From df2b81c3737de7553d74613f10cf39c5a045b8d8 Mon Sep 17 00:00:00 2001 From: jps6 Date: Mon, 29 Nov 2021 19:16:33 +0000 Subject: Update on Overleaf. --- ising_scaling.bib | 22 ++++++++++++++++++++++ ising_scaling.tex | 32 ++++++++++++++++++++++++-------- 2 files changed, 46 insertions(+), 8 deletions(-) diff --git a/ising_scaling.bib b/ising_scaling.bib index a971fbd..ffc714c 100644 --- a/ising_scaling.bib +++ b/ising_scaling.bib @@ -340,4 +340,26 @@ doi = {10.1103/physrevd.95.085001} } +@ARTICLE{ChenPMSnn, + author = {Yan-Jiun Chen and Natalie Paquette and Benjamin + B. Machta and James P. Sethna}, + title = {Universal scaling function for the two-dimensional + {I}sing model in an external field: A pragmatic approach}, + journal = {\url{http://arxiv.org/abs/1307.6899}}, + year = {2013}, + volume = {}, + pages = {} +} + +@article{BarmaFisherPRB, + title={Two-dimensional Ising-like systems: Corrections to scaling in the Klauder and double-Gaussian models}, + author={Barma, Mustansir and Fisher, Michael E}, + journal={Physical Review B}, + volume={31}, + number={9}, + pages={5954}, + year={1985}, + publisher={APS} +} + diff --git a/ising_scaling.tex b/ising_scaling.tex index 8488f9d..ab5ea1b 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -67,10 +67,11 @@ universality class. The continuous phase transition in the two-dimensional Ising model is the most well studied, and its universal thermodynamic functions have likewise received -the most attention. Without a field, an exact solution is known for some -lattice models \cite{Onsager_1944_Crystal}. Precision numeric work both on +the most attention. Onsager provided an exact solution in the absence of an external field \cite{Onsager_1944_Crystal}. Here we provide a high-precision, rapidly converging calculation of the universal scaling function for the 2D Ising model in a field. Our solution is not an exact formula in terms of well-known special functions (as is Onsager's result). Indeed, it seems likely that there is no such formula. The critical exponents for the 3D Ising model have recently been determined to high-precision calculations using conformal bootstrap methods, which should be viewed as a solution to that outstanding problem. The universal scaling function for the 2D Ising model in a field is a well-defined function with known singularities; in analogy, we tentatively suggest that our convergent, high-precision approximation for the function can be viewed as the complete solution to the universal part of the 2D Ising free energy in an external field. + +Precision numeric work both on lattice models and on the ``Ising'' conformal field theory (related by -universality) have yielded high-order polynomial expansions of those functions, +universality) have yielded high-order polynomial expansions of the free energy and other universal thermodynamic functions, along with a comprehensive understanding of their analytic properties \cite{Fonseca_2003_Ising, Mangazeev_2008_Variational, Mangazeev_2010_Scaling}. In parallel, smooth approximations of the Ising equation of state produce @@ -83,7 +84,7 @@ This paper attempts to find the best of both worlds: a smooth approximate universal thermodynamic function that respects the global analytic properties of the Ising free energy. By constructing approximate functions with the correct singularities, corrections converge \emph{exponentially} to the true -function. To make the construction, we review the analytic properties of the +function. To make the construction, we review the analytic properties of the Ising scaling function. Parametric coordinates are introduced that remove unnecessary singularities that are a remnant of the coordinate choice. The singularities known to be present in the scaling function are incorporated in @@ -102,6 +103,7 @@ what advances in conformal bootstrap did for critical exponents \cite{Gliozzi_2014_Critical}. Mathematica and Python implementations will be provided in the supplemental material. \section{Universal scaling functions} +\label{sec:UniversalScalingFunctions} A renormalization group analysis predicts that certain thermodynamic functions will be universal in the vicinity of \emph{any} critical point in the Ising @@ -130,8 +132,12 @@ $\delta=15$ are dimensionless constants. The combination $\Delta=\beta\delta=\frac{15}8$ will appear often. The flow equations are truncated here, but in general all terms allowed by the symmetries of the parameters are present on their righthand side. By making a near-identity -transformation to the coordinates and the free energy of the form $u_t(t, -h)=t+\cdots$, $u_h(t, h)=h+\cdots$, and $u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h)$, one can bring +transformation to the coordinates and the free energy of the form +\begin{equation} + \label{eq:AnalyticCOV} +u_t(t,h)=t+\cdots, ~~~~u_h(t, h)=h+\cdots,~~~\mathrm{and}~u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h), +\end{equation} +one can bring the flow equations into the agreed upon simplest normal form \begin{align} \label{eq:flow} \frac{du_t}{d\ell}=\frac1\nu u_t @@ -149,6 +155,7 @@ matter of convention, fixing the scale of $u_t$. Here the free energy $f=u_f+f_a Solving these equations for $u_f$ yields \begin{equation} +\label{eq:FpmF0eqns} \begin{aligned} u_f(u_t, u_h) &=|u_t|^{D\nu}\mathcal F_\pm(u_h|u_t|^{-\Delta})+\frac{|u_t|^{D\nu}}{8\pi}\log u_t^2 \\ @@ -594,6 +601,7 @@ Because the real part of $\mathcal F$ is even, the imaginary part must be odd. T \end{equation} Evaluating these ordinary integrals, we find for $\theta\in\mathbb R$ \begin{equation} +\label{eq:FfromFoFYLG} \operatorname{Re}\mathcal F(\theta)=\operatorname{Re}\mathcal F_0(\theta)+\mathcal F_\mathrm{YL}(\theta)+G(\theta) \end{equation} where @@ -611,6 +619,7 @@ where $\mathcal R$ is given by the function \end{equation} and \begin{equation} +\label{eq:FYL} \mathcal F_{\mathrm{YL}}(\theta)=2C_\mathrm{YL}\left[2(\theta^2+\theta_\mathrm{YL}^2)^{(1+\sigma)/2}\cos\left((1+\sigma)\tan^{-1}\frac\theta{\theta_\mathrm{YL}}\right)-\theta_\mathrm{YL}^{1+\sigma}\right] \end{equation} We have also included the analytic part $G$, which we assume has a simple @@ -1195,14 +1204,21 @@ the ratio. We have introduced explicit approximate functions forms for the two-dimensional Ising universal scaling function in the relevant variables. These functions are smooth to all orders, include the correct singularities, and appear to converge -exponentially to the function as they are fixed to larger polynomial order. +exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in both Mathematica and Python in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta}$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. This method, although spectacularly successful, could be improved. It becomes difficult to fit the unknown functions at progressively higher order due to the complexity of the chain-rule derivatives, and we find an inflation of predicted coefficients in our higher-precision fits. These problems may be related to the precise form and method of truncation for the unknown functions. -The successful smooth description of the Ising free energy produced in part by +It would be natural to extend our approach to the 3D Ising model, where enough high-precision information is available to provide the first few levels of approximation. In 3D, there is an important singular correction to scaling, which could be incorporated as a third invariant scaling variable in the universal scaling function. Indeed, it is believed that there are singular corrections to scaling also in 2D, which happen to vanish for the exactly solvable models~\cite{BarmaFisherPRB}. + +Derivatives of our Ising free energy provides most bulk thermodynamic properties, but not the correlation functions. The 2D Ising correlation function has been estimated~\cite{ChenPMSnn}, but without incorporating the effects of the essential singularity as one crosses the abrupt transition line. This correlation function would be experimentally useful, for example, in analyzing FRET data for two-dimensional membranes. + +It is interesting to note the close analogy between our analysis and the incorporation of analytic corrections to scaling discussed in section~\ref{sec:UniversalScalingFunctions}. Here the added function $G(\theta)$ corresponds to the analytic part of the free energy $f_a(t,h)$, and the coordinate change $g(\theta)$ corresponds to the scaling field change of variables $u_t(t,h)$ and $u_h(t,h)$ +(Eqs.~\ref{eq:AnalyticCOV} and~\ref{eq:FpmF0eqns}). One might view the universal scaling form for the Ising free energy as a scaling function describing the crossover scaling between the universal essential singularities at the two abrupt, `first-order' transition at $\pm H$, $T Date: Wed, 19 Jan 2022 13:50:41 +0100 Subject: Fixed an incorrect brace. --- ising_scaling.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index ab5ea1b..cbd1945 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -1204,7 +1204,7 @@ the ratio. We have introduced explicit approximate functions forms for the two-dimensional Ising universal scaling function in the relevant variables. These functions are smooth to all orders, include the correct singularities, and appear to converge -exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in both Mathematica and Python in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta}$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. +exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in both Mathematica and Python in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta)$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. This method, although spectacularly successful, could be improved. It becomes difficult to fit the unknown functions at progressively higher order due to the complexity of the -- cgit v1.2.3-70-g09d2 From 6abbd2bace696c746166324bbef4b1d9ba123bb3 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Wed, 19 Jan 2022 13:50:56 +0100 Subject: Removed trailing whitespace. --- ising_scaling.