ is a universal quantity. % The constant $B$ should be universal near the critical point given careful % definition of the variable $X$. @@ -205,7 +201,7 @@ universal quantity. The decay rate of the metastable state is proportional to the Boltzmann factor for the creation of a critical droplet, yielding \[ - \im F\sim\Gamma\propto e^{-\beta\Delta F_\c}=e^{-\fG(h|t|^{-\beta\delta})^{-(\dim-1)}}. + \im F\sim\Gamma\propto e^{-\Delta F_\c/T}=e^{-\fG(h|t|^{-\beta\delta})^{-(\dim-1)}}. \] For $d>1$ this function has an essential singularity in the invariant combination $h|t|^{-\beta\delta}$. @@ -283,7 +279,7 @@ 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'. \] -With $\chi=|t|^{-\gamma}\fX(h|t|^{-\Delta})$, this yields +With $\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, this yields \[ \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] \label{eq:sus_scaling} @@ -300,12 +296,18 @@ and their constants of integration fixed by known zero field values, yielding \end{align} with $F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})+t^{2-\alpha}\log|t|$ in two dimensions. -Previous work at zero field suggests that -$\fX(0)=\frac{2AB^2}\pi=C_{0-}/T_\c$, with $C_{0-}=0.025\,536\,971\,9$ -\cite{barouch.1973.susceptibility}. -Exact results for the \twodee Ising -model have $\fS(0)=4$ and $\fM(0)=(2^{5/2}\arcsinh1)^\beta$, so -$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\sinh^{-1}1)^{15/8})^{-1}$. +How are these functional forms to be interpreted? They are not asymptotic +forms in any sense, as there is no limit of $t$ or $h$ in which they become +aribitrarily correct. It is well established that this method of using +unphysical or metastable elements of a theory to extract properties of the +stable or equilibrium theory only works for high moments of those predictions +\cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. +These functions should be understood as possessing exactly the correct +singularity at the coexistence line, but requiring polynomial corrections, +especially for smaller integer powers. Using these forms in conjunction with +existing methods of describing the critical equation of state or critical +properties with analytic functions in $h$ will incorporate these low-order +corrections while preserving the correct singular structure. How predictive are these scaling forms in the proximity of the critical point and the abrupt transition line? We simulated the \twodee Ising model on square lattice using a form of the Wolff algorithm modified @@ -313,31 +315,30 @@ to remain efficient in the presence of an external field. Briefly, the external \cite{dimitrovic.1991.finite}. A quickly converging estimate for the magnetization in the finite-size system was then made by taking $M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin. For the \twodee Ising model on a square lattice, exact results at zero temperature have $\fS(0)=4/T_c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ \cite{onsager.1944.crystal}, and $\fX(0)=C_{0-}/T_\c$ with $C_{0-}=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that $B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$ and $A=\frac\pi2\fX(0)/B^2=2^{11/8}\pi^3(\arcsinh1)^{19/4}C_{0-}$. Data was then taken for susceptibility and -magnetization for $T_\c-T,H\leq0.1$. This data is plotted in -Fig.~\ref{fig:scaling_fits}, along with collapses of data onto a single universal curve -in the insets of those figures. As can be seen, there is very good agreement +magnetization for $T_\c-T,H\leq0.1$. The resulting scaling functions $\fX$ and +$\fM$ are plotted using this data 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 are systematic differences that can be seen most clearly in the magnetization. Since our method is known to only be accurate for high moments of the free energy, we should expect that low moments require corrections. -Therefore, we also fit those corrections of the form +Therefore, we also fit those corrections using \begin{align} - \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\\ - \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX) + \fX^{\twodee\prime}(X)&=\fX^\twodee(X)+\sum_{n=1}^Nf_n(BX)\label{eq:sus_scaling_poly}\\ + \fM^{\twodee\prime}(X)&=\fM^\twodee(X)+\frac{T_\c}B\sum_{n=1}^NF_n(BX)\label{eq:mag_scaling_poly} \end{align} where $F_n'(x)=f_n(x)$ and \[ \begin{aligned} f_n(x)&=\frac{C_nx^n}{1+(\lambda x)^{n+1}}\\ - F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}) + F_n(x)&=\frac{C_n\lambda^{-(n+1)}}{n+1}\log(1+(\lambda x)^{n+1}). \end{aligned} \label{eq:poly} \] +The functions $f_n$ have been chosen to be pure integer power laws for small +argument, but vanish appropriately at large argument. We fit these functions to our numeric data for $N=3$. The resulting curves are -also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. - - - +also plotted in Fig.