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Diffstat (limited to 'ising_scaling.tex')
| -rw-r--r-- | ising_scaling.tex | 37 |
1 files changed, 22 insertions, 15 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex index 32085ab..6b89cb2 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -142,16 +142,21 @@ to constant rescaling of $u_h$). The invariant scaling combinations that appear as the arguments to the universal scaling functions will come up often, and we will 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. 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 $\mathcal F_0(\eta)=\eta^{D\nu}\mathcal -F_\pm(\eta^{-1/\Delta})$ has a power-law expansion about zero, $\mathcal -F_\pm(\xi)\sim \xi^{D\nu/\Delta}$ for large $\xi$. The nonanalyticity of -these functions at infinite argument can therefore be understood as an artifact -of the chosen coordinates. +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 +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 +\begin{equation} + \mathcal F_\pm(\xi) + =\xi^{D\nu/\Delta}\mathcal F_0(\pm \xi^{-1/\Delta})+\frac1{8\pi}\log\xi^{2/\Delta} +\end{equation} +and $\mathcal F_0$ has a power-law expansion about zero, $\mathcal +F_\pm$ has a series like $\xi^{D\nu/\Delta-j/\Delta}$ for $j\in\mathbb N$ at +large $\xi$, along with logarithms. The nonanalyticity of these functions at +infinite argument can be understood as an artifact of the chosen coordinates. For the scale of $u_t$ and $u_h$, we adopt the same convention as used by \cite{Fonseca_2003_Ising}. The dependence of the nonlinear scaling variables on @@ -215,7 +220,8 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant. To lowest order, this singularity is a function of the scaling invariant $\xi$ alone. It is therefore suggestive that this should be considered a part of the singular free energy and moreover part of the scaling function that composes -it. We will therefore make the ansatz that +it. There is substantial numeric evidence for this as well. We will therefore +make the ansatz that \begin{equation} \label{eq:essential.singularity} \operatorname{Im}\mathcal F_-(\xi+i0)=A_0\Theta(-\xi)\xi e^{-1/b|\xi|}\left[1+O(\xi)\right] \end{equation} @@ -281,7 +287,7 @@ functions with different asymptotic expansions in different limits, we adopt another coordinate system, in terms of which a scaling function can be defined that has polynomial expansions in \emph{all} limits. -In all dimensions, the Schofield coordinates $R$ and $\theta$ will be implicitly defined by +The Schofield coordinates $R$ and $\theta$ are implicitly defined by \begin{align} \label{eq:schofield} u_t(R, \theta) = R(1-\theta^2) && @@ -700,13 +706,14 @@ accuracy of the fit results can be checked against the known values here. dat = 'data/phi_comparison.dat' set xlabel '$n$' - set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$' + set xrange [1.5:7.5] + set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$' set format y '$10^{%T}$' - set style data linespoints set logscale y + + set style data linespoints set key title '\raisebox{0.5em}{$m$}' bottom left - set xrange [1.5:9.5] plot \ dat using 1:2 title '0', \ |