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index cbd1945..819ad16 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -46,7 +46,7 @@ linkcolor=purple the low- and high-temperature zero-field limits fixes the parametric coordinate transformation. For the two-dimensional Ising model, we show that this procedure converges exponentially with the order to which the series are - matched, up to seven digits of accuracy. + matched, up to seven digits of accuracy. To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy. %We speculate that with appropriately modified parametric coordinates, the method may converge even deep into the metastable phase. \end{abstract} @@ -132,7 +132,7 @@ $\delta=15$ are dimensionless constants. The combination $\Delta=\beta\delta=\frac{15}8$ will appear often. The flow equations are truncated here, but in general all terms allowed by the symmetries of the parameters are present on their righthand side. By making a near-identity -transformation to the coordinates and the free energy of the form +transformation to the coordinates and the free energy of the form \begin{equation} \label{eq:AnalyticCOV} u_t(t,h)=t+\cdots, ~~~~u_h(t, h)=h+\cdots,~~~\mathrm{and}~u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h), @@ -170,7 +170,7 @@ $\mathcal F_\pm(\xi)=G_{\mathrm{high}/\mathrm{low}}(\xi)$.}. The scaling functions are universal in the sense that any system in the same universality class will share the free energy \eqref{eq:flow}, for suitable analytic functions $u_t$, $u_h$, and analytic background $f_a$ -- the singular behavior is universal up to an analytic coordinate change. %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 the units of $u_t$ and $u_h$. +%form \eqref{eq:flow}, one will see the same functional form, up to the units of $u_t$ and $u_h$. 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 $\eta=u_t|u_h|^{-1/\Delta}$. @@ -424,7 +424,7 @@ entirely fixed, and it will be truncated at finite order. [0:20:0.1] '+' u ($1*f(12*-t0/16)):($1**del*g(12*-t0/16)) dt 2 lc black lw 2 , \ [0:20:0.1] '+' u ($1*f(13*-t0/16)):($1**del*g(13*-t0/16)) dt 2 lc black lw 2 , \ [0:20:0.1] '+' u ($1*f(14*-t0/16)):($1**del*g(14*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(15*-t0/16)):($1**del*g(15*-t0/16)) dt 2 lc black lw 2 + [0:20:0.1] '+' u ($1*f(15*-t0/16)):($1**del*g(15*-t0/16)) dt 2 lc black lw 2 \end{gnuplot} \caption{ Example of the parametric coordinates. Solid lines are of constant @@ -1110,7 +1110,7 @@ Notice that this infelicity does not appear to cause significant errors in the f function of polynomial order $m$, rescaled by their asymptotic limit $\mathcal F_-^\infty(m)$ from \eqref{eq:low.asymptotic}. 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}. Note that our $n=6$ fit generates significant deviations in polynomial coefficients $m$ above around 10. + from the most accurate scaling function listed in \cite{Caselle_2001_The}. Note that our $n=6$ fit generates significant deviations in polynomial coefficients $m$ above around 10. } \label{fig:glow.series.scaled} \end{figure} @@ -1216,7 +1216,7 @@ It would be natural to extend our approach to the 3D Ising model, where enough h Derivatives of our Ising free energy provides most bulk thermodynamic properties, but not the correlation functions. The 2D Ising correlation function has been estimated~\cite{ChenPMSnn}, but without incorporating the effects of the essential singularity as one crosses the abrupt transition line. This correlation function would be experimentally useful, for example, in analyzing FRET data for two-dimensional membranes. It is interesting to note the close analogy between our analysis and the incorporation of analytic corrections to scaling discussed in section~\ref{sec:UniversalScalingFunctions}. Here the added function $G(\theta)$ corresponds to the analytic part of the free energy $f_a(t,h)$, and the coordinate change $g(\theta)$ corresponds to the scaling field change of variables $u_t(t,h)$ and $u_h(t,h)$ -(Eqs.~\ref{eq:AnalyticCOV} and~\ref{eq:FpmF0eqns}). One might view the universal scaling form for the Ising free energy as a scaling function describing the crossover scaling between the universal essential singularities at the two abrupt, `first-order' transition at $\pm H$, $T Date: Wed, 19 Jan 2022 13:56:41 +0100 Subject: Changed PDF figures to EPS. --- figs/F_higher_singularities.eps | 343 ++++++ figs/F_higher_singularities.pdf | Bin 8097 -> 0 bytes figs/F_lower_singularities.eps | 270 +++++ figs/F_lower_singularities.pdf | Bin 6542 -> 0 bytes figs/F_theta_singularities.eps | 376 +++++++ figs/F_theta_singularities.pdf | Bin 8379 -> 0 bytes figs/contour_path.eps | 440 ++++++++ figs/contour_path.pdf | Bin 9494 -> 0 bytes figs/figures.nb | 2371 +++++++++++++++++++++++++++++++++++++-- 9 files changed, 3713 insertions(+), 87 deletions(-) create mode 100644 figs/F_higher_singularities.eps delete mode 100644 figs/F_higher_singularities.pdf create mode 100644 figs/F_lower_singularities.eps delete mode 100644 figs/F_lower_singularities.pdf 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changed, 4 insertions(+), 4 deletions(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index 819ad16..210f63c 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -242,7 +242,7 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \begin{figure} - \includegraphics{figs/F_lower_singularities.pdf} + \includegraphics{figs/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 @@ -289,7 +289,7 @@ branch cuts beginning at $\pm i\xi_{\mathrm{YL}}$ for a universal constant $\xi_{\mathrm{YL}}$. \begin{figure} - \includegraphics{figs/F_higher_singularities.pdf} + \includegraphics{figs/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 @@ -491,7 +491,7 @@ $\theta$. Therefore, The location $\theta_0$ is not fixed by any principle. \begin{figure} - \includegraphics{figs/F_theta_singularities.pdf} + \includegraphics{figs/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 @@ -548,7 +548,7 @@ 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.pdf} + \includegraphics{figs/contour_path} \caption{ Integration contour over the global scaling function $\mathcal F$ in the complex $\theta$ plane used to produce the dispersion relation. The -- cgit v1.2.3-70-g09d2 From 9ff4fff4dbe01eab0d11b4dbc4c8a8ab748c88ad Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 14 Apr 2022 14:49:43 +0200 Subject: Added mathematica files. --- IsingScalingFunction.wl | 182 + IsingScalingFunctionExamples.nb | 11602 ++++++++++++++++++++++++++++++++++++++ 2 files changed, 11784 insertions(+) create mode 100644 IsingScalingFunction.wl create mode 100644 IsingScalingFunctionExamples.nb diff --git a/IsingScalingFunction.wl b/IsingScalingFunction.wl new file mode 100644 index 0000000..edc46c2 --- /dev/null +++ b/IsingScalingFunction.wl @@ -0,0 +1,182 @@ + +BeginPackage["IsingScalingFunction`"] + +g::usage = "g[θ0, gs][θ] gives the Schofield coordinate transformation defined in (14)." + +ut::usage = "ut[θ] gives the scaling field u_t as a function of Schofield coordinates." + +uh::usage = "uh[θ0, gs][θ] gives the scaling field u_h as a function of Schofield coordinates." + +η::usage = "η[θ0, gs][θ] gives the invariant scaling combination η." + +ξ::usage = "ξ[θ0, gs][θ] gives the invariant scaling combination ξ." + +ReScriptF::usage = "ReScriptF[θ0, θYL, B, C0, CYL, Gs][θ] gives the free energy scaling function defined in (19)." + +ScriptF::usage = "ScriptF[θ0, θYL, B, C0, CYL, Gs][θ] gives the free energy scaling function defined in (35)." + +DScriptFPlusMinusDξθ0List::usage = + "DScriptFPlusMinusDξθ0List computes the first m derivatives of the scaling function F_- evaluated at θ_0." + +DScriptFPlusMinusDξList::usage = + "DScriptFPlusMinusDξList computes the first m derivatives of the scaling function F_+/-." + +DScriptF0DηList::usage = + "DScriptF0DηList computes the first m derivatives of the scaling function F_0." + +uf::usage = "uf computes the singular free energy u_f." + +DufDut::usage = + "DufDut computes derivatives of the singular free energy u_f with respect to the temperature-like scaling field u_t." + +DufDuh::usage = + "DufDuh computes derivatives of the singular free energy u_f with respect to the temperature-like scaling field u_h." +ruleB::usage = "" + +ruleAL::usage = "" + +Begin["Private`"] + +β := 1/8 + +δ := 15 + +Δ := β δ + +OverlineS := 2^(1/12) Exp[-1/8] Glaisher^(3/2) + +Φs := { + -Gamma[1/3]Gamma[1/5]Gamma[7/15]/(2 π Gamma[2/3]Gamma[4/5]Gamma[8/15])(4 π^2 Gamma[13/16]^2 Gamma[3/4]/(Gamma[3/16]^2 Gamma[1/4]))^(8/15), + -0.31881012489061, + Around[0.110886196683, 2.0 10^-12], + Around[0.01642689465, 2.0 10^-11], + Around[-2.639978 10^-4, 1.0 10^-10], + Around[-5.140526 10^-4, 1.0 10^-10], + Around[2.08865 10^-4, 1.0 10^-9], + Around[-4.4819 10^-5, 1.0 10^-9], + Around[3.16 10^-7, 1.0 10^-9], + Around[4.31 10^-6, 0.01 10^-6], + Around[-1.99 10^-6, 0.01 10^-6] +} + +Gls := { + 0, + -OverlineS, + −1.000960328725262189480934955172097320572505951770117 Sqrt[2]/((2 )^(-7/8) (2^(3/16)/OverlineS)^2)/2/(12 \[Pi]), + Around[ 0.038863932, 3.0 10^(-9)], + Around[−0.068362119, 2.0 10^(-9)], + Around[ 0.18388370, 1.0 10^(-8)], + Around[-0.6591714, 1.0 10^(-7)], + Around[ 2.937665, 3.0 10^(-6)], + Around[-15.61, 1.0 10^(-2)], + Around[ 96.76, 1.0 10^(-2)], + Around[-6.79 10^2, 1.0], + Around[ 5.34 10^3, 10.], + Around[-4.66 10^4, 0.01 10^4], + Around[ 4.46 10^5, 0.01 10^5], + Around[-4.66 10^6, 0.01 10^6] +} + +Ghs := { + 0, + 0, + -1.000815260440212647119476363047210236937534925597789 Sqrt[2]/((2 )^(-7/8) (2^(3/16)/OverlineS)^2)/2, + 0, + Around[ 8.333711750, 5.0 10^(-9)], + 0, + Around[-95.16896, 1.0 10^(-5)], + 0, + Around[1457.62, 3.0 10^(-2)], + 0, + Around[-2.5891 10^4, 2.0], + 0, + Around[5.02 10^5, 0.01 10^5], + 0, + Around[-1.04 