~\ref{fig:scaling_fits} as dashed lines. %\begin{table} % \centering @@ -357,13 +358,16 @@ also plotted in Fig.~\ref{fig:scaling_fits} as a dashed line. \begin{figure} \input{figs/fig-susmag} \caption{ - Comparisons of scaling forms \eqref{eq:sus_scaling} and - \eqref{eq:mag_scaling} to numeric data. Data with - sampling error taken from Monte Carlo simulations of a $4096\times4096$ + Scaling functions for (top) the susceptibility and (bottom) the + magnetization plotted in terms of the invarient combination + $h|t|^{-\beta\delta}$. Points with error bars show data with + sampling error taken from simulations of a $4096\times4096$ square-lattice Ising model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ and - $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid lines show our - analytic results, while the dashed lines have polynomial corrections of - the form \eqref{eq:poly} fit to the data for $N=3$, with $C_1=-0.00368$, + $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. Color denotes the value of + $T$. The solid lines show our + analytic results \eqref{eq:sus_scaling} and \eqref{eq:mag_scaling}, while + the dashed lines show fits of \eqref{eq:sus_scaling_poly} and + \eqref{eq:mag_scaling_poly} to the data for $N=3$, with $C_1=-0.00368$, $C_2=-0.0191$, $C_3=0.0350$, and $\lambda=2.42$. } \label{fig:scaling_fits} @@ -374,7 +378,7 @@ and the analytic nature of the free energy to derive the universal scaling functions for the free energy, and in \twodee the magnetization and susceptibility, in the limit of small $t$ and $h$. Because of an essential singularity in these functions at $h=0$---the abrupt transition line---their -form cannot be modified by analytic redefinition of control or thermodynamic +form cannot be brought into that of regular functions 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 singularity leads to observable effects. we hope that these functional diff --git a/figs/fig-susmag.gplot b/figs/fig-susmag.gplot index dd002a4..39a3fdf 100644 --- a/figs/fig-susmag.gplot +++ b/figs/fig-susmag.gplot @@ -50,7 +50,7 @@ set origin 0,0 set xrange [-20:1200] set yrange [1.18:1.75] set ylabel offset 1,0 '$M|t|^{-\beta}$' -set xlabel '$h|t|^{-\Delta}$' +set xlabel '$h|t|^{-\beta\delta}$' set xtics format '%g' plot num using (X($2, $3)):($6 * t($2)**(-beta)):($7 * t($2)**(-beta)):(hsv2rgb(20 * t($2), 1, 1)) with yerrorbars pt 0 lc rgb variable, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B) with linespoints pt 0 lw 2 lc black, magfunc using (10**$1 / B):(M0 + 10**($2) * Tc * A * B + (polyint(Tc * C1, lamb, 1, 10**$1) + polyint(Tc * C2, lamb, 2, 10**$1) + polyint(Tc * C3, lamb, 3, 10**$1)) / B) smooth csplines with lines dt 2 lw 2 lc black @@ -62,7 +62,7 @@ set origin 0.31,0.5 + 0.29 / 2 set xrange [-3:3] set yrange [-4.5:-1.5] set ylabel offset 2.5,0 '\footnotesize$\chi|t|^\gamma$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize$10^{%g}$' -4,1,-2 @@ -73,7 +73,7 @@ set origin 0.31,0.23 / 2 set xrange [-3:3] set yrange [1.18:1.75] set ylabel offset 4,0 '\footnotesize$M|t|^{-\beta}$' -set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\Delta}$' +set xlabel offset 0,0.5 '\footnotesize$h|t|^{-\beta\delta}$' set xtics format '\footnotesize$10^{%g}$' -2,2,3 set ytics format '\footnotesize {%g}' 1.2,.2,1.6 @@ -4,13 +4,15 @@ FIGS=fig-susmag DATA=data_square-4096 FUNCS=fig-mag_scaling-func fig-sus_scaling-func -all: ${FIGS:%=figs/%.tex} - rubber $(DOC).tex - dvipdf $(DOC).dvi +all: ${DOC}.pdf figs/%.tex: figs/%.gplot ${DATA:%=data/%.dat} ${FUNCS:%=figs/%.dat} gnuplot $< > $@ +${DOC}.pdf: ${DOC}.tex ${DOC}.bib ${FIGS:%=figs/%.tex} + rubber $(DOC).tex + dvipdf $(DOC).dvi + clean: rubber --clean $(DOC) rm -f $(DOC).pdf |