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diff --git a/ising_scaling.tex b/ising_scaling.tex index 5066ea5..e1b8589 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -23,7 +23,7 @@ \begin{document} -\title{Smooth Ising universal scaling functions} +\title{Smooth and global Ising universal scaling functions} \author{Jaron Kent-Dobias} \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} @@ -110,29 +110,30 @@ bring the flow equations into an agreed upon simplest normal form && \frac{du_f}{d\ell}=Du_f+g(u_t), \end{align} -which are exact as written. The flow of the parameters is made exactly linear, +which are exact as written \cite{Raju_2019_Normal}. The flow of the parameters is made exactly linear, while that of the free energy is linearized as nearly as possible. Solving these equations for $u_f$ yields \begin{equation} \begin{aligned} - u_f(u_t, u_h) - &=|u_t|^{D\nu}\mathcal F_\pm(u_h|u_t|^{-\beta\delta})+|u_t|^{D\nu}\int_1^{u_t}dx\,\frac{g(x)}{x^{1+D\nu}} \\ - &=|u_h|^{D\nu/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta})+|u_t|^{D\nu}\int_1^{u_t}dx\,\frac{g(x)}{x^{1+D\nu}} \\ + &u_f(u_t, u_h) + =|u_t|^{D\nu}\mathcal F_\pm(u_h|u_t|^{-\beta\delta})+|u_t|^{D\nu}\int_1^{u_t}dx\,\frac{g(x)}{x^{1+D\nu}} \\ + &=|u_h|^{D\nu/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta})+|u_t|^{D\nu}\int_1^{u_h^{1/\beta\delta}}dx\,\frac{g(x)}{x^{1+D\nu}} \\ \end{aligned} \end{equation} -where $\mathcal F_\pm$ and $\mathcal F_0$ are undetermined scaling functions. The scaling functions are universal in the sense that if -another system whose critical point belongs to the same universality class has -its parameters brought to the form \eqref{eq:flow}, one will see the same -functional form (up to constant rescaling of $u_t$ and $u_h$ and choice of -$g$). +where $\mathcal F_\pm$ and $\mathcal F_0$ are undetermined scaling functions. +The scaling functions are universal in the sense that if another system whose +critical point belongs to the same universality class has its parameters +brought to the form \eqref{eq:flow}, one will see the same functional form (up +to constant rescaling of $u_t$ and $u_h$ and choice of $g$). The analyticity of the free energy at finite size implies that the functions -$\mathcal F_\pm$ have power-law expansions of their arguments about zero. This -is not the case at infinity, and in fact $\mathcal -F_0(\eta)=\eta^{2/\beta\delta}\mathcal F_\pm(\eta^{-1/\beta\delta})$ itself has a power-law -expansion about zero, implying that $\mathcal F_\pm(\xi)\sim \xi^{2\beta\delta}$ for large $x$. +$\mathcal F_\pm$ and $\mathcal F_0$ have power-law expansions of their +arguments about zero. This is not the case at infinity: since $\mathcal +F_0(\eta)=\eta^{D\nu}\mathcal F_\pm(\eta^{-1/\beta\delta})$ has +a power-law expansion about zero, $\mathcal F_\pm(\xi)\sim +\xi^{D\nu/\beta\delta}$ for large $\xi$. The free energy flow equation of the 3D Ising model can be completely linearised, giving $g(x)=0$. This is not the case for the 2D Ising model, where a term proportional to $u_t^2$ cannot be removed by a smooth change of coordinates. The scale of this term sets the relative size of $u_f$ and $u_t$. -For the constant scale of $u_t$ and $u_h$, we adopt the same convention as used by +For the scale of $u_t$ and $u_h$, we adopt the same convention as used by \cite{Fonseca_2003_Ising}. This gives $g(u_t)=-\frac1{4\pi}u_t^2$. The dependence of the nonlinear scaling variables on the parameters $t$ and $h$ is system-dependent, and their form can be found for common model systems (the square- and triangular-lattice Ising @@ -228,6 +229,12 @@ $\xi_{\mathrm{YL}}$. The Yang--Lee singularities are critical points in their own right, with their own universality class different from that of the Ising model \cite{Fisher_1978_Yang-Lee}. +\begin{equation} + \mathcal F_+(\xi) + =A(\xi) +B(\xi)[1+(\xi/\xi_{\mathrm{YL}})^2]^{1+\sigma}+C(\xi)+\cdots +\end{equation} +for edge exponent $\sigma$. + \cite{Cardy_1985_Conformal} \cite{Connelly_2020_Universal} \cite{An_2016_Functional} @@ -286,9 +293,23 @@ both will have polynomial expansions in $\theta$ at all three places above. Therefore, in Schofield coordinates one expects to be able to define a global scaling function $\mathcal F(\theta)$ which has a polynomial expansion in its -argument for all real $\theta$. For small $\theta$ $\mathcal F(\theta)$ will +argument for all real $\theta$ by +\begin{equation} + u_f(R,\theta)=R^{D\nu}\mathcal F(\theta)+|Rt(\theta)|^{D\nu}\int_1^Rdx\,\frac{g(x)}{x^{1+D\nu}} +\end{equation} +For small $\theta$ $\mathcal F(\theta)$ will resemble $\mathcal F_+$, for $\theta$ near one it will resemble $\mathcal F_0$, -and for $\theta$ near $\theta_c$ it will resemble $\mathcal F_-$. This leads us +and for $\theta$ near $\theta_c$ it will resemble $\mathcal F_-$. This can be seen explicitly using the definitions \eqref{eq:schofield} to relate the above form to the original scaling functions, giving +\begin{equation} \label{eq:scaling.function.equivalences.2d} + \begin{aligned} + &\mathcal F(\theta) + =|t(\theta)|^{D\nu}\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] + +|t(\theta)|^{D\nu}\int_1^{t(\theta)} dx\,\frac{g(x)}{x^{1+D\nu}}\\ + &=|h(\theta)|^{D\nu/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] + +|t(\theta)|^{D\nu}\int_1^{h(\theta)^{1/\beta\delta}} dx\,\frac{g(x)}{x^{1+D\nu}} + \end{aligned} +\end{equation} +This leads us to expect that the singularities present in these functions will likewise be present in $\mathcal F(\theta)$. This is shown in Figure \ref{fig:schofield.singularities}. Two copies of the Langer branch cut stretch @@ -372,15 +393,23 @@ In principle one would need to account for the residue of the pole at zero, but \end{equation} Because the real part of $\mathcal F$ is even, the imaginary part must be odd. Therefore \begin{equation} - \operatorname{Re}\mathcal F(\theta) - =\frac{\theta^2}{\pi} - \int_{\theta_c}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2}\left(\frac1{\vartheta-\theta}+\frac1{\vartheta+\theta}\right) - -\frac{2\theta^2}\pi\int_{\theta_{\mathrm{YL}}}^{\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(i\vartheta)}{\vartheta(\vartheta^2+\theta^2)} + \begin{aligned} + \operatorname{Re}\mathcal F(\theta) + &=\frac{\theta^2}{\pi} + \int_{\theta_c}^\infty d\vartheta\,\frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2}\left(\frac1{\vartheta-\theta}+\frac1{\vartheta+\theta}\right) \\ + &-\frac{2\theta^2}\pi\int_{\theta_{\mathrm{YL}}}^{\infty}d\vartheta\,\frac{\operatorname{Im}\mathcal F(i\vartheta)}{\vartheta(\vartheta^2+\theta^2)} + \end{aligned} \end{equation} Now we must make our assertion of the form of the imaginary part of $\operatorname{Im}\mathcal F(\theta)$. Since both of the limits we are interested in---\eqref{eq:langer.sing} along the real axis and \eqref{eq:yang.lee.sing} along the imaginary axis---have symmetries which make their imaginary contribution vanish in the domain of the other limit, we do not need to construct a sophisticated combination to have the correct asymptotics: a simple sum will do! - +For $\theta\in\mathbb C$, +\begin{equation} + \mathcal F(\theta)=\mathcal F_c(\theta)+\mathcal F_{\mathrm{YL}}(\theta)+\sum_{i=0}^\infty F_{2i}\theta^{2+2i} +\end{equation} +\begin{equation} + \mathcal F_{\mathrm{YL}}(\theta)=F_{\mathrm{YL}}[1+(\theta/\theta_c)^2]^{1+\sigma} +\end{equation} \section{The 2D Ising model} @@ -390,13 +419,13 @@ The scaling function for the two-dimensional Ising model is the most exhaustively studied universal forms in statistical physics and quantum field theory. \begin{equation} \label{eq:free.energy.2d.low} - f_s(u_t, u_h) + u_f(u_t, u_h) = |u_t|^2\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta}) +\frac{u_t^2}{8\pi}\log u_t^2 \end{equation} where the functions $\mathcal F_\pm$ have expansions in nonnegative integer powers of their arguments. \begin{equation} \label{eq:free.energy.2d.mid} - f_s(u_t, u_h) + u_f(u_t, u_h) = |u_h|^{2/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta}) +\frac{u_t^2}{8\pi}\log u_h^{2/\beta\delta} \end{equation} @@ -410,45 +439,46 @@ Schofield coordinates allow us to define a global scaling function $\mathcal F$ The scaling function $\mathcal F$ can be defined in terms of the more conventional ones above by substituting \eqref{eq:schofield} into \eqref{eq:free.energy.2d.low} and \eqref{eq:free.energy.2d.mid}, yielding -\begin{equation} \label{eq:scaling.function.equivalences.2d} - \begin{aligned} - &\mathcal F(\theta) - =t(\theta)^2\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] - +\frac{t(\theta)^2}{8\pi}\log t(\theta)^2 \\ - &=|h(\theta)|^{2/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] - +\frac{t(\theta)^2}{8\pi}\log h(\theta)^{2/\beta\delta} - \end{aligned} -\end{equation} Examination of \eqref{eq:scaling.function.equivalences.2d} finds that $\mathcal F$ has expansions in integer powers in the entire domain $-\theta_c\leq0\leq\theta_c$. - -\begin{equation} \label{eq:im.f.func.2d} - f(x)=\Theta(-x) |x| e^{-1/|x|} +For $\theta\in\mathbb R$, +\begin{equation} + \begin{aligned} + \operatorname{Im}\mathcal F(\theta+0i)&=F_c[\Theta(\theta-\theta_c)\mathcal I(\theta)-\Theta(-\theta-\theta_c)\mathcal I(-\theta)] + \end{aligned} \end{equation} \begin{equation} - \operatorname{Im}\mathcal F(\theta)=A\left\{f\left[\tilde B(\theta_c-\theta)\right]+f\left[b(\theta_c+\theta)\right]\right\} + \mathcal I(\theta)=(\theta-\theta_c)e^{-1/B(\theta-\theta_c)} \end{equation} - +The dispersion integral \eqref{} can be used to find the real part of $\mathcal F_c$ for $\theta\in\mathbb R$, or +\begin{equation} \label{eq:2d.real.Fc} + \operatorname{Re}\mathcal F_c(\theta)=F_c[\mathcal R(\theta)+\mathcal R(-\theta)] +\end{equation} +where $\mathcal R$ is given by the function \begin{equation} \begin{aligned} - \operatorname{Re}\mathcal F(\theta) - &=G(\theta^2)-\frac{\theta^2}\pi\int d\vartheta\, \frac{\operatorname{Im}\mathcal F(\vartheta)}{\vartheta^2(\vartheta-\theta)} \\ - &=G(\theta^2)+\frac A\pi\left\{f[\tilde B(\theta_c-\theta)]+f[\tilde B(\theta_c+\theta)]\right\} + \mathcal R(\theta) + &=\frac1\pi\left[ + \theta_ce^{1/B\theta_c}\operatorname{Ei}(-1/B\theta_c) + \right.\\ + &\left. + +(\theta-\theta_c)e^{-1/B(\theta-\theta_c)}\operatorname{Ei}(1/B(\theta-\theta_c)) + \right] \end{aligned} \end{equation} -for arbitrary analytic function $G$ with -\begin{equation} - G(x)=\sum_{i=0}^\infty G_ix^i -\end{equation} -and $f$ is +When analytically continued to complex $\theta$, \eqref{eq:2d.real.Fc} has branch cuts in the incorrect places. The real and imaginary parts can be combined to yield the function \begin{equation} - f(x)=xe^{1/x}\operatorname{Ei}(-1/x) + \mathcal F_c(\theta)=F_c\left\{ + \mathcal R(\theta)+\mathcal R(-\theta)+i\operatorname{sgn}(\operatorname{Im}\theta)[\mathcal I(\theta)-\mathcal I(-\theta)] + \right\} \end{equation} -the Kramers--Kronig transformation of \eqref{eq:im.f.func.2d}, where $\operatorname{Ei}$ is the exponential integral. +analytic for all $\theta\in\mathbb C$ outside the Langer branch cuts. -\subsection{Iterative fitting} -\subsection{Comparison with other smooth forms} + +\subsection{Fitting} + +\subsection{Comparison} \section{The three-dimensional Ising model} @@ -457,31 +487,25 @@ the Kramers--Kronig transformation of \eqref{eq:im.f.func.2d}, where $\operatorn The three-dimensional Ising model is easier in some ways, since its hyperbolic critical point lacks stray logarithms. \begin{equation} \label{eq:free.energy.3d.low} - f_s(u_t, u_h) + u_f(u_t, u_h) = |u_t|^{D\nu}\mathcal F_{\pm}(u_h|u_t|^{-\beta\delta}) \end{equation} \begin{equation} \label{eq:free.energy.3d.mid} - f_s(u_t, u_h) + u_f(u_t, u_h) = |u_h|^{D\nu/\beta\delta}\mathcal F_0(u_t|u_h|^{-1/\beta\delta}) \end{equation} \begin{equation} \label{eq:schofield.3d.free.energy} - f_s(R, \theta) = R^2\mathcal F(\theta) + u_f(R, \theta) = R^{D\nu}\mathcal F(\theta) \end{equation} \begin{equation} \label{eq:scaling.function.equivalences.3d} \begin{aligned} \mathcal F(\theta) - &=t(\theta)^{2-\alpha}\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] \\ - &=|h(\theta)|^{(2-\alpha)/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] + &=t(\theta)^{D\nu}\mathcal F_\pm\left[h(\theta)|t(\theta)|^{-\beta\delta}\right] \\ + &=|h(\theta)|^{D\nu/\beta\delta}\mathcal F_0\left[t(\theta)|h(\theta)|^{-1/\beta\delta}\right] \end{aligned} \end{equation} -\begin{equation} \label{eq:im.f.func.3d} - f(x)=\Theta(-x) |x|^{-7/3} e^{-1/|x|^2} -\end{equation} \begin{equation} - f(x)=\frac{e^{-1/x^2}}{12}\left[ - \frac4x\Gamma\big(\tfrac23\big)\operatorname{E}_{\frac53}(-x^{-2}) - -\frac1{x^2}\Gamma\big(\tfrac16\big)\operatorname{E}_{\frac76}(-x^{-2}) - \right] + \mathcal F_c(\theta)=F_c(\theta_c^2-\theta^2)^{-7/3}e^{-1/[B(\theta_c^2-\theta^2)]^2} \end{equation} \section{Outlook} |