10^7, 0.01 10^7] +} + +t[θ_] := θ^2 - 1 + +g[θ0_, gs_][θ_] := (1 - (θ/θ0)^2) Total[MapIndexed[Function[{gi, i}, gi θ^(2*i[[1]]-1)], gs]] + +ut[R_, θ_] := R t[θ] + +uh[θ0_, gs_][R_, θ_] := R^Δ g[θ0, gs][θ] + +η[θ0_, gs_][θ_] := t[θ] / RealAbs[g[θ0, gs][θ]]^(1 / Δ) + +ξ[θ0_, gs_][θ_] := g[θ0, gs][θ] / RealAbs[t[θ]]^Δ + +ScriptR[θc_, B_][θ_] := (θc Exp[1/(B θc)] ExpIntegralEi[-1/(B θc)] + (θ - θc) Exp[-1/(B (θ - θc))] ExpIntegralEi[1/(B (θ - θc))]) / π + +ReScriptF0[C0_, θc_, B_][θ_] := C0 (ScriptR[θc, B][θ] + ScriptR[θc, B][-θ]) + +ScriptFYL[θYL_, CYL_][θ_] := CYL ((-I θ + θYL)^(5/6) + (I θ + θYL)^(5/6) - 2 θYL^(5/6)) + +ReScriptFRegular[θ0_, θYL_, B_, C0_, CYL_, Gs_][θ_] := C0 ScriptR[θ0, B][-θ] + ScriptFYL[θYL, CYL][θ] + Total[MapIndexed[Function[{G, i}, G θ^(2*i[[1]])], Gs]] + +ReScriptF[θ0_, θYL_, B_, C0_, CYL_, Gs_][θ_] := ReScriptFRegular[θ0, θYL, B, C0, CYL, Gs][θ] + C0 ScriptR[θ0, B][θ] + +DReScriptFIrregular[θ0_, B_, C0_][m_] := Piecewise[{{C0 m! Gamma[m - 1] B^(m - 1) / π, m > 1}, {C0 θ0 Exp[1/(B θ0)] ExpIntegralEi[-1/(B θ0)] / π, m == 0}}, 0] + +ScriptF[θ0_, θYL_, B_, C0_, CYL_, Gs_][θ_] := ReScriptF[θ0, θYL, B, C0, CYL, Gs][θ] + C0 I Sign[Im[θ]] ((θ-θ0)Exp[-1/(B(θ-θ0))]-(-θ-θ0)Exp[-1/(B(-θ-θ0))]) + +ScriptFPlusMinus[ScriptF_][θ_] := ScriptF[θ] / t[θ]^2 - 1/(8 \[Pi]) Log[t[θ]^2] + +ScriptF0[θ0_, gs_][ScriptF_][θ_] := RealAbs[g[θ0, gs][θ]]^(-2 / Δ) ScriptF[θ] - η[θ0, gs][θ]^2 Log[g[θ0, gs][θ]^2] / (8 π Δ) + +uf[params__][R_, θ_] := R^2 ReScriptF[params][θ] + t[θ]^2 R^2 / (8 π) Log[R^2] + +EfficientDerivativeList[n_][f_][x_] := Module[ + {xp}, NestList[Function[g, D[g, xp]], f[xp], n] /. xp -> x +] + +InverseDerivativeList[n_][f_][x_] := Module[ + {xp, dfs, fp, Pns}, + dfs = Rest[EfficientDerivativeList[n][f][x]]; + Pns = FoldList[Function[{Pm, m}, + fp'[xp] D[Pm, xp] - (2 m - 1) fp''[xp] Pm], 1, Range[n - 1]] /. + Derivative[m_][fp][xp] :> dfs[[m]]; + MapIndexed[{Pn, i} \[Function] Pn/dfs[[1]]^(2 i[[1]] - 1), Pns] + ] + +CompositeFunctionDerivativeList[G_, F_, X_, FSupp_:(0&)][m_, θ_] := Module[ + { ds, dF, df, fp }, + ds = InverseDerivativeList[m+1][X][θ]; + dF = EfficientDerivativeList[m][F][θ] + FSupp /@ Range[0, m]; + df = EfficientDerivativeList[m][G[fp]][θ] /. + Map[Derivative[#][fp][θ] -> dF[[# + 1]] &, Range[0, m]]; + Table[Sum[df[[k+1]] BellY[j, k, ds[[;; j - k + 1]]], {k, 0, j}]/(j!), {j, 0, m}] +] + +DScriptFPlusMinusDξθ0List[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_] := CompositeFunctionDerivativeList[ + ScriptFPlusMinus, ReScriptFRegular[θ0, θYL, B, C0, CYL, Gs], + ξ[θ0, gs], DReScriptFIrregular[θ0, B, C0] + ][m, θ0] + +DScriptFPlusMinusDξList[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_, θ_] := CompositeFunctionDerivativeList[ + ScriptFPlusMinus, ReScriptF[θ0, θYL, B, C0, CYL, Gs], ξ[θ0, gs] + ][m, θ] + +DScriptF0DηList[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_, θ_] := CompositeFunctionDerivativeList[ + ScriptF0[θ0, gs], ReScriptF[θ0, θYL, B, C0, CYL, Gs], η[θ0, gs] + ][m, θ] + +DScriptFPlusMinusDξθ0[params__][m_] := Last[DScriptFPlusMinusDξθ0List[params][m]] + +DScriptFPlusMinusDξ[params__][m_, θ_] := Last[DScriptFPlusMinusDξList[params][m, θ]] + +DScriptF0Dη[params__][m_, θ_] := Last[DScriptF0DηList[params][m, θ]] + +DufDut[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := RealAbs[uh[θ0, gs][R, θ]]^(2 / Δ - m / Δ) DScriptF0Dη[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Log[uh[θ0, gs][R, θ]^2] / (8 π Δ) Derivative[m][Function[utp, utp^2]][ut[R, θ]] + +DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := RealAbs[ut[R, θ]]^(2-m Δ) DScriptFPlusMinusDξ[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + ut[R, θ]^2 / (8 π) Log[ut[R, θ]^2] + +ruleB[θ0_, gs_] := (2 * OverlineS / π) * (- g[θ0, gs]'[θ0] / t[θ0]^Δ) + +ruleAL[θ0_, gs_] := Exp[Δ t[θ0]^(Δ - 1) t'[θ0] / (2 OverlineS / π g[θ0, gs]'[θ0]) - t[θ0]^Δ g[θ0, gs]''[θ0] / (4 OverlineS / π g[θ0, gs]'[θ0]^2)] t[θ0]^(1/8) OverlineS / (2 π) * g[θ0, gs]'[θ0] + +End[] + +EndPackage[] + diff --git a/IsingScalingFunctionExamples.nb 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insertions(+), 2282 deletions(-) diff --git a/IsingScalingFunction.wl b/IsingScalingFunction.wl index edc46c2..0d5b38c 100644 --- a/IsingScalingFunction.wl +++ b/IsingScalingFunction.wl @@ -33,7 +33,11 @@ DufDuh::usage = "DufDuh computes derivatives of the singular free energy u_f with respect to the temperature-like scaling field u_h." ruleB::usage = "" -ruleAL::usage = "" +ruleC0::usage = "" + +Data::usage = "" + +PrepareArgument::usage = "" Begin["Private`"] @@ -95,6 +99,62 @@ Ghs := { Around[-1.04 10^7, 0.01 10^7] } +Data[2] = <| + "θ0" -> 1.148407773492004`, + "θYL" -> 0.9896669889911205`, + "CYL" -> -0.172823989504767`, + "Gs" -> {-0.31018352388662596`, 0.2474537923130002`}, + "gs" -> {0.37369093055254343`, -0.021636313152585823`} +|> + +Data[3] = <| + "θ0" -> 1.2542120477507488`, + "θYL" -> 0.6020557328641167`, + "CYL" -> -0.38566364361428684`, + "Gs" -> {-0.3527514794812415`, 0.2582430860166863`}, + "gs" -> {0.4483788209731592`, -0.022032295172535358`, 0.00022200608228654115`} +|> + +Data[4] = <| + "θ0" -> 1.3164928721109121`, + "θYL" -> 0.6400189996493497`, + "CYL" -> -0.3563974694580203`, + "Gs" -> {-0.3550547624920263`, 0.23465947408509413`, -0.0019083731028066697`}, + "gs" -> {0.4410742751152714`, -0.034817777358116885`, 0.000678172648789985`, -0.00004305140578834467`} +|> + +Data[5] = <| + "θ0" -> 1.3403205742656135`, + "θYL" -> 0.6238113973493433`, + "CYL" -> -0.38002950945224295`, + "Gs" -> {-0.35127522582179693`, 0.23704589676915347`, -0.007319731639727028`}, + "gs" -> {0.44371885415894785`, -0.04609943321005163`, -0.0007458341071947777`, 0.00005966875622885447`, -4.403083529955303`*^-6} +|> + +Data[6] = <| + "θ0" -> 1.3626103817690176`, + "θYL" -> 0.6462147447024515`, + "CYL" -> -0.35576386594103865`, + "Gs" -> {-0.3520586281920383`, 0.23316561297622435`, -0.006649030656179257`, -0.0016899077640685814`}, + "gs" -> {0.43845335615925396`, -0.05312704168994819`, -0.003914782631377569`, -0.0004080160912692574`, 0.000026262906640471588`, -1.0974538440746828`*^-6} +|> + +PrepareArgument[data_] := With[ + { + θ0 = data["θ0"], + gs = data["gs"] + }, + { + θ0, + data["θYL"], + ruleB[θ0, gs], + ruleC0[θ0, gs], + data["CYL"], + data["Gs"], + gs + } +] + t[θ_] := θ^2 - 1 g[θ0_, gs_][θ_] := (1 - (θ/θ0)^2) Total[MapIndexed[Function[{gi, i}, gi θ^(2*i[[1]]-1)], gs]] @@ -168,13 +228,19 @@ DScriptFPlusMinusDξ[params__][m_, θ_] := Last[DScriptFPlusMinusDξList[params] DScriptF0Dη[params__][m_, θ_] := Last[DScriptF0DηList[params][m, θ]] -DufDut[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := RealAbs[uh[θ0, gs][R, θ]]^(2 / Δ - m / Δ) DScriptF0Dη[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Log[uh[θ0, gs][R, θ]^2] / (8 π Δ) Derivative[m][Function[utp, utp^2]][ut[R, θ]] +DufDut[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! RealAbs[uh[θ0, gs][R, θ]]^(2 / Δ - m / Δ) DScriptF0Dη[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Log[uh[θ0, gs][R, θ]^2] / (8 π Δ) Derivative[m][Function[utp, utp^2]][ut[R, θ]] -DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := RealAbs[ut[R, θ]]^(2-m Δ) DScriptFPlusMinusDξ[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + ut[R, θ]^2 / (8 π) Log[ut[R, θ]^2] +DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! RealAbs[ut[R, θ]]^(2-m Δ) DScriptFPlusMinusDξ[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + ut[R, θ]^2 / (8 π) Log[ut[R, θ]^2] ruleB[θ0_, gs_] := (2 * OverlineS / π) * (- g[θ0, gs]'[θ0] / t[θ0]^Δ) -ruleAL[θ0_, gs_] := Exp[Δ t[θ0]^(Δ - 1) t'[θ0] / (2 OverlineS / π g[θ0, gs]'[θ0]) - t[θ0]^Δ g[θ0, gs]''[θ0] / (4 OverlineS / π g[θ0, gs]'[θ0]^2)] t[θ0]^(1/8) OverlineS / (2 π) * g[θ0, gs]'[θ0] +ruleC0[θ0_, gs_] := Exp[Δ t[θ0]^(Δ - 1) t'[θ0] / (2 OverlineS / π g[θ0, gs]'[θ0]) - t[θ0]^Δ g[θ0, gs]''[θ0] / (4 OverlineS / π g[θ0, gs]'[θ0]^2)] t[θ0]^(1/8) OverlineS / (2 π) * g[θ0, gs]'[θ0] + +Unprotect[Derivative] + + + +Protect[Derivative] End[] diff --git a/IsingScalingFunctionExamples.nb b/IsingScalingFunctionExamples.nb index 507bc20..b5b6bd0 100644 --- a/IsingScalingFunctionExamples.nb +++ b/IsingScalingFunctionExamples.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 661071, 11594] -NotebookOptionsPosition[ 653812, 11470] 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2022 13:30:26 +0000 Subject: Update on Overleaf. --- ising_scaling.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index 210f63c..58cd767 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -47,7 +47,7 @@ linkcolor=purple coordinate transformation. For the two-dimensional Ising model, we show that this procedure converges exponentially with the order to which the series are matched, up to seven digits of accuracy. - To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy. + To facilitate use, we provide a Mathematica implementation of the code at both lowest order (three digit) and high accuracy. %We speculate that with appropriately modified parametric coordinates, the method may converge even deep into the metastable phase. \end{abstract} @@ -100,7 +100,7 @@ With six derivatives, it is accurate to about $10^{-7}$. We hope that with some refinement, this idea might be used to establish accurate scaling functions for critical behavior in other universality classes, doing for scaling functions what advances in conformal bootstrap did for critical exponents -\cite{Gliozzi_2014_Critical}. Mathematica and Python implementations will be provided in the supplemental material. +\cite{Gliozzi_2014_Critical}. A Mathematica implementation will be provided in the supplemental material. \section{Universal scaling functions} \label{sec:UniversalScalingFunctions} @@ -1204,7 +1204,7 @@ the ratio. We have introduced explicit approximate functions forms for the two-dimensional Ising universal scaling function in the relevant variables. These functions are smooth to all orders, include the correct singularities, and