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-rw-r--r--.gitignore1
-rw-r--r--IsingScalingFunction.wl292
-rw-r--r--IsingScalingFunctionExamples.nb10814
-rw-r--r--figs/F_higher_singularities.eps343
-rw-r--r--figs/F_higher_singularities.pdfbin8097 -> 0 bytes
-rw-r--r--figs/F_lower_singularities.eps270
-rw-r--r--figs/F_lower_singularities.pdfbin6542 -> 0 bytes
-rw-r--r--figs/F_theta_singularities.eps374
-rw-r--r--figs/F_theta_singularities.pdfbin8379 -> 0 bytes
-rw-r--r--figs/contour_path.eps440
-rw-r--r--figs/contour_path.pdfbin9494 -> 0 bytes
-rw-r--r--figs/figures.nb2399
-rw-r--r--ising_scaling.bib36
-rw-r--r--ising_scaling.tex62
-rw-r--r--referee_response.tex240
15 files changed, 15151 insertions, 120 deletions
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/*
diff --git a/IsingScalingFunction.wl b/IsingScalingFunction.wl
new file mode 100644
index 0000000..81d5e47
--- /dev/null
+++ b/IsingScalingFunction.wl
@@ -0,0 +1,292 @@
+BeginPackage["IsingScalingFunction`"]
+
+InverseCoordinates::usage = "Numerically convert Schofield coordinates to t and h."
+
+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."
+
+DScriptF0Dη::usage =
+ "DScriptF0Dη computes the mth derivative 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 =
+ "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 = "Fixes B given other data as in (38)."
+
+ruleC0::usage = "Fixes C0 given other data as in (39)."
+
+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 = "θ0 from Caselle et al."
+
+gsCas::usage = "g function coefficients from Caselle et al."
+
+Φs::usage = "List of numeric coefficients for the scaling function F_0"
+
+Gls::usage = "List of numeric coefficients for the scaling function F_-"
+
+Ghs::usage = "List of numeric coefficients for the scaling function F_+"
+
+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]
+}
+
+Data[2] = Rationalize[#, 10^-20] & /@ <|
+ "θ0" -> 1.148407773492004`,
+ "θYL" -> 0.9896669889911205`,
+ "CYL" -> -0.172823989504767`,
+ "Gs" -> {-0.31018352388662596`, 0.2474537923130002`},
+ "gs" -> {0.37369093055254343`, -0.021636313152585823`}
+|>
+
+Data[3] = Rationalize[#, 10^-20] & /@ <|
+ "θ0" -> 1.2542120477507488`,
+ "θYL" -> 0.6020557328641167`,
+ "CYL" -> -0.38566364361428684`,
+ "Gs" -> {-0.3527514794812415`, 0.2582430860166863`},
+ "gs" -> {0.4483788209731592`, -0.022032295172535358`, 0.00022200608228654115`}
+|>
+
+Data[4] = Rationalize[#, 10^-20] & /@ <|
+ "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] = Rationalize[#, 10^-20] & /@ <|
+ "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] = Rationalize[#, 10^-20] & /@ <|
+ "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]]
+
+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
+
+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_] := 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][1&][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)
+
+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]
+
+θ0Cas := Sqrt[1.16951];
+
+h0Cas := a b ρ /. {
+ a -> c2^2/c4,
+ b -> (-c4/c2^3)^(1/2),
+ ρ -> 2.00881
+ } /. {
+ c2->Ghs[[3]] 2!,
+ c4->Ghs[[5]]["Value"] 4!
+ }
+
+gsCas := h0Cas * {
+ 1,
+ -0.222389,
+ -0.043547,
+ -0.014809,
+ -0.007168
+ }
+
+m0Cas := -Ghs[[3]]2! h0Cas
+
+DScriptMCasDξList[m_, θ_] := CompositeFunctionDerivativeList[
+ Identity, Function[θp, m0Cas * θp / RealAbs[θp^2 - 1]^β], ξ[θ0Cas, gsCas]
+ ][m, θ]
+
+End[]
+
+EndPackage[]
+
diff --git a/IsingScalingFunctionExamples.nb b/IsingScalingFunctionExamples.nb
new file mode 100644
index 0000000..c522108
--- /dev/null
+++ b/IsingScalingFunctionExamples.nb
@@ -0,0 +1,10814 @@
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+
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+Cell[CellGroupData[{
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diff --git a/ising_scaling.bib b/ising_scaling.bib
index 4185f2f..11ae107 100644
--- a/ising_scaling.bib
+++ b/ising_scaling.bib
@@ -340,6 +340,42 @@
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}
+}
+
+
+@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}
+}
info: 'Griffiths_1967' has been autocompleted into 'Griffiths_1967_Thermodynamic'.
@article{Griffiths_1967_Thermodynamic,
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 2b1fb62..7fa1ae6 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -26,10 +26,10 @@ 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}
+\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}
@@ -46,8 +46,8 @@ 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.
- To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy.
+ matched, up to seven digits of 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}
@@ -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
@@ -99,9 +100,10 @@ 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}
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 +132,16 @@ $\delta=15$ 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{align}
+ \label{eq:AnalyticCOV}
+ 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}
\frac{du_t}{d\ell}=\frac1\nu u_t
@@ -149,6 +159,7 @@ 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 \\
@@ -163,7 +174,7 @@ $\mathcal F_\pm(\xi)=G_{\mathrm{high}/\mathrm{low}}(\xi)$.}. 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 $\eta=u_t|u_h|^{-1/\Delta}$.
@@ -196,7 +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
@@ -235,7 +246,7 @@ 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
@@ -282,7 +293,7 @@ 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
@@ -417,7 +428,7 @@ 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
@@ -484,7 +495,7 @@ $\theta$. 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
@@ -541,7 +552,7 @@ 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
@@ -594,6 +605,7 @@ 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 +623,7 @@ 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
@@ -1101,7 +1114,7 @@ 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}
@@ -1197,14 +1210,21 @@ 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 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
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 $\pm H$, $T<T_c$.
+
+Finally, the successful smooth description of the Ising free energy produced in part by
analytically continuing the singular imaginary part of the metastable free
energy inspires an extension of this work: a smooth function that captures the
universal scaling \emph{through the coexistence line and into the metastable
diff --git a/referee_response.tex b/referee_response.tex
new file mode 100644
index 0000000..0f2ce95
--- /dev/null
+++ b/referee_response.tex
@@ -0,0 +1,240 @@
+\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}
+
+\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. 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}
+----------------------------------------------------------------------
+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?
+\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
+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?
+
+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
+$\theta_c$ 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.
+\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
+----------------------------------------------------------------------
+
+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.
+\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.
+
+\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
+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}
+
+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
+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.
+\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 approaches the numeric values
+of the scaling function and its derivatives measured in the aforementioned
+works exponentially with iterative fitting.
+
+\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.
+
+The paper should be rejected.
+\end{verbatim}
+\end{document}