Please note that before starting you need to fix the Working precision (wpc). Normaly 30 is enough;

Same for AccuracyGoal (acc = 5 is enough).

wpc=40;
$MinPrecision=wpc;
acc=7;

You may load our results (see section results).

Get["EE_GPPZ_.mx"];

Understanding GPPZ background

Background geometry

SetDirectory[NotebookDirectory[]]

In the first part of this notebook I check that the (second order) equations of motion are solved by the BPS ones. Then we find the background corresponding to a given superpotential.

dρdz=-L/z/.z->(E^(-(ρ/L))) L;
changecoordzToρ={func_''[z]->dρdz D[dρdz func'[ρ],ρ],func_'[z]->dρdz func'[ρ],func_[z]->func[ρ]};

dzdρ=(((-L/z))^(-1));
changecoordρToz={func_''[ρ]->dzdρ D[dzdρ func'[z],z],func_'[ρ]->dzdρ func'[z],func_[ρ]->func[z]};

{f''[z]->(f[z] (-171 (z^3) g[z] ((g'[z])^3)+18 (z^4) ((g'[z])^4)+6 (z^2) ((g[z])^2) ((g'[z])^2) (102+6 (L^2) V[ϕ[z]]-5 (z^2) ((ϕ'[z])^2))+12 z ((g[z])^3) g'[z] (-81-18 (L^2) V[ϕ[z]]+13 (z^2) ((ϕ'[z])^2))+8 ((g[z])^4) (72+4 (L^4) ((V[ϕ[z]])^2)-24 (z^2) ((ϕ'[z])^2)+(z^4) ((ϕ'[z])^4)-4 (L^2) V[ϕ[z]] (-9+(z^2) ((ϕ'[z])^2)))))/(9 (z^2) ((g[z])^2) (((-2 g[z]+z g'[z]))^2)),g''[z]->((9 z g'[z]-2 g[z] (6+2 (L^2) V[ϕ[z]]+(z^2) ((ϕ'[z])^2)))/(3 (z^2))),ϕ''[z]->(-3 (z^3) ((g'[z])^2) ϕ'[z]+3 z g[z] g'[z] ((L^2) V'[ϕ[z]]+3 z ϕ'[z])-2 ((g[z])^2) (3 (L^2) V'[ϕ[z]]+z ϕ'[z] (3-2 (L^2) V[ϕ[z]]+(z^2) ((ϕ'[z])^2))))/(3 (z^2) g[z] (-2 g[z]+z g'[z]))}/.Rule[x_,y_]->x-y/.f->(g[#]&)/.V->Function[ϕ,Simplify[( (-4)(1/(3 )) ((W[ϕ])^2)+(1/2) ((W'[ϕ])^2))]]/.g->Function[z,((z^2)/(L^2))Exp[2A[z]]]/.changecoordzToρ/.z->  (E^(-(ρ/L))) L//.{A''[ρ]->-2/3 W'[ϕ[ρ]] ϕ'[ρ],
ϕ''[ρ]-> ϕ'[ρ] W''[ϕ[ρ]],
A'[ρ]->-2/3 W[ϕ[ρ]],
ϕ'[ρ]->   D[W[ϕ[ρ]],ϕ[ρ]]}//Simplify

The following superpotential allows to interpolate between confining and non-confining IR, see https://arxiv.org/abs/2411.17806
For a particular choice of γ =  1/(Sqrt[3]) it gives GPPZ.

ruleW={W->Function[ϕ,((1-6 (γ^2)-Cosh[2 γ ϕ])/(4 L (γ^2)))]};

Series[V[ϕ]/.V->Function[ϕ,Simplify[( (-4)(1/(3 )) ((W[ϕ])^2)+(1/2) ((W'[ϕ])^2))]]/.ruleW,{ϕ,0,3}]

