ORIGINAL ARTICLE: Gökçe Başar, Gerald V. Dunne, and Zelong Yin (2022), Uniformizing Lee-Yang singularities, Phys. Rev. D 105, 105002. https://doi.org/10.1103/PhysRevD.105.105002
Article Abstract: Motivated by the search for the QCD critical point, we discuss how to obtain the singular behavior of a thermodynamic system near a critical point, namely the Lee-Yang singularities, from a limited amount of local data generated in a different region of the phase diagram. We show that by using a limited number of Taylor series coefficients, it is possible to reconstruct the equation of state past the radius of convergence, in particular in the critical region. Furthermore we also show that it is possible to extend this reconstruction to go from a crossover region to the first-order transition region in the phase diagram, using a uniformizing map to pass between Riemann sheets. We illustrate these ideas via the chiral random matrix model and the Ising model.
Based on work in collaboration with G. V. Dunne and Z. Yin (references below)

Overview

The main goal of this work is to improve the Pade resummation which approximately reconstructs the underlying singularities of a function f(x) from a truncated Taylor series expansion with finite number of Taylor coefficients:
f(x)≃
c
0
+
c
1
x+
c
2
2
x
+…+
c
N
N
x
. Pade resummation approximates this Polynomial by a rational function whose coefficients are determined by matching with
c
n
s
. However Pade resummation cannot approximate the function beyond its radius of convergence.
​
The strategy we employ is to conformally map a multi-sheeted Riemann surface where the function is assumed to be defined over, into a compact space, such as the interior of the unit disk, and perform Pade approximation in the new domain.
​
This file visualizes the conformal maps we have used in the references below. The function that we focus on is the equation of state associated with a strongly interacting many-body system. The singularity structure of the equation of state is determined by the Lee-Yang edge singularities that appear as a complex conjugate pair in the complex plane of our variable, the chemical potential. The conformal maps we choose are tailored to make maximum use of this physical input.
​
Throughout the file
μ2
, which stands for the variable
2
μ
,
denotes the argument of the original function. The equation of state is naturally defined as a function of
2
μ
.
z or ζ denotes the coordinate of the mapped domain. The conformal map is defined by
2
μ
=ϕ(z).

Definitions

MyBlue=RGBColor[{78,99,160}/255];​​MyRed=RGBColor[{125,55,47}/255];​​MyGreen=RGBColor[{57,102,100}/255];​​MyBluePale=RGBColor[{168,177,197}/255];​​MyRedPale=RGBColor[{185,157,152}/255];​​MyGreenPale=RGBColor[{181,194,193}/255];

Conformal maps

Here we define the conformal maps we employ.
ϕ
2
(z)
stands for the 2-cut map and
ϕ
uni
(z)
stands for the uniformizing map. We denote the location of the singularity in the original domain as
ω
c
.
The image of the singularity is denoted by
ζ
2
(
ζ
uni
)
for the two-cut (uniformizing) map.
(*choosethesingularityintheμ2planeforthetwo-cutmapasanillustration*)​​θc=π/6;​​μ2c=
432
100
θc

;​​(*thetwo-cutmap*)​​ϕ2[z_,ωc_:μ2c]=4Abs@ωc
Arg@ωc/π
Arg@ωc
π
1-Arg@ωc/π
1-
Arg@ωc
π
z
2
(1+z)
2Arg@ωc/π
1+z
1-z
;​​ϕ2inv[ω_,ωc_:μ2c]:=z/.NSolve[ϕ2[z,ωc]ω,z];​​​​ζ2={{ϕ2inv[μ2c,μ2c]〚2〛},{ϕ2inv[Conjugate@μ2c,μ2c]〚2〛}};​​thetac=Arg@ζ2〚1,1〛;​​​​(*theuniformizingmap*)​​ϕuni[z_]=-1+2ModularLambda
1-(-)z
1+(-)z
;​​ϕuniinv[p_]=
EllipticK
1-p
2
-EllipticK
1+p
2

EllipticK
1-p
2
+EllipticK
1+p
2

;​​​​(*centersof1stcircles*)​​z01={1+,1-,-1-,-1+};​​(*boundariesofthe1stsheet(1stcircles)*)​​c1[θ_]=z01+
θ

,
(θ-π/2)

,
-(θ+π/2)

,
(θ+π/2)

;​​(*schwarzreflection*)​​zR[z_,z0_]:=z0+Conjugate
1
z-z0
;​​(*Schwartzreflectionwrt1stcircles*)​​c2[θ_]=Table[zR[#,z01〚i〛]&@c1[θ],{i,4}];​​​​(*Mobiustransformationanditsinverse*)​​fmob[z_]=
1-(-)z
1+(-)z
;​​fmobinv[f_]=
-f
+f
;​​​​S[τ_]:=-1/τ;​​​​(*Cutsandintheoriginaldomain,colorconventionsetc...*)​​cuts=Show[ParametricPlot[{{0,y},{0,-y}},{y,-11,-1},PlotStyle{{MyBlue,Thickness[0.01]},{MyRed,Thickness[0.01]}}],PlotRange{{-5,5},{-5,5}},AspectRatio->Automatic];​​cuts2=Show[ParametricPlot[{{0,y},{0,-y}},{y,-11,-1},PlotStyle{{MyBlue,Thickness[0.007],Dashed},{MyRed,Thickness[0.01]}}],PlotRange{{-5,5},{-5,5}},AspectRatio->Automatic];​​cuts3=Show[ParametricPlot[{{0,y},{0,-y}},{y,-11,-1},PlotStyle{{MyBlue,Thickness[0.01]},{MyRed,Thickness[0.007],Dashed}}],PlotRange{{-5,5},{-5,5}},AspectRatio->Automatic];​​​​stylec1={{MyRed,Thickness[0.007]},{MyBlue,Thickness[0.007]},{MyBlue,Thickness[0.005]},{MyRed,Thickness[0.007]}};​​stylec12={{MyRed,Thickness[0.007]},{MyBlue,Thickness[0.007],Dashed},{MyBlue,Thickness[0.005],Dashed},{MyRed,Thickness[0.007]}};​​stylec13={{MyRed,Thickness[0.005],Dashed},{MyBlue,Thickness[0.007]},{MyBlue,Thickness[0.007]},{MyRed,Thickness[0.005],Dashed}};​​stylereim={{Black,Thickness[0.007],Dashed},{Darker@Green,Thickness[0.007],Dashed}};​​range[x_,y_]=x,-1+
2y-
2
y
,1-
2y-
2
y
;

