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| -rw-r--r-- | ising_scaling.tex | 20 |
1 files changed, 10 insertions, 10 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index c25dfe2..fb01780 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -373,7 +373,7 @@ entirely fixed, and it will be truncated at finite order. \caption{ Example of the parametric coordinates. Lines are of constant $R$ from $-\theta_0$ to $\theta_0$ for $g(\theta)$ taken from the $n=6$ entry of - Table \ref{tab:fits}. {\color{blue} \bf XXX Can we have lines of constant $\theta$ as well? Maybe dashed? Also maybe smaller radii, $R=1/4$, 1/2, and 1? Legend could be + Table \ref{tab:fits}. {\color{blue} \bf XXX Can we have lines of constant $\theta$ as well? Maybe dashed? Also maybe smaller radii, $R=1/4$, 1/2, and 1?} } \label{fig:schofield} \end{figure} @@ -421,8 +421,8 @@ present in $\mathcal F(\theta)$. The analytic structure of this function is shown in Fig.~\ref{fig:schofield.singularities}. Two copies of the Langer branch cut stretch out from $\pm\theta_0$, where the equilibrium phase ends, and the Yang--Lee edge singularities are present on the imaginary-$\theta$ -line, where they must be since $\mathcal F$ has the same symmetry in $\theta$ -as $\mathcal F_+$ has in $\xi$. +line (because $\mathcal F$ has the same symmetry in $\theta$ +as $\mathcal F_+$ has in $\xi$). The location of the Yang--Lee edge singularities can be calculated directly from the coordinate transformation \eqref{eq:schofield}. Since $g(\theta)$ is @@ -451,13 +451,13 @@ part of the free energy. Our strategy follows. First, we take the singular imaginary parts of the scaling functions $\mathcal F_{\pm}(\xi)$ and truncate them to the lowest order -accessible under polynomial coordinate changes of $\xi$. Then, we assert that -the imaginary part of $\mathcal F(\theta)$ must have this simplest form, -implicitly defining the parametric coordinate change. Third, we perform a +accessible under polynomial coordinate changes of $\xi$. Then, we constrain +the imaginary part of $\mathcal F(\theta)$ to have this simplest form, +implicitly defining the analytic parametric coordinate change $g(\theta)$. Third, we perform a Kramers--Kronig type transformation to establish an explicit form for the real -part of $\mathcal F(\theta)$. Finally, we make good on the assertion made in -the second step by finding the coordinate transformation that produces the -correct series coefficients of $\mathcal F_{\pm}$. +part of $\mathcal F(\theta)$ involving a second analytic function $G(\theta)$. Finally, we make good on the constraint made in +the second step by fitting the coefficients of $g(\theta)$ and $G(\theta)$ to reproduce the +correct known series coefficients of $\mathcal F_{\pm}$. This success of this stems from the commutative diagram below. So long as the application of Schofield coordinates and the Kramers--Kronig relation can be @@ -589,7 +589,7 @@ The scaling function has a number of free parameters: the position $\theta_0$ of the abrupt transition, prefactors in front of singular functions from the abrupt transition and the Yang--Lee point, the coefficients in the analytic part $G$ of the scaling function, and the coefficients in the undetermined -coordinate function $g$. Other parameters are determined by known properties. +coordinate function $g$. The other parameters $B$, $C_0$, $\theta_{YL$ are determined by known properties. For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be expanded around $\theta=\theta_0$ to yield |