The system can be solved analytically for any γ

ϕ'[ρ]== D[W[ϕ[ρ]],ϕ[ρ]]/.ruleW;
DSolve[%,ϕ,ρ]//Flatten
Assuming[0<z&&L>0&&ϕ1>0,ϕ[ρ]/.%[[1]]/.ρ->L Log[(L/z)]//Simplify]
Assuming[0<z&&L>0&&ϕ1>0,Series[%,{z,0,2}]//Simplify]

Series[W[ϕ]/.ruleW,{ϕ,0,4}]
Series[W[ϕ]/.W-> Function[ϕ,-((1-6 (γ^2)-Cosh[2 γ ϕ])/(4 L (γ^2)))],{ϕ,0,4}]

Also in the z coordinate

or=5;

ϕ'[ρ]== D[W[ϕ[ρ]],ϕ[ρ]]/.changecoordρToz/.ruleW//Simplify
DSolve[%,ϕ,z]//Simplify//Flatten
Assuming[ϕ0>0&&z>0&&Λ>0,ϕ[z]/.%/.(C[1])->((Log[γ Λ])/(2 γ))//FullSimplify]
Assuming[ϕ0>0&&z>0&&Λ>0,Series[%,{z,0,3}]]

Block[{Λ=1,γ=1},Plot[((ArcTanh[z γ Λ])/γ),{z,0,Λ/γ}]]

A'[ρ]+2/3 W[ϕ[ρ]]/.changecoordρToz/.ruleW/.ϕ->Function[z,((ArcTanh[z γ Λ])/γ)]//Simplify
DSolve[%==0,A,z]/.(C[1])->(1/2) Log[(L^2) (((γ Λ))^(1/(3 (γ^2))))]//Flatten//Simplify

Assuming[z>0,Series[Exp[2 A[z]]/.%,{z,0,1}]//FullSimplify]

Assuming[γ>0&&z>0&&Λ>0&&L>0,Simplify[Exp[2A[z]]/.{A->Function[{z},(1/2) Log[(L^2) (((γ Λ))^(1/(3 (γ^2))))]-Log[z]+((Log[z])/(6 (γ^2)))-((Log[((z γ Λ)/(Sqrt[1-(z^2) (γ^2) (Λ^2)]))])/(6 (γ^2)))+((Log[1-(z^2) (γ^2) (Λ^2)])/(12 (γ^2)))]}]//FullSimplify]

So we have

ruleWarpFactor={e2A->Function[{z},(((L^2) (((1-(z^2) (γ^2) (Λ^2)))^(1/(3 (γ^2)))))/(z^2))]}

Assuming[ϕ0>0&&z>0&&Λ>0&&LIR>0,Simplify[Series[e2A[z]/.ruleWarpFactor,{z,0,4}]]]

((dρdz)^2)/.ρ->L Log[(L/z)]//Simplify

Expressions for GPPZ

Now we focus in the case of GPPZ:

W[ϕ]/.ruleW/.γ-> 1/(Sqrt[3])

ϕ[z]/.ϕ->Function[z,((ArcTanh[z γ Λ])/γ)]/.γ-> 1/(Sqrt[3])//Simplify//Simplify
Series[%,{z,0,3}]

e2A[z]/.ruleWarpFactor/.γ-> 1/(Sqrt[3])//Simplify

Spheres

Equation that we want to solve

Assuming[R[z]>0&&e2A[z]>0,
ℒ=(((Exp[4A[z]]((R[z])^4)(Exp[2A[z]]((R'[z])^2)+((L^2)/(z^2)))))^(1/2))/.A->( (1/2)Log[e2A[#]]&)//Simplify]

D[ℒ,R[z]]-D[D[ℒ,R'[z]],z]//Simplify
eomR=(Solve[%==0,R''[z]]//Simplify//Flatten)[[1]]/.Rule[x_,y_]->x-y