appear to converge -exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in both Mathematica and Python in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta)$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. +exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in Mathematica in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta)$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. This method, although spectacularly successful, could be improved. It becomes difficult to fit the unknown functions at progressively higher order due to the complexity of the -- cgit v1.2.3-70-g09d2 From 3ec595d0e594e94b57db07feb6329a772d190cdf Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 8 Sep 2022 14:34:37 +0200 Subject: Small style changes, and title change. --- ising_scaling.tex | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/ising_scaling.tex b/ising_scaling.tex index 210f63c..1933983 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -26,7 +26,7 @@ linkcolor=purple \begin{document} -\title{Smooth and global Ising universal scaling functions} +\title{Precision approximation of the universal scaling functions for the 2D Ising model in an external field} \author{Jaron Kent-Dobias} \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} @@ -133,10 +133,14 @@ $\Delta=\beta\delta=\frac{15}8$ will appear often. The flow equations are truncated here, but in general all terms allowed by the symmetries of the parameters are present on their righthand side. By making a near-identity transformation to the coordinates and the free energy of the form -\begin{equation} +\begin{align} \label{eq:AnalyticCOV} -u_t(t,h)=t+\cdots, ~~~~u_h(t, h)=h+\cdots,~~~\mathrm{and}~u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h), -\end{equation} + u_t(t,h)=t+\cdots + && + u_h(t, h)=h+\cdots + && + u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h), +\end{align} one can bring the flow equations into the agreed upon simplest normal form \begin{align} \label{eq:flow} -- cgit v1.2.3-70-g09d2 From d89d6e1ac00609fc59be5df89e41e5ac3f004617 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 8 Sep 2022 14:39:01 +0200 Subject: Fixed stupid mistake in inverse function. --- IsingScalingFunctionExamples.nb | 119 +++++++++++++++++----------------------- 1 file changed, 51 insertions(+), 68 deletions(-) diff --git a/IsingScalingFunctionExamples.nb b/IsingScalingFunctionExamples.nb index ea6931e..7f37ad0 100644 --- a/IsingScalingFunctionExamples.nb +++ b/IsingScalingFunctionExamples.nb @@ 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temperature-like scaling field u_h." -ruleB::usage = "" -ruleC0::usage = "" +ruleB::usage = "Fixes B given other data as in (38)." + +ruleC0::usage = "Fixes C0 given other data as in (39)." -Data::usage = "" +Data::usage = "Data[n] gives data from the fit to nth order from Table II." -PrepareArgument::usage = "" +PrepareArgument::usage = "Converts scaling function data into appropriate argument to function interfaces." Begin["Private`"] -- cgit v1.2.3-70-g09d2 From 49799190a6d089ba58758c76b7a2194934af2de5 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 6 Feb 2023 17:05:47 +0100 Subject: Added an inverse coordinate function. --- IsingScalingFunction.wl | 20 ++++++++++++++------ 1 file changed, 14 insertions(+), 6 deletions(-) diff --git a/IsingScalingFunction.wl b/IsingScalingFunction.wl index 29370eb..b119b9f 100644 --- a/IsingScalingFunction.wl +++ b/IsingScalingFunction.wl @@ -99,7 +99,7 @@ Ghs := { Around[-1.04 10^7, 0.01 10^7] } -Data[2] = <| +Data[2] = Rationalize[#, 10^-20] & /@ <| "θ0" -> 1.148407773492004`, "θYL" -> 0.9896669889911205`, "CYL" -> -0.172823989504767`, @@ -107,7 +107,7 @@ Data[2] = <| "gs" -> {0.37369093055254343`, -0.021636313152585823`} |> -Data[3] = <| +Data[3] = Rationalize[#, 10^-20] & /@ <| "θ0" -> 1.2542120477507488`, "θYL" -> 0.6020557328641167`, "CYL" -> -0.38566364361428684`, @@ -115,7 +115,7 @@ Data[3] = <| "gs" -> {0.4483788209731592`, -0.022032295172535358`, 0.00022200608228654115`} |> -Data[4] = <| +Data[4] = Rationalize[#, 10^-20] & /@ <| "θ0" -> 1.3164928721109121`, "θYL" -> 0.6400189996493497`, "CYL" -> -0.3563974694580203`, @@ -123,7 +123,7 @@ Data[4] = <| "gs" -> {0.4410742751152714`, -0.034817777358116885`, 0.000678172648789985`, -0.00004305140578834467`} |> -Data[5] = <| +Data[5] = Rationalize[#, 10^-20] & /@ <| "θ0" -> 1.3403205742656135`, "θYL" -> 0.6238113973493433`, "CYL" -> -0.38002950945224295`, @@ -131,7 +131,7 @@ Data[5] = <| "gs" -> {0.44371885415894785`, -0.04609943321005163`, -0.0007458341071947777`, 0.00005966875622885447`, -4.403083529955303`*^-6} |> -Data[6] = <| +Data[6] = Rationalize[#, 10^-20] & /@ <| "θ0" -> 1.3626103817690176`, "θYL" -> 0.6462147447024515`, "CYL" -> -0.35576386594103865`, @@ -163,6 +163,14 @@ ut[R_, θ_] := R t[θ] uh[θ0_, gs_][R_, θ_] := R^Δ g[θ0, gs][θ] +InverseCoordinates[\[Theta]0_, gs_, wp_:20][tn_, hn_] := + ({Exp[logR], \[Theta]0 Tanh[x]} /. + FindRoot[{ + Rationalize[tn, 10^-30] == ut[Exp[logR], \[Theta]0 Tanh[x]], + Rationalize[hn, 10^-30] == uh[\[Theta]0, gs][Exp[logR], \[Theta]0 Tanh[x]] + }, {{logR, 2}, {x, Sign[hn]/2}}, WorkingPrecision -> wp]) /; + NumericQ[tn] && NumericQ[hn] + η[θ0_, gs_][θ_] := t[θ] / RealAbs[g[θ0, gs][θ]]^(1 / Δ) ξ[θ0_, gs_][θ_] := g[θ0, gs][θ] / RealAbs[t[θ]]^Δ @@ -198,7 +206,7 @@ InverseDerivativeList[n_][f_][x_] := Module[ fp'[xp] D[Pm, xp] - (2 m - 1) fp''[xp] Pm], 1, Range[n - 1]] /. Derivative[m_][fp][xp] :> dfs[[m]]; MapIndexed[{Pn, i} \[Function] Pn/dfs[[1]]^(2 i[[1]] - 1), Pns] - ] +] CompositeFunctionDerivativeList[G_, F_, X_, FSupp_:(0&)][m_, θ_] := Module[ { ds, dF, df, fp }, -- cgit v1.2.3-70-g09d2 From 0c9f4d522b557a0c9f1436e80bf666d9d2630592 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 10 Feb 2023 17:27:13 +0100 Subject: Some progress. --- IsingScalingFunction.wl | 39 +- IsingScalingFunctionExamples.nb | 8964 ++++++++++++++++++++++++++++++++++++--- 2 files changed, 8368 insertions(+), 635 deletions(-) diff --git a/IsingScalingFunction.wl b/IsingScalingFunction.wl index b119b9f..8545567 100644 --- a/IsingScalingFunction.wl +++ b/IsingScalingFunction.wl @@ -1,5 +1,7 @@ BeginPackage["IsingScalingFunction`"] +InverseCoordinates::usage = "" + g::usage = "g[θ0, gs][θ] gives the Schofield coordinate transformation defined in (14)." ut::usage = "ut[θ] gives the scaling field u_t as a function of Schofield coordinates." @@ -23,6 +25,8 @@ DScriptFPlusMinusDξList::usage = DScriptF0DηList::usage = "DScriptF0DηList computes the first m derivatives of the scaling function F_0." +DScriptMCasDξList::usage = "Computes the first m derivatives of the scaling function M given by Caselle et al." + uf::usage = "uf computes the singular free energy u_f." DufDut::usage = @@ -39,6 +43,16 @@ Data::usage = "Data[n] gives data from the fit to nth order from Table II." PrepareArgument::usage = "Converts scaling function data into appropriate argument to function interfaces." +θ0Cas::usage = "" + +gsCas::usage = "" + +Φs::usage = "" + +Gls::usage = "" + +Ghs::usage = "" + Begin["Private`"] β := 1/8 @@ -238,17 +252,36 @@ DScriptF0Dη[params__][m_, θ_] := Last[DScriptF0DηList[params][m, θ]] DufDut[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! RealAbs[uh[θ0, gs][R, θ]]^(2 / Δ - m / Δ) DScriptF0Dη[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Log[uh[θ0, gs][R, θ]^2] / (8 π Δ) Derivative[m][Function[utp, utp^2]][ut[R, θ]] -DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! RealAbs[ut[R, θ]]^(2-m Δ) DScriptFPlusMinusDξ[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + ut[R, θ]^2 / (8 π) Log[ut[R, θ]^2] +DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! 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+1NzN3Qw49JO/cuodQTnJsRxKxM9nuZIn6R8JSu+laKrGCQb8mzXnd/lMUC9v +6xgKnGYA48qxl/fqCapySlFXAq8fJhZ05PsPgjJhdvPQ8f7rMpM4eFsJSrko +vK8Of56YzvOCre0EFaWro7XdjwFH333dG9hNUK5X91kluDIgZ/fIcmE/QRU+ +3eHTdoABE90cj4eGsX6LmF6rPBlg6Cu7U+YfQfnk8H387xwDIlm3zDpOEhSL +1br1Kr4M+Ba7Py12lqD8rxxKD3FngKS0//aaRXy/qZWlwR4M+P/fF6k1sEVS +AD//f6I0RVc= + "]]}, Annotation[#, "Charting`Private`Tag$1241108#4"]& ]}}, {}}, { + DisplayFunction -> Identity, + Method -> { + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> + None}, DisplayFunction -> Identity, DisplayFunction -> Identity, + Ticks -> {Automatic, + Charting`ScaledTicks[{Log, Exp}, {Log, Exp}, "Nice", WorkingPrecision -> + MachinePrecision, RotateLabel -> 0]}, + AxesOrigin -> {0, -24.4628263145363}, FrameTicks -> {{ + Charting`ScaledTicks[{Log, Exp}, {Log, Exp}, "Nice", WorkingPrecision -> + MachinePrecision, RotateLabel -> 0], + Charting`ScaledFrameTicks[{Log, Exp}]}, {Automatic, Automatic}}, + GridLines -> {None, None}, DisplayFunction -> Identity, + PlotRangePadding -> {{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.05], + Scaled[0.05]}}, PlotRangeClipping -> True, ImagePadding -> All, + DisplayFunction -> Identity, + Method -> { + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "ClippingRange" -> {{{-5.999999755102041, + 5.999999755102041}, 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singular free energy u_f." @@ -45,6 +48,8 @@ PrepareArgument::usage = "Converts scaling function data into appropriate argume θ0Cas::usage = "" +h0Cas::usage = "" + gsCas::usage = "" Φs::usage = "" @@ -252,7 +257,7 @@ DScriptF0Dη[params__][m_, θ_] := Last[DScriptF0DηList[params][m, θ]] DufDut[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! RealAbs[uh[θ0, gs][R, θ]]^(2 / Δ - m / Δ) DScriptF0Dη[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Log[uh[θ0, gs][R, θ]^2] / (8 π Δ) Derivative[m][Function[utp, utp^2]][ut[R, θ]] -DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! RealAbs[ut[R, θ]]^(2-m Δ) DScriptFPlusMinusDξ[θ0, θYL, B, C0, CYL, Gs, gs][m, θ] + Derivative[m][Identity][θ] ut[R, θ]^2 / (8 π) Log[ut[R, θ]^2] +DufDuh[θ0_, θYL_, B_, C0_, CYL_, Gs_, gs_][m_][R_, θ_] := m! 