diff options
| -rw-r--r-- | data/phi_comparison.dat | 14 | ||||
| -rw-r--r-- | data/yl_comparison.dat | 14 | ||||
| -rw-r--r-- | ising_scaling.tex | 37 |
3 files changed, 34 insertions, 31 deletions
diff --git a/data/phi_comparison.dat b/data/phi_comparison.dat index d58e96c..14c1885 100644 --- a/data/phi_comparison.dat +++ b/data/phi_comparison.dat @@ -1,8 +1,6 @@ -2 0.002860955318525926 0.004496459219585747 0.0025781014469987568 0.0004361990091461404 0.0015990766131468608 -3 0.0005720429508622171 0.0010847239134089692 0.000805365486839224 0.00006427186448818359 0.0003917599113948043 -4 0.00003961608489011503 0.0001278039316774393 0.00018174532718064074 0.00013408467605927413 0.00002358497641938859 -5 0.0000622987443403833 0.00016919055775577174 0.0002085264051783775 0.0001300161350704411 3.732946060521912e-6 -6 0.00005016392362722222 0.00014150435296594877 0.00016732830408854038 0.00007562595035311148 0.00005202003334785774 -7 0.000015703311988302104 0.00005762691693961264 0.00009304388663239349 0.00007579125219901034 0.000010561158237231718 -8 8.173890766238756e-6 0.000021607891761421527 0.000022883111337773654 6.9772437447900015e-6 0.000010861362342928695 -9 2.4873158455118727e-6 7.768088409076945e-6 9.816343265717231e-6 3.88602466879287e-6 5.648983756909563e-6
\ No newline at end of file +2 0.0015185988806263317 0.0016424512339846542 0.00010931409748554666 0.0014019918834638455 0.0009799159023033766 +3 0.0007353808953864949 0.0014547698806950948 0.0011618922642676333 0.0001926030528925822 0.0004716780917582721 +4 0.0006832169448554026 0.0015493096608610868 0.001202074914739018 0.00024563197897510894 0.0011902522778089069 +5 0.00003083151013294483 0.000024777126610253664 0.00004067889303242811 0.00008916575299173363 0.00006668594729061653 +6 0.000012555575489514581 0.000011541389969949023 0.00003418912424536791 0.00009040132752613159 0.00009049768674665578 +7 3.513158584711462e-6 5.987223218928417e-6 4.50941314238118e-6 0.000026308155365337843 0.000039381306641833976
\ No newline at end of file diff --git a/data/yl_comparison.dat b/data/yl_comparison.dat index f1ece63..3001c98 100644 --- a/data/yl_comparison.dat +++ b/data/yl_comparison.dat @@ -1,8 +1,6 @@ -2 0.0156377454168716 0.00005 -3 0.0026092903452304694 0.00005 -4 0.00287653656882067 0.00005 -5 0.0030257297041298703 0.00005 -6 0.0006769621913844392 0.00005 -7 0.0005281357188132441 0.00005 -8 0.0000685310321167365 0.00005 -9 0.000035091061848013805 0.00005
\ No newline at end of file +2 2.5824766344828554e-7 0.00005 +3 7.646795701476972e-6 0.00005 +4 5.935120440947461e-7 0.00005 +5 0.0013932555059845142 0.00005 +6 0.0005014284531485447 0.00005 +7 0.0002780661068152446 0.00005
\ No newline at end of file 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', \ |