Boundary conditions: close to the UV

Note the appearance of logarithms in the UV expansions:

orUV=10;
Assuming[ϕ0>0,eomR/.ruleWarpFactor/.γ-> 1/(Sqrt[3])/.R->Function[z,( (∑_(kk=0)^orUV)(cR[kk]+cRl[kk]Log[z]+cRl2[kk]((Log[z])^2)+cRl3[kk]((Log[z])^3))(z^kk)+((O[z,0])^(orUV+1)))/.{cR[1]->0,cR[2]->-(1/(2 cR[0])),cR[3]->0,cRl[4]->-((Λ^2)/(6 cR[0])),cR[5]->0,cRl[5]->0,cR[6]->((81+8 (Λ^4) ((cR[0])^4)+972 ((cR[0])^3) cR[4]+9 (Λ^2) ((cR[0])^2) (9+32 ((cR[0])^3) cR[4]))/(648 ((cR[0])^5))),cRl[6]->-(((Λ^2) (27+8 (Λ^2) ((cR[0])^2)))/(108 ((cR[0])^3))),cR[7]->0,cRl[7]->0,cR[8]->(1/(20736 ((cR[0])^7)))(120 (Λ^6) ((cR[0])^6)+9 (Λ^2) ((cR[0])^2) (405+1376 ((cR[0])^3) cR[4])+2 (Λ^4) ((cR[0])^4) (631+1728 ((cR[0])^3) cR[4])+162 (17+120 ((cR[0])^3) cR[4]-448 ((cR[0])^6) ((cR[4])^2))),cRl[8]->-(((Λ^2) (135+86 (Λ^2) ((cR[0])^2)+24 (Λ^4) ((cR[0])^4)-1008 ((cR[0])^3) cR[4]))/(864 ((cR[0])^5))),cRl2[8]->-((7 (Λ^4))/(72 ((cR[0])^3))),cR[9]->0,cRl[9]->0,cRl2[9]->0,cR[10]->(1/(25920000 ((cR[0])^9)))(57600 (Λ^8) ((cR[0])^8)+24 (Λ^6) ((cR[0])^6) (24491+64000 ((cR[0])^3) cR[4])+10 (Λ^4) ((cR[0])^4) (174123+344704 ((cR[0])^3) cR[4])-225 (Λ^2) ((cR[0])^2) (-11613+63072 ((cR[0])^3) cR[4]+355328 ((cR[0])^6) ((cR[4])^2))+20250 (73-808 ((cR[0])^3) cR[4]-12864 ((cR[0])^6) ((cR[4])^2)+4096 ((cR[0])^9) ((cR[4])^3))),cRl[10]->-(1/(1944000 ((cR[0])^7)))(43088 (Λ^6) ((cR[0])^4)+19200 (Λ^8) ((cR[0])^6)-90 (Λ^4) ((cR[0])^2) (1971+22208 ((cR[0])^3) cR[4])+2025 (Λ^2) (-101-3216 ((cR[0])^3) cR[4]+1536 ((cR[0])^6) ((cR[4])^2))),cRl2[10]->-(((Λ^4) (9045+2776 (Λ^2) ((cR[0])^2)-8640 ((cR[0])^3) cR[4]))/(32400 ((cR[0])^5))),cRl3[10]->-((2 (Λ^6))/(135 ((cR[0])^3)))}/.Table[cRl[kk]->0,{kk,0,3}]/.Table[cRl2[kk]->0,{kk,0,7}]/.Table[cRl3[kk]->0,{kk,0,9}]]//Simplify]//Normal//CoefficientList[#,Log[z]]&
Solve[%==0,{cR[orUV],cRl[orUV],cRl2[orUV],cRl3[orUV]}]//Simplify