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Exp}, {Log, Exp}, "Nice", WorkingPrecision -> + MachinePrecision, RotateLabel -> 0]}, + AxesOrigin -> {0, -25.205799457979587`}, FrameTicks -> {{ + Charting`ScaledTicks[{Log, Exp}, {Log, Exp}, "Nice", WorkingPrecision -> + MachinePrecision, RotateLabel -> 0], + Charting`ScaledFrameTicks[{Log, Exp}]}, {Automatic, Automatic}}, + GridLines -> {None, None}, DisplayFunction -> Identity, + PlotRangePadding -> {{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.05], + Scaled[0.05]}}, PlotRangeClipping -> True, ImagePadding -> All, + DisplayFunction -> Identity, + Method -> { + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "ClippingRange" -> {{{-5.999999755102041, + 5.999999755102041}, {-25.205799457979644`, -7.661000555932736}}, \ +{{-5.999999755102041, + 5.999999755102041}, {-25.205799457979644`, -7.661000555932736}}}}, + DisplayFunction -> Identity, AspectRatio -> + NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, + AxesLabel -> { + FormBox[ + TagBox[ + "\"\\!\\(\\*StyleBox[\\\"t\\\",FontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\ +StyleBox[\\\" \ +\\\",FontSlant->\\\"Italic\\\"]\\)\\!\\(\\*SuperscriptBox[StyleBox[\\\"h\\\",\ +FontSlant->\\\"Italic\\\"], RowBox[{RowBox[{RowBox[{\\\"-\\\", \\\"1\\\"}], \ +\\\"/\\\", \\\"\[Beta]\\\"}], \\\" \\\", \\\"\[Delta]\\\"}]]\\)\"", HoldForm], + TraditionalForm], + FormBox[ + TagBox[ + "\"| \\!\\(\\*SuperscriptBox[SubscriptBox[\\\"\[ScriptCapitalF]\\\", \ +\\\"0\\\"], StyleBox[RowBox[{\\\"[\\\", \\\"n\\\", \ +\\\"]\\\"}],FontSlant->\\\"Italic\\\"]]\\) - \ +\\!\\(\\*SuperscriptBox[SubscriptBox[\\(\[ScriptCapitalF]\\), \\(0\\)], \ +\\([6]\\)]\\) |\"", HoldForm], TraditionalForm]}, + AxesOrigin -> {0, -25.205799457979644`}, + CoordinatesToolOptions -> {"DisplayFunction" -> ({ + Part[#, 1], + Exp[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ + Part[#, 1], + Exp[ + Part[#, 2]]}& )}, DisplayFunction :> Identity, + Frame -> {{False, False}, {False, False}}, + FrameLabel -> {{None, None}, {None, None}}, + FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, + GridLines -> {None, None}, GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], LabelStyle -> { + GrayLevel[0], FontSize -> 12}, + Method -> { + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> + None}, PlotRange -> {{-6, + 6}, {-25.205799457979644`, -7.661000555932736}}, PlotRangeClipping -> + True, PlotRangePadding -> {{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.02], + Scaled[0.02]}}, Ticks -> {Automatic, Automatic}}], + FormBox[ + FormBox[ + TemplateBox[{"2", "3", "4", "5"}, "LineLegend", + DisplayFunction -> (FormBox[ + StyleBox[ + StyleBox[ + PaneBox[ + TagBox[ + GridBox[{{ + StyleBox["\"n\"", { + GrayLevel[0], FontSize -> 12, FontFamily -> "Arial"}, + Background -> Automatic, StripOnInput -> False]}, { + TagBox[ + GridBox[{{ + TagBox[ + GridBox[{{ + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.368417, 0.506779, 0.709798], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.368417, 0.506779, 0.709798], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.880722, 0.611041, 0.142051], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.880722, 0.611041, 0.142051], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #2}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.560181, 0.691569, 0.194885], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.560181, 0.691569, 0.194885], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #3}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.922526, 0.385626, 0.209179], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.922526, 0.385626, 0.209179], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #4}}, + GridBoxAlignment -> { + "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, + AutoDelete -> False, + GridBoxDividers -> { + "Columns" -> {{False}}, "Rows" -> {{False}}}, + GridBoxItemSize -> { + "Columns" -> {{All}}, "Rows" -> {{All}}}, + GridBoxSpacings -> { + "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, + GridBoxAlignment -> { + "Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> + False, GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], + "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}}, + AutoDelete -> False, + GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"], + Alignment -> Left, AppearanceElements -> None, + ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> + "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { + GrayLevel[0], FontSize -> 12, FontFamily -> "Arial"}, Background -> + Automatic, StripOnInput -> False], TraditionalForm]& ), + InterpretationFunction :> (RowBox[{"LineLegend", "[", + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> RGBColor[0.368417, 0.506779, 0.709798]|>, + "RGBColorSwatchTemplate"], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], + ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> RGBColor[0.880722, 0.611041, 0.142051]|>, + "RGBColorSwatchTemplate"], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], + ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> RGBColor[0.560181, 0.691569, 0.194885]|>, + "RGBColorSwatchTemplate"], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], + ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> 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Charting`ScaledTicks[{Log, Exp}, {Log, Exp}, "Nice", WorkingPrecision -> + MachinePrecision, RotateLabel -> 0]}, + AxesOrigin -> {0, -24.4628263145363}, FrameTicks -> {{ + Charting`ScaledTicks[{Log, Exp}, {Log, Exp}, "Nice", WorkingPrecision -> + MachinePrecision, RotateLabel -> 0], + Charting`ScaledFrameTicks[{Log, Exp}]}, {Automatic, Automatic}}, + GridLines -> {None, None}, DisplayFunction -> Identity, + PlotRangePadding -> {{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.05], + Scaled[0.05]}}, PlotRangeClipping -> True, ImagePadding -> All, + DisplayFunction -> Identity, + Method -> { + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> None, + "ClippingRange" -> {{{-5.999999755102041, + 5.999999755102041}, {-24.462826314536354`, -8.136370804376675}}, \ +{{-5.999999755102041, + 5.999999755102041}, {-24.462826314536354`, -8.136370804376675}}}}, + DisplayFunction -> Identity, AspectRatio -> + NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, + AxesLabel -> { + FormBox[ + TagBox[ + "\"\\!\\(\\*StyleBox[\\\"t\\\",FontSlant->\\\"Italic\\\"]\\)\\!\\(\\*\ +StyleBox[\\\" \ +\\\",FontSlant->\\\"Italic\\\"]\\)\\!\\(\\*SuperscriptBox[StyleBox[\\\"h\\\",\ +FontSlant->\\\"Italic\\\"], RowBox[{RowBox[{RowBox[{\\\"-\\\", \\\"1\\\"}], \ +\\\"/\\\", \\\"\[Beta]\\\"}], \\\" \\\", \\\"\[Delta]\\\"}]]\\)\"", HoldForm], + TraditionalForm], + FormBox[ + TagBox[ + "\"| \\!\\(\\*SubscriptBox[\\(\[ScriptCapitalF]\\), \\(0\\)]\\)\\!\\(\ +\\*SuperscriptBox[\\\"'\\\", StyleBox[RowBox[{\\\"[\\\", \\\"n\\\", \ +\\\"]\\\"}],FontSlant->\\\"Italic\\\"]]\\) - \\!\\(\\*SubscriptBox[\\(\ +\[ScriptCapitalF]\\), \\(0\\)]\\)\\!\\(\\*SuperscriptBox[\\('\\), \ +\\([6]\\)]\\) |\"", HoldForm], TraditionalForm]}, + AxesOrigin -> {0, -24.462826314536354`}, + CoordinatesToolOptions -> {"DisplayFunction" -> ({ + Part[#, 1], + Exp[ + Part[#, 2]]}& ), "CopiedValueFunction" -> ({ + Part[#, 1], + Exp[ + Part[#, 2]]}& )}, DisplayFunction :> Identity, + Frame -> {{False, False}, {False, False}}, + FrameLabel -> {{None, None}, {None, None}}, + FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, + GridLines -> {None, None}, GridLinesStyle -> Directive[ + GrayLevel[0.5, 0.4]], LabelStyle -> { + GrayLevel[0], FontSize -> 12}, + Method -> { + "DefaultBoundaryStyle" -> Automatic, + "DefaultGraphicsInteraction" -> { + "Version" -> 1.2, "TrackMousePosition" -> {True, False}, + "Effects" -> { + "Highlight" -> {"ratio" -> 2}, "HighlightPoint" -> {"ratio" -> 2}, + "Droplines" -> { + "freeformCursorMode" -> True, + "placement" -> {"x" -> "All", "y" -> "None"}}}}, + "DefaultMeshStyle" -> AbsolutePointSize[6], "ScalingFunctions" -> + None}, PlotRange -> {{-6, + 6}, {-24.462826314536354`, -8.136370804376675}}, PlotRangeClipping -> + True, PlotRangePadding -> {{ + Scaled[0.02], + Scaled[0.02]}, { + Scaled[0.02], + Scaled[0.02]}}, Ticks -> {Automatic, Automatic}}], + FormBox[ + FormBox[ + TemplateBox[{"2", "3", "4", "5"}, "LineLegend", + DisplayFunction -> (FormBox[ + StyleBox[ + StyleBox[ + PaneBox[ + TagBox[ + GridBox[{{ + StyleBox["\"n\"", { + GrayLevel[0], FontSize -> 12, FontFamily -> "Arial"}, + Background -> Automatic, StripOnInput -> False]}, { + TagBox[ + GridBox[{{ + TagBox[ + GridBox[{{ + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.368417, 0.506779, 0.709798], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.368417, 0.506779, 0.709798], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.880722, 0.611041, 0.142051], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.880722, 0.611041, 0.142051], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #2}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.560181, 0.691569, 0.194885], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.560181, 0.691569, 0.194885], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #3}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.922526, 0.385626, 0.209179], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.922526, 0.385626, 0.209179], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.038000000000000006`] -> + Baseline)], #4}}, + GridBoxAlignment -> { + "Columns" -> {Center, Left}, "Rows" -> {{Baseline}}}, + AutoDelete -> False, + GridBoxDividers -> { + "Columns" -> {{False}}, "Rows" -> {{False}}}, + GridBoxItemSize -> { + "Columns" -> {{All}}, "Rows" -> {{All}}}, + GridBoxSpacings -> { + "Columns" -> {{0.5}}, "Rows" -> {{0.8}}}], "Grid"]}}, + GridBoxAlignment -> { + "Columns" -> {{Left}}, "Rows" -> {{Top}}}, AutoDelete -> + False, GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> {"Columns" -> {{1}}, "Rows" -> {{0}}}], + "Grid"]}}, GridBoxAlignment -> {"Columns" -> {{Center}}}, + AutoDelete -> False, + GridBoxItemSize -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, + GridBoxSpacings -> { + "Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"], + Alignment -> Left, AppearanceElements -> None, + ImageMargins -> {{5, 5}, {5, 5}}, ImageSizeAction -> + "ResizeToFit"], LineIndent -> 0, StripOnInput -> False], { + GrayLevel[0], FontSize -> 12, FontFamily -> "Arial"}, Background -> + Automatic, 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fit to nth order from Table II." 