diff options
Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 18 |
1 files changed, 10 insertions, 8 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index a1c8205..5881786 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -181,7 +181,7 @@ use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$. The analyticity of the free energy at places away from the critical point implies that the functions $\mathcal F_\pm$ and $\mathcal F_0$ have power-law expansions of their arguments about zero, the result of so-called Griffiths -analyticity. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is +analyticity \cite{Griffiths_1967_Thermodynamic}. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is no phase transition, and the free energy must be an analytic function of its arguments. It follows that $\mathcal F_0$ is analytic about zero. This is not the case at infinity: since @@ -245,7 +245,7 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \begin{figure} - \includegraphics{figs/F_lower_singularities} + \includegraphics{F_lower_singularities} \caption{ Analytic structure of the low-temperature scaling function $\mathcal F_-$ in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The circle @@ -292,7 +292,7 @@ branch cuts beginning at $\pm i\xi_{\mathrm{YL}}$ for a universal constant $\xi_{\mathrm{YL}}$. \begin{figure} - \includegraphics{figs/F_higher_singularities} + \includegraphics{F_higher_singularities} \caption{ Analytic structure of the high-temperature scaling function $\mathcal F_+$ in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The squares @@ -421,7 +421,7 @@ $\theta$. Therefore, The location $\theta_0$ is not fixed by any principle. \begin{figure} - \includegraphics{figs/F_theta_singularities} + \includegraphics{F_theta_singularities} \caption{ Analytic structure of the global scaling function $\mathcal F$ in the complex $\theta$ plane. The circles depict essential singularities of the @@ -478,7 +478,7 @@ Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes its real part up to an analytic even function $G(\theta)$, real for real $\theta$. \begin{figure} - \includegraphics{figs/contour_path} + \includegraphics{contour_path} \caption{ Integration contour over the global scaling function $\mathcal F$ in the complex $\theta$ plane used to produce the dispersion relation. The @@ -835,7 +835,7 @@ values of both are plotted. Free parameters in the fit of the parametric coordinate transformation and scaling form to known values of the scaling function series coefficients for $\mathcal F_\pm$. The fit at stage $n$ matches those coefficients up to - and including order $n$. Error estimates are difficult to quantify directly. + and including order $n$. Uncertainty estimates are difficult to quantify directly. } \label{tab:fits} \end{table} @@ -880,7 +880,7 @@ Fig.~\ref{fig:phi.series}. The series coefficients for the scaling function $\mathcal F_-$ as a function of polynomial order $m$. The numeric values are from Table \ref{tab:data}, and those of Caselle \textit{et al.} are from the most - accurate scaling function listed in \cite{Caselle_2001_The}. The deviation at high polynomial order illustrates the lack of the essential singularity in Caselle's form. + accurate scaling function listed in \cite{Caselle_2001_The}. The deviation at high polynomial order illustrates the lack of the essential singularity in the form of Caselle \textit{et al.}. } \label{fig:glow.series} \end{figure} @@ -951,7 +951,9 @@ the ratio. Sequential ratios of the series coefficients of the scaling function $\mathcal F_-$ as a function of inverse polynomial order $m$. The extrapolated $y$-intercept of this plot gives the radius of convergence of - the series, which should be zero due to the essential singularity (as seen in the known numeric values and in this work). Cassel {\em et al} do not incorporate the essential singularity. + the series, which should be zero due to the essential singularity (as seen + in the known numeric values and in this work). Caselle \textit{et al.} do + not incorporate the essential singularity. } \label{fig:glow.radius} \end{figure} |