R->Function[z,( (∑_(kk=0)^4)(cR[kk]+cRl[kk]Log[z]+cRl2[kk]((Log[z])^2)+cRl3[kk]((Log[z])^3))(z^kk)+((O[z,0])^(orUV+1)))/.{cR[1]->0,cR[2]->-(1/(2 cR[0])),cR[3]->0,cRl[4]->-((Λ^2)/(6 cR[0])),cR[5]->0,cRl[5]->0}/.Table[cRl[kk]->0,{kk,0,3}]/.Table[cRl2[kk]->0,{kk,0,7}]/.Table[cRl3[kk]->0,{kk,0,9}]];
(((24 R[z]))^(-1))(R[z]R''''[z]+4  (Λ^2) Log[z]+((25 (Λ^2))/3))//Simplify
Series[%/.%%,{z,0,0}]//Simplify

{R[z],R'''' [z]}/.R->Function[z,( (∑_(kk=0)^4)(cR[kk]+cRl[kk]Log[z]+cRl2[kk]((Log[z])^2)+cRl3[kk]((Log[z])^3))(z^kk)+((O[z,0])^(orUV+1)))/.{cR[1]->0,cR[2]->-(1/(2 cR[0])),cR[3]->0,cRl[4]->-((Λ^2)/(6 cR[0])),cR[5]->0,cRl[5]->0}/.Table[cRl[kk]->0,{kk,0,3}]/.Table[cRl2[kk]->0,{kk,0,7}]/.Table[cRl3[kk]->0,{kk,0,9}]]/.cR[4]->R4/.cR[0]->R0//Normal
relationsR0R4=Solve[{R[z],R' [z]}==%,{R0,R4}]//Simplify

{R[z],R' [z]}=.

Clear[RUV]
RUV[Λ_,R0_,R4_,z_]=( (∑_(kk=0)^orUV)(cR[kk]+cRl[kk]Log[z]+cRl2[kk]((Log[z])^2)+cRl3[kk]((Log[z])^3))(z^kk))/.{cR[1]->0,cR[2]->-(1/(2 cR[0])),cR[3]->0,cRl[4]->-((Λ^2)/(6 cR[0])),cR[5]->0,cRl[5]->0,cR[6]->((81+8 (Λ^4) ((cR[0])^4)+972 ((cR[0])^3) cR[4]+9 (Λ^2) ((cR[0])^2) (9+32 ((cR[0])^3) cR[4]))/(648 ((cR[0])^5))),cRl[6]->-(((Λ^2) (27+8 (Λ^2) ((cR[0])^2)))/(108 ((cR[0])^3))),cR[7]->0,cRl[7]->0,cR[8]->(1/(20736 ((cR[0])^7)))(120 (Λ^6) ((cR[0])^6)+9 (Λ^2) ((cR[0])^2) (405+1376 ((cR[0])^3) cR[4])+2 (Λ^4) ((cR[0])^4) (631+1728 ((cR[0])^3) cR[4])+162 (17+120 ((cR[0])^3) cR[4]-448 ((cR[0])^6) ((cR[4])^2))),cRl[8]->-(((Λ^2) (135+86 (Λ^2) ((cR[0])^2)+24 (Λ^4) ((cR[0])^4)-1008 ((cR[0])^3) cR[4]))/(864 ((cR[0])^5))),cRl2[8]->-((7 (Λ^4))/(72 ((cR[0])^3))),cR[9]->0,cRl[9]->0,cRl2[9]->0,cR[10]->(1/(25920000 ((cR[0])^9)))(57600 (Λ^8) ((cR[0])^8)+24 (Λ^6) ((cR[0])^6) (24491+64000 ((cR[0])^3) cR[4])+10 (Λ^4) ((cR[0])^4) (174123+344704 ((cR[0])^3) cR[4])-225 (Λ^2) ((cR[0])^2) (-11613+63072 ((cR[0])^3) cR[4]+355328 ((cR[0])^6) ((cR[4])^2))+20250 (73-808 ((cR[0])^3) cR[4]-12864 ((cR[0])^6) ((cR[4])^2)+4096 ((cR[0])^9) ((cR[4])^3))),cRl[10]->-(1/(1944000 ((cR[0])^7)))(43088 (Λ^6) ((cR[0])^4)+19200 (Λ^8) ((cR[0])^6)-90 (Λ^4) ((cR[0])^2) (1971+22208 ((cR[0])^3) cR[4])+2025 (Λ^2) (-101-3216 ((cR[0])^3) cR[4]+1536 ((cR[0])^6) ((cR[4])^2))),cRl2[10]->-(((Λ^4) (9045+2776 (Λ^2) ((cR[0])^2)-8640 ((cR[0])^3) cR[4]))/(32400 ((cR[0])^5))),cRl3[10]->-((2 (Λ^6))/(135 ((cR[0])^3)))}/.Table[cRl[kk]->0,{kk,0,3}]/.Table[cRl2[kk]->0,{kk,0,7}]/.Table[cRl3[kk]->0,{kk,0,9}]/.cR[0]->R0/.cR[4]->R4;