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PointSize[0.5], + Opacity[1.], + RGBColor[0.880722, 0.611041, 0.142051], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.880722, 0.611041, 0.142051], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.1] -> Baseline)], #2}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.560181, 0.691569, 0.194885], + AbsoluteThickness[1.6]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + Opacity[1.], + RGBColor[0.560181, 0.691569, 0.194885], + AbsoluteThickness[1.6]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + 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RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], + ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> RGBColor[0.880722, 0.611041, 0.142051]|>, + "RGBColorSwatchTemplate"], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], + ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> RGBColor[0.560181, 0.691569, 0.194885]|>, + "RGBColorSwatchTemplate"], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}], + ",", + RowBox[{"Directive", "[", + RowBox[{ + RowBox[{"Opacity", "[", "1.`", "]"}], ",", + + TemplateBox[<| + "color" -> RGBColor[0.922526, 0.385626, 0.209179]|>, + "RGBColorSwatchTemplate"], ",", + RowBox[{"AbsoluteThickness", "[", "1.6`", "]"}]}], "]"}]}], + "}"}], ",", + RowBox[{"{", + RowBox[{#, ",", #2, ",", #3, ",", #4}], "}"}], ",", + RowBox[{"LegendMarkers", "\[Rule]", "None"}], ",", + RowBox[{"LabelStyle", 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IsingScalingFunctionExamples.nb | 3487 +++++++-------------------------------- 2 files changed, 612 insertions(+), 2877 deletions(-) diff --git a/IsingScalingFunction.wl b/IsingScalingFunction.wl index 074c451..81d5e47 100644 --- a/IsingScalingFunction.wl +++ b/IsingScalingFunction.wl @@ -1,6 +1,6 @@ BeginPackage["IsingScalingFunction`"] -InverseCoordinates::usage = "" +InverseCoordinates::usage = "Numerically convert Schofield coordinates to t and h." g::usage = "g[θ0, gs][θ] gives the Schofield coordinate transformation defined in (14)." diff --git a/IsingScalingFunctionExamples.nb b/IsingScalingFunctionExamples.nb index 39ec940..5327e14 100644 --- a/IsingScalingFunctionExamples.nb +++ b/IsingScalingFunctionExamples.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 667710, 12917] -NotebookOptionsPosition[ 661856, 12816] -NotebookOutlinePosition[ 662252, 12832] -CellTagsIndexPosition[ 662209, 12829] 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119.328 @@ -362,12 +360,12 @@ BT 7.2 0 0 -7.2 186.125859 72.102703 Tm /f-0-0 1 Tf (YL)Tj -10.8 0 0 -10.8 285.142031 104.944219 Tm +10.8 0 0 -10.8 285.038906 104.944219 Tm /f-1-1 1 Tf <01>Tj -7.2 0 0 -7.2 289.882031 106.691719 Tm -/f-2-0 1 Tf -(c)Tj +7.2 0 0 -7.2 289.778906 106.650469 Tm +/f-0-0 1 Tf +(0)Tj ET Q Q showpage -- cgit v1.2.3-70-g09d2 From 62deaa8b298eed218603462d27ffe516c1474b6c Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 16 May 2023 17:21:51 +0200 Subject: Also updated Mathematica files for generating figure. --- figs/figures.nb | 92 ++++++++++++++++++++++++++++++--------------------------- 1 file changed, 48 insertions(+), 44 deletions(-) diff --git a/figs/figures.nb b/figs/figures.nb index dd5ee3b..f2fddc9 100644 --- a/figs/figures.nb +++ b/figs/figures.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 710009, 12491] -NotebookOptionsPosition[ 706824, 12432] 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Date: Tue, 16 May 2023 17:52:47 +0200 Subject: Added explanations for the control variable figures. --- IsingScalingFunctionExamples.nb | 242 ++++++++++++++++++++++++++++++++-------- 1 file changed, 197 insertions(+), 45 deletions(-) diff --git a/IsingScalingFunctionExamples.nb b/IsingScalingFunctionExamples.nb index 5327e14..3d355ca 100644 --- a/IsingScalingFunctionExamples.nb +++ b/IsingScalingFunctionExamples.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 548992, 10652] -NotebookOptionsPosition[ 543628, 10559] -NotebookOutlinePosition[ 544025, 10575] -CellTagsIndexPosition[ 543982, 10572] +NotebookDataLength[ 554300, 10804] +NotebookOptionsPosition[ 548009, 10695] +NotebookOutlinePosition[ 548406, 10711] +CellTagsIndexPosition[ 548363, 10708] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -3965,14 +3965,14 @@ FontSlant->\\\"Italic\\\"], RowBox[{RowBox[{RowBox[{\\\"-\\\", \\\"1\\\"}], \ 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of control variables (", + StyleBox["t", + FontSlant->"Italic"], + " and ", + StyleBox["h", + FontSlant->"Italic"], + ")" +}], "Section", CellChangeTimes->{{3.887175666126995*^9, 3.887175672719225*^9}, - 3.8871757098402243`*^9},ExpressionUUID->"7bcdac80-37e1-4f66-bc64-\ + 3.8871757098402243`*^9, {3.893240714208487*^9, + 3.8932407443131866`*^9}},ExpressionUUID->"7bcdac80-37e1-4f66-bc64-\ b0d2db5bf4c3"], Cell[BoxData[{ @@ -7395,6 +7404,32 @@ Cell[BoxData[{ Cell[CellGroupData[{ +Cell["Entropy", "Subsection", + CellChangeTimes->{{3.893240903148159*^9, + 3.893240903699958*^9}},ExpressionUUID->"732869ab-e280-4316-b799-\ +ffab64c308f3"], + +Cell[TextData[{ + "In this plot, we show ", + Cell[BoxData[ + FormBox[ + FractionBox[ + RowBox[{"\[PartialD]", + SubscriptBox["u", "f"]}], + RowBox[{"\[PartialD]", + SubscriptBox["u", "t"]}]], TraditionalForm]], + FormatType->TraditionalForm,ExpressionUUID-> + "20c22e44-ae22-4cf1-ba9e-ec412d2155ac"], + ", which is the singular part of the entropy (modulo a 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file changed, 1 insertion(+) diff --git a/.gitignore b/.gitignore index 80d6041..089f9bd 100644 --- a/.gitignore +++ b/.gitignore @@ -9,5 +9,6 @@ *.fls *.out /*.pdf +figs/*eps-converted-to.pdf *.gnuploterrors gnuplottex/* -- cgit v1.2.3-70-g09d2 From 8280facaa0ce71199011ebd3101de02c89601798 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 25 May 2023 15:23:03 +0200 Subject: Updated affiliation and started referee response. --- ising_scaling.tex | 2 +- referee_response.tex | 148 +++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 149 insertions(+), 1 deletion(-) create mode 100644 referee_response.tex diff --git a/ising_scaling.tex b/ising_scaling.tex index 53f76ad..d7d2474 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -29,7 +29,7 @@ linkcolor=purple \title{Precision approximation of the universal scaling functions for the 2D Ising model in an external field} \author{Jaron Kent-Dobias} -\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\affiliation{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma} \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} diff --git a/referee_response.tex b/referee_response.tex new file mode 100644 index 0000000..7990a4f --- /dev/null +++ b/referee_response.tex @@ -0,0 +1,148 @@ +\documentclass[a4paper]{article} + +\usepackage{fullpage} +\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? +\usepackage[T1]{fontenc} % vector fonts plz +\usepackage{fullpage,amsmath,amssymb,latexsym,graphicx} +\usepackage{newtxtext,newtxmath} % Times for PR + +\begin{document} + +Response to referees + +\begin{verbatim} +---------------------------------------------------------------------- +Report of Referee A -- LK15589/Kent-Dobias +---------------------------------------------------------------------- + +New expressions of the scaling function of free energy, magnetization, +and magnetic susceptibility of the Ising model in a magnetic field are +proposed. These expressions are obtained by combining: + +- an essential singularity at zero magnetic field (as predicted by the +critical droplet theory), obtained by applying the Kramers-Kronig +relation to a scaling ansatz of the 'metastable free energy', + +- a parameterization (in the spirit of Schofield parameterization) in +terms of new scaling fields of the analytical part of the scaling +function. + +Even though both approaches have been introduced in the late 1960s, I +am not aware of any other attempt to combine them. This is the great +originality of this paper. The agreement of the proposed scaling +functions with the Monte Carlo data presented on figure 1 is +impressive. The improvement compared to the series expansion (8th +order plotted on figure 1) is undeniable. It seems to me that this +work constitutes a real progress in the field of critical phenomena. +In the presentation, the focus is put on the 2D Ising model but the +ideas could be applied to a broad class of systems where a continuous +transition lies at the end of first-order transition line. For these +reasons, I recommend the publication in Physical Review Letters. +Questions and comments follow. + +1. I did not find in Ref [3] the statement that the essential +singularity is not observable, as written by the authors. Could the +authors tell me at which page they found this statement? + +2. Before equation (1), some factors are missing in the expression of +the critical droplet size that should read $R_c={(d-1)\over d}{\Sigma +S_d\over M|H|V_d}$. + +3. The steps leading to the scaling functions (7) and (8) does not +seem to depend on any particular model but only on the dimension $d$ +and on the exponent $b$ describing the fluctuations of the spherical +critical droplet. I am therefore wondering if the same scaling +functions would also hold for models in different universality +classes, the 3-state Potts model for example. Could the authors +comment on this? + +4. In the particular case of the Ising model, $d=4$ is the upper +critical dimension. Could this affect the scaling function (8), for +example by the presence of logarithmic corrections? + +5. After equation (12), in the expression of $F(t,h)$, the term +$t^2\ln t^2$ cannot come from the integration of (10). Its presence +should be motivated. + +6. Did the authors try to produce the same comparison as in figure 1 +in the case of the 3D and 4D Ising model? + +7. There is no function $f$ in equation (13) as mentioned in the +sentence that follows. + +8. The presentation of the Schofield-like parameterization (page 3) is +really minimalist compared to the rest of the paper. I think that the +presentation of this part could (should?) be improved. What does +$\theta_c$ correspond to? Is it a free parameter? Why is (15) analytic +in the range $-\theta_c<\theta <\theta_c$? What is the interest? Why +this parameterization is more useful than the original scaling +variable? I understand that details will be given in a forthcoming +publication but more details would help the non-expert reader to +appreciate the interest of the approach. + +9. In the conclusion, the authors wrote ``We have developed a Wolff +algorithm for the Ising model in a field''. The idea of introducing a +ghost spin is not new. It is mentioned in R.H. Swendsen and J.S. Wang +(1987) \textit{Phys. Rev. Lett.