Graphical representation of the conformal maps

2-cut map

Here we show the singularity structure in the
2
μ
plane where there are two radial cuts emanating from the a pair of complex conjugate singularities and how they are mapped into the unit disk via the two-cut map.
In[]:=
μ2plane=ShowParametricPlotReIm[
(θc)

r],ReIm[
-θc

r],{r,1,20},TicksNone,PlotStyle{{Dashed,MyBlue,Thickness[0.01]},{Dashed,MyRed,Thickness[0.01]}},PlotRange->{{-2.5,2.5},{-2.3,2.3}},​​​​EpilogInset[Style["
2
μ
B
",16],{-2.2,1.85}],​​InsetStyle"⏌",28,{-1.8,1.7},​​Inset[Style["
2
μ
LY
",14],{0.9,1.2}],​​Inset[Style["
*
2
μ
LY
",14],{0.9,-1.2}]​​,​​ParametricPlotReIm[
(θc+0.1)

+
(θc)

r],ReIm[
-(θc+0.1)

+
-(θc)

r],{r,0,20},TicksNone,PlotStyle{{MyBlue,Thickness[0.01]},{MyRed,Thickness[0.01]}},PlotRange->{{-3.2,3.2},{-2.5,2.5}},​​ListPlot[{{0,0}},PlotStyle{{Black,PointSize[0.03]}}],​​ListPlotReIm@{
(θc+0.05)

},{
-(θc+0.05)

},PlotStyle{{MyBlue,PointSize[0.035]},{MyRed,PointSize[0.035]}},​​Plot[0,{x,-10,10},PlotStyle{Black}],ImageSize200;​​​​z2plane=ShowGraphics[{Opacity[0],Circle[{0,0},1]}],​​ParametricPlotReIm[
θ

],ReIm[
-θ

],{θ,0,thetac},TicksNone,PlotStyle​​{{Dashed,MyBlue,Thickness[0.01]},{Dashed,MyRed,Thickness[0.01]}},PlotRange->{{-1.01,1.01},{-1.01,1.01}},AxesFalse,​​​​EpilogInset[Style["ζ",16],{-0.9,0.9}],​​InsetStyle"⏌",25,{-0.8,0.86},​​Inset[Style["
ζ
LY
",14],{0.9,0.85}],​​Inset[Style["
*
ζ
LY
",14],{0.9,-0.85}]​​​​​​,​​ParametricPlotReIm[
θ

],ReIm[
-θ

],{θ,thetac,π},TicksNone,PlotStyle{{MyBlue,Thickness[0.01]},{MyRed,Thickness[0.01]}},​​ListPlotReIm@
2ArcSin
θc/π


,
-2ArcSin
θc/π


,PlotStyle{{MyBlue,PointSize[0.03]},{MyRed,PointSize[0.03]}},​​ListPlot[{{0,0}},PlotStyle{{Black,PointSize[0.03]}}],​​ParametricPlot[ReIm[x],{x,-1,1},PlotStyle{Black}],​​ImageSize210,​​EpilogInset[Style["ζ",16],{-0.9,0.9}],​​InsetStyle"⏌",25,{-0.8,0.86},​​InsetStyle"
ζ
LY
=

ψ
2

",14,{1,.9},​​Inset[Style["
*
ζ
LY
",14],{0.85,-0.87}],ImagePadding->{{0,25},{0,0}};

Uniformizing map

1st Riemann Sheet

Here we show the mapping of the first Riemann sheet into a subspace of the fundamental domain (and the unit disk) . The red and blue solid lines denote the cuts and the color coded curves are drawn to illustrate the map .

2nd Riemann Sheet

The image of the second Riemann sheet can be obtained from a Schwarz reflection from the first Riemann sheet
Here we show the images of both Riemann sheets

Analytical continuation from the first to the second sheet

Here we show two different trajectories that pass through a branch cut to a different Riemann sheet in the original domain. Their image on the unit disk is a smooth curve with no discontinuity. This property of the conformal map is the reason for its name. The process of jumping to a different Riemann sheet is represented by a smooth trajectory in the unit disk. In other words the entire multi-sheeted domain is mapped inside the unit disk in a uniform way.

References

G . Basar, G.V. Dunne, Z. Yin, Uniformizing Lee-Yang singularities, Phys. Rev. D 105 (2022) 10, 105002 , e - Print : arXiv: 2112.14269[hep - th]
G . Basar, On the QCD critical point, Lee - Yang edge singularities and Pade resummations, e - Print : : arXiv: 2312.06952[hep - th]
G . Basar, Universality, Lee - Yang Singularities, and Series Expansions, Phys. Rev. Lett. 127 (2021) 17, 171603, e - Print : arXiv: 2105.08080 [hep - th]

CITE THIS NOTEBOOK

Uniformizing Lee-Yang singularities​
by Gökçe Başar, Gerald V. Dunne, and Zelong Yin
Wolfram Community, STAFF PICKS, February 2, 2024
​https://community.wolfram.com/groups/-/m/t/3115040