orUV=10;
Assuming[ϕ0>0&&z>0&&L>0,ℒ/.ruleWarpFactor/.γ-> 1/(Sqrt[3])/.R->Function[z,( (∑_(kk=0)^orUV)(cR[kk]+cRl[kk]Log[z])(z^kk)+((O[z,0])^(orUV+1)))/.{cR[1]->0,cR[2]->-(1/(2 cR[0])),cR[3]->0,cRl[4]->-((Λ^2)/(6 cR[0])),cR[5]->0,cRl[5]->0,cR[6]->((81+8 (Λ^4) ((cR[0])^4)+972 ((cR[0])^3) cR[4]+9 (Λ^2) ((cR[0])^2) (9+32 ((cR[0])^3) cR[4]))/(648 ((cR[0])^5))),cRl[6]->-(((Λ^2) (27+8 (Λ^2) ((cR[0])^2)))/(108 ((cR[0])^3))),cR[7]->0,cRl[7]->0,cR[8]->(1/(20736 ((cR[0])^7)))(120 (Λ^6) ((cR[0])^6)+9 (Λ^2) ((cR[0])^2) (405+1376 ((cR[0])^3) cR[4])+2 (Λ^4) ((cR[0])^4) (631+1728 ((cR[0])^3) cR[4])+162 (17+120 ((cR[0])^3) cR[4]-448 ((cR[0])^6) ((cR[4])^2))),cRl[8]->-(((Λ^2) (135+86 (Λ^2) ((cR[0])^2)+24 (Λ^4) ((cR[0])^4)-1008 ((cR[0])^3) cR[4]))/(864 ((cR[0])^5))),cRl2[8]->-((7 (Λ^4))/(72 ((cR[0])^3))),cR[9]->0,cRl[9]->0,cRl2[9]->0,cR[10]->(1/(25920000 ((cR[0])^9)))(57600 (Λ^8) ((cR[0])^8)+24 (Λ^6) ((cR[0])^6) (24491+64000 ((cR[0])^3) cR[4])+10 (Λ^4) ((cR[0])^4) (174123+344704 ((cR[0])^3) cR[4])-225 (Λ^2) ((cR[0])^2) (-11613+63072 ((cR[0])^3) cR[4]+355328 ((cR[0])^6) ((cR[4])^2))+20250 (73-808 ((cR[0])^3) cR[4]-12864 ((cR[0])^6) ((cR[4])^2)+4096 ((cR[0])^9) ((cR[4])^3))),cRl[10]->-(1/(1944000 ((cR[0])^7)))(43088 (Λ^6) ((cR[0])^4)+19200 (Λ^8) ((cR[0])^6)-90 (Λ^4) ((cR[0])^2) (1971+22208 ((cR[0])^3) cR[4])+2025 (Λ^2) (-101-3216 ((cR[0])^3) cR[4]+1536 ((cR[0])^6) ((cR[4])^2))),cRl2[10]->-(((Λ^4) (9045+2776 (Λ^2) ((cR[0])^2)-8640 ((cR[0])^3) cR[4]))/(32400 ((cR[0])^5))),cRl3[10]->-((2 (Λ^6))/(135 ((cR[0])^3)))}/.Table[cRl[kk]->0,{kk,0,3}]]//Simplify]
Limit[%-((((L^3) ((cR[0])^2))/(z^3))-(((L^3) (3+2 (Λ^2) ((cR[0])^2)))/(6 z))),z->0]