} \textbf{58} 86 where it is attributed +to the original Fortuin-Kastelyn work from 1969. + +10. There is a minor typo in the acknowledgment: I guess that you want +to thank Jacques Perk. + +---------------------------------------------------------------------- +Report of Referee B -- LK15589/Kent-Dobias +---------------------------------------------------------------------- + +There are a variety of problems with this paper and it should not be +published. Since the authors will not agree with this I will attempt +to detail my objections: + +This paper appears to combine the droplet model picture from the 60's +with some renormalization group language and a computer computation +which is not explained and it is not clear what the authors are +willing to call an actual result. + +The two dimensional Ising model in a magnetic field has been studied +for decades and any further study must relate to these extensive +computations. This paper fails completely to do this. + +1. Several references are missing: + +S. N. Isakov, Comm. Math. Phys. (1984) 427-443 where the essential +singularity are the phase boundary is demonstrated. + +P. Fonseca and A. Zamolodchikov, J. Stat. Phys. 110 (2002) 527-590 +which gives a comprehensive scenario for the scaled free energy in the +critical region. + +A. Zamolodchikov and I Ziyaldinov, Nuclear Physics B849 (2011) 654-674 +where scattering in the Ising field theory is extensively discussed. + +2. Several references are clearly not understood. The authors state +the references 15-20 deal with an essential singularity in the +magnetic susceptibility whereas papers 15-20 are concerned with a +natural boundary in the susceptibility. Essential singularities are +isolated singularities, natural boundaries are not. The authors say +nothing about this natural boundary which is a major feature of the +analyticity of the model that must be explained. + +3. There are completely unsubstantiated claims made at the end of the +paper. It is said that "Our methods should allow improved +high-precision forms for the free energy." The results of references +15 and 16 have generated, used and analyzed series of hundreds and +thousands of terms. There is no reason to believe that anything in +this present paper will improve on this monumental work or on the work +of ref. 43. Statements such as "Our methods might be generalized to +predict similar singularities..." have no place in a scientific paper. + +4. The statement "Our forms both exhibit incorrect low-order +coefficients at the transition (Fig. 2) and incorrect asymptotics as +h|t|^{-\beta delta} becomes very large" does not inspire confidence in +the paper. + +In short, I cannot find anything in this paper which makes an advance +over the previous literature of 50 years. + +The paper should be rejected. +\end{verbatim} +\end{document} -- cgit v1.2.3-70-g09d2 From fd0b72927daa9df68e0f0c83437b1392a18830a5 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 25 May 2023 15:48:48 +0200 Subject: Draft of referee response complete. --- ising_scaling.bib | 16 ++++++++++- ising_scaling.tex | 2 +- referee_response.tex | 80 +++++++++++++++++++++++++++++++++++++++++++++++++++- 3 files changed, 95 insertions(+), 3 deletions(-) diff --git a/ising_scaling.bib b/ising_scaling.bib index ffc714c..0d0dc51 100644 --- a/ising_scaling.bib +++ b/ising_scaling.bib @@ -352,7 +352,7 @@ } @article{BarmaFisherPRB, - title={Two-dimensional Ising-like systems: Corrections to scaling in the Klauder and double-Gaussian models}, + title={Two-dimensional {Ising}-like systems: Corrections to scaling in the {Klauder} and double-{Gaussian} models}, author={Barma, Mustansir and Fisher, Michael E}, journal={Physical Review B}, volume={31}, @@ -363,3 +363,17 @@ } +@article{Isakov_1984_Nonanalytic, + author = {Isakov, S. N.}, + title = {Nonanalytic features of the first order phase transition in the Ising model}, + journal = {Communications in Mathematical Physics}, + publisher = {Springer Science and Business Media LLC}, + year = {1984}, + month = {12}, + number = {4}, + volume = {95}, + pages = {427--443}, + url = {https://doi.org/10.1007%2Fbf01210832}, + doi = {10.1007/bf01210832} +} + diff --git a/ising_scaling.tex b/ising_scaling.tex index d7d2474..00de5e3 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -207,7 +207,7 @@ literature \cite{Mangazeev_2010_Scaling, Clement_2019_Respect}. In the low temperature phase, the free energy has an essential singularity at zero field, which becomes a branch cut along the negative-$h$ axis when -analytically continued to negative $h$ \cite{Langer_1967_Theory}. The origin +analytically continued to negative $h$ \cite{Langer_1967_Theory, Isakov_1984_Nonanalytic}. The origin can be schematically understood to arise from a singularity that exists in the imaginary free energy of the metastable phase of the model. When the equilibrium Ising model with positive magnetization is subjected to a small diff --git a/referee_response.tex b/referee_response.tex index 7990a4f..01ece7a 100644 --- a/referee_response.tex +++ b/referee_response.tex @@ -8,7 +8,12 @@ \begin{document} -Response to referees +\section*{Response to referees for \texttt{LK15589/Kent-Dobias}} + +First, we would like to apologize for the large delay in resubmission. As is +evident, the manuscript has undergone a significant transformation as a result +of the reviews we received. We would like to thank the reviewers for their +helpful notes on the original manuscript. \begin{verbatim} ---------------------------------------------------------------------- @@ -43,11 +48,19 @@ Questions and comments follow. 1. I did not find in Ref [3] the statement that the essential singularity is not observable, as written by the authors. Could the authors tell me at which page they found this statement? +\end{verbatim} + +The comment has been removed. +\begin{verbatim} 2. Before equation (1), some factors are missing in the expression of the critical droplet size that should read $R_c={(d-1)\over d}{\Sigma S_d\over M|H|V_d}$. +\end{verbatim} +These equations are completely changed in the new manuscript. + +\begin{verbatim} 3. The steps leading to the scaling functions (7) and (8) does not seem to depend on any particular model but only on the dimension $d$ and on the exponent $b$ describing the fluctuations of the spherical @@ -55,7 +68,12 @@ critical droplet. I am therefore wondering if the same scaling functions would also hold for models in different universality classes, the 3-state Potts model for example. Could the authors comment on this? +\end{verbatim} +The observation of the referee is true, and these models could be studied with +a similar technique if sufficient data on their scaling functions is measured. + +\begin{verbatim} 4. In the particular case of the Ising model, $d=4$ is the upper critical dimension. Could this affect the scaling function (8), for example by the presence of logarithmic corrections? @@ -63,13 +81,31 @@ example by the presence of logarithmic corrections? 5. After equation (12), in the expression of $F(t,h)$, the term $t^2\ln t^2$ cannot come from the integration of (10). Its presence should be motivated. +\end{verbatim} + +We have now clarified both of these questions in part II, where the +relationship between flow equations and singularities in the free energy is +discussed. For the 4D model, the presence of a marginal variable dramatically +changes the analytic structure of the scaling function. +\begin{verbatim} 6. Did the authors try to produce the same comparison as in figure 1 in the case of the 3D and 4D Ising model? +\end{verbatim} + +We do not, though it would not be difficult to apply these techniques to the 3D +model. For the 4D, as mentioned above, some substantial changes would need to +be made to the parametric form. In addition, less data on the scaling functions +are available in 3D and especially 4D. +\begin{verbatim} 7. There is no function $f$ in equation (13) as mentioned in the sentence that follows. +\end{verbatim} + +This is no longer relevant to the modified manuscript. +\begin{verbatim} 8. The presentation of the Schofield-like parameterization (page 3) is really minimalist compared to the rest of the paper. I think that the presentation of this part could (should?) be improved. What does @@ -79,16 +115,29 @@ this parameterization is more useful than the original scaling variable? I understand that details will be given in a forthcoming publication but more details would help the non-expert reader to appreciate the interest of the approach. +\end{verbatim} +In the new manuscript, the treatment of the Schofield parameterization has now +been made central. + +\begin{verbatim} 9. In the conclusion, the authors wrote ``We have developed a Wolff algorithm for the Ising model in a field''. The idea of introducing a ghost spin is not new. It is mentioned in R.H. Swendsen and J.S. Wang (1987) \textit{Phys. Rev. Lett.} \textbf{58} 86 where it is attributed to the original Fortuin-Kastelyn work from 1969. +\end{verbatim} +Indeed true, numeric references have since been removed. + +\begin{verbatim} 10. There is a minor typo in the acknowledgment: I guess that you want to thank Jacques Perk. +\end{verbatim} + +The name has been corrected. +\begin{verbatim} ---------------------------------------------------------------------- Report of Referee B -- LK15589/Kent-Dobias ---------------------------------------------------------------------- @@ -117,7 +166,14 @@ critical region. A. Zamolodchikov and I Ziyaldinov, Nuclear Physics B849 (2011) 654-674 where scattering in the Ising field theory is extensively discussed. +\end{verbatim} + +We thank the referee for their helpful references, and we have cited the first +two. The second one was especially relevant to our study. We don't see the +relevance of the scattering to the scaling functions we study here, but perhaps +future work may examine it as well. +\begin{verbatim} 2. Several references are clearly not understood. The authors state the references 15-20 deal with an essential singularity in the magnetic susceptibility whereas papers 15-20 are concerned with a @@ -125,7 +181,14 @@ natural boundary in the susceptibility. Essential singularities are isolated singularities, natural boundaries are not. The authors say nothing about this natural boundary which is a major feature of the analyticity of the model that must be explained. +\end{verbatim} + +The natural boundary mentioned is purported to exist in the complex temperature +dependence susceptibility of the lattice Ising model. It is not clear to us +why it should be present in the scaling function of the free energy for the +Ising universality class. We have removed the inaccurate comments. +\begin{verbatim} 3. There are completely unsubstantiated claims made at the end of the paper. It is said that "Our methods should allow improved high-precision forms for the free energy." The results of references @@ -134,12 +197,27 @@ thousands of terms. There is no reason to believe that anything in this present paper will improve on this monumental work or on the work of ref. 43. Statements such as "Our methods might be generalized to predict similar singularities..." have no place in a scientific paper. +\end{verbatim} +We believe that our transformed technique and manuscript can substantiate this +claim, in a specific sense. Though the free energy computed point by point in +our references by Mangazeev et al. and Fonseca et al. are more accurate, they +are not functional forms: they are tables of data. We now show in the +manuscript that our functional form exponentially approaches the numeric values +of the scaling function and its derivatives measured in the aforementioned +works. + +\begin{verbatim} 4. The statement "Our forms both exhibit incorrect low-order coefficients at the transition (Fig. 2) and incorrect asymptotics as h|t|^{-\beta delta} becomes very large" does not inspire confidence in the paper. +\end{verbatim} +The asymptotic problems of the old manuscript have been repaired by treating +more carefully the parametric coordinates. + +\begin{verbatim} In short, I cannot find anything in this paper which makes an advance over the previous literature of 50 years. -- cgit v1.2.3-70-g09d2 From 90e8393b4069ace54c73975e3159aaf0db782ab7 Mon Sep 17 00:00:00 2001 From: jps6 Date: Thu, 25 May 2023 15:35:15 +0000 Subject: Update on Overleaf. --- referee_response.tex | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/referee_response.tex b/referee_response.tex index 01ece7a..b33d3b1 100644 --- a/referee_response.tex +++ b/referee_response.tex @@ -183,10 +183,11 @@ nothing about this natural boundary which is a major feature of the analyticity of the model that must be explained. \end{verbatim} -The natural boundary mentioned is purported to exist in the complex temperature -dependence susceptibility of the lattice Ising model. It is not clear to us -why it should be present in the scaling function of the free energy for the -Ising universality class. We have removed the inaccurate comments. +Our scaling function indeed does not show any evidence of a natural boundary or logarithmic corrections at complex temperatures in a field: we see only the branch cut of the dominant logarithmic singularity in the free energy. This is to be expected, because our calculation focuses on the universal scaling function as it depends upon the relevant variables $t$ and $h$, and does not incorporate singular corrections to scaling from irrelevant operators. + +The logarithmic corrections seen in the susceptibility are thought by these authors to come from singular corrections to scaling from these irrelevant operators. Furthermore, these logarithms are thought by Perk (private communication) to be associated with the lattice models, so they should not be seen in (say) the $\phi^4$ theory or membrane Ising phase transitions. + +We expect that a natural boundary in the susceptibility in the complex plane in the lattice model is due to these corrections to scaling, and thus should not be expected to manifest itself in the universal scaling function we calculate. \begin{verbatim} 3. There are completely unsubstantiated claims made at the end of the -- cgit v1.2.3-70-g09d2 From aab3c8e2b5fcad46e41f2f76e1dd342031f8afe0 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 26 May 2023 10:49:08 +0200 Subject: Edited the referee response. --- referee_response.tex | 27 ++++++++++++++++++++------- 1 file changed, 20 insertions(+), 7 deletions(-) diff --git a/referee_response.tex b/referee_response.tex index b33d3b1..0f2ce95 100644 --- a/referee_response.tex +++ b/referee_response.tex @@ -13,7 +13,22 @@ First, we would like to apologize for the large delay in resubmission. As is evident, the manuscript has undergone a significant transformation as a result of the reviews we received. We would like to thank the reviewers for their -helpful notes on the original manuscript. +helpful notes on the original manuscript. The first reviewer was supportive and +asked instructive questions. The second reviewer, though critical, led us to +some great insights. + +The manuscript now focuses on the approximation of the 2D Ising universal +scaling function by a smooth functional form. Though the singularity discussed +in the original manuscript still plays an important role, our approximation now +encompasses the whole parameter space of the relevant scaling fields. We +compare this form to the values of the universal scaling function and its +derivatives previously measured in the literature, and find exponential +convergence with the amount of data fit. + +We believe that the substantial changes to our manuscript merit its +reconsideration for publication. Though the new manuscript is so different from +the old one as to likely deserve a new reviewing cycle, we respond to the +original reviews here, to make clear how the revised manuscript addresses them. \begin{verbatim} ---------------------------------------------------------------------- @@ -169,9 +184,7 @@ where scattering in the Ising field theory is extensively discussed. \end{verbatim} We thank the referee for their helpful references, and we have cited the first -two. The second one was especially relevant to our study. We don't see the -relevance of the scattering to the scaling functions we study here, but perhaps -future work may examine it as well. +two. The second one was especially relevant to our study. \begin{verbatim} 2. Several references are clearly not understood. The authors state @@ -202,11 +215,11 @@ predict similar singularities..." have no place in a scientific paper. We believe that our transformed technique and manuscript can substantiate this claim, in a specific sense. Though the free energy computed point by point in -our references by Mangazeev et al. and Fonseca et al. are more accurate, they +our references by Mangazeev et al.\ and Fonseca et al.\ are more accurate, they are not functional forms: they are tables of data. We now show in the -manuscript that our functional form exponentially approaches the numeric values +manuscript that our functional form approaches the numeric values of the scaling function and its derivatives measured in the aforementioned -works. +works exponentially with iterative fitting. \begin{verbatim} 4. The statement "Our forms both exhibit incorrect low-order -- cgit v1.2.3-70-g09d2 From 6242cca8eb8c28e12c085cc67181f5a36a1cf844 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Sat, 27 May 2023 16:27:41 +0200 Subject: Some last changes to example file. --- IsingScalingFunctionExamples.nb | 48 +++++++++++++++++++++-------------------- 1 file changed, 25 insertions(+), 23 deletions(-) diff --git a/IsingScalingFunctionExamples.nb b/IsingScalingFunctionExamples.nb index 3d355ca..c522108 100644 --- a/IsingScalingFunctionExamples.nb +++ b/IsingScalingFunctionExamples.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 554300, 10804] -NotebookOptionsPosition[ 548009, 10695] -NotebookOutlinePosition[ 548406, 10711] -CellTagsIndexPosition[ 548363, 10708] +NotebookDataLength[ 554332, 10806] +NotebookOptionsPosition[ 548041, 10697] +NotebookOutlinePosition[ 548438, 10713] +CellTagsIndexPosition[ 548395, 10710] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -7399,8 +7399,10 @@ Cell[BoxData[{ RowBox[{"DufDut", "@@", RowBox[{"PrepareArgument", "[", RowBox[{"Data", "[", "6", "]"}], "]"}]}]}], ";"}]}], "Input", - 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