ℒct=((((L^3) ((cR[0])^2))/(z^3))-(((L^3) (3+2 (Λ^2) ((cR[0])^2)))/(6 z)))/.cR[0]->R0

integrandentropySpheress//Clear
integrandentropySpheress[Λ_,z_]=ℒ-ℒct/.ruleWarpFactor/.γ-> 1/(Sqrt[3])/.L->1//Simplify

Assuming[z>0,integrandentropySpheress[Λ,z]/.R->Function[z,( (∑_(kk=0)^orUV)(cR[kk]+cRl[kk]Log[z])(z^kk)+((O[z,0])^(orUV+1)))/.{cR[1]->0,cR[2]->-(1/(2 cR[0])),cR[3]->0,cRl[4]->-((Λ^2)/(6 cR[0])),cR[5]->0,cRl[5]->0,cR[6]->(81+8 (Λ^4) ((cR[0])^4)+972 ((cR[0])^3) cR[4]+9 (Λ^2) ((cR[0])^2) (9+32 ((cR[0])^3) cR[4]))/(648 ((cR[0])^5)),cRl[6]->-(((Λ^2) (27+8 (Λ^2) ((cR[0])^2)))/(108 ((cR[0])^3))),cR[7]->0,cRl[7]->0,cR[8]->(1/(20736 ((cR[0])^7)))(120 (Λ^6) ((cR[0])^6)+9 (Λ^2) ((cR[0])^2) (405+1376 ((cR[0])^3) cR[4])+2 (Λ^4) ((cR[0])^4) (631+1728 ((cR[0])^3) cR[4])+162 (17+120 ((cR[0])^3) cR[4]-448 ((cR[0])^6) ((cR[4])^2))),cRl[8]->-(((Λ^2) (135+86 (Λ^2) ((cR[0])^2)+24 (Λ^4) ((cR[0])^4)-1008 ((cR[0])^3) cR[4]))/(864 ((cR[0])^5))),cRl2[8]->-((7 (Λ^4))/(72 ((cR[0])^3))),cR[9]->0,cRl[9]->0,cRl2[9]->0,cR[10]->(1/(25920000 ((cR[0])^9)))(57600 (Λ^8) ((cR[0])^8)+24 (Λ^6) ((cR[0])^6) (24491+64000 ((cR[0])^3) cR[4])+10 (Λ^4) ((cR[0])^4) (174123+344704 ((cR[0])^3) cR[4])-225 (Λ^2) ((cR[0])^2) (-11613+63072 ((cR[0])^3) cR[4]+355328 ((cR[0])^6) ((cR[4])^2))+20250 (73-808 ((cR[0])^3) cR[4]-12864 ((cR[0])^6) ((cR[4])^2)+4096 ((cR[0])^9) ((cR[4])^3))),cRl[10]->-(1/(1944000 ((cR[0])^7)))(43088 (Λ^6) ((cR[0])^4)+19200 (Λ^8) ((cR[0])^6)-90 (Λ^4) ((cR[0])^2) (1971+22208 ((cR[0])^3) cR[4])+2025 (Λ^2) (-101-3216 ((cR[0])^3) cR[4]+1536 ((cR[0])^6) ((cR[4])^2))),cRl2[10]->-(((Λ^4) (9045+2776 (Λ^2) ((cR[0])^2)-8640 ((cR[0])^3) cR[4]))/(32400 ((cR[0])^5))),cRl3[10]->-((2 (Λ^6))/(135 ((cR[0])^3)))}/.γ-> 1/(Sqrt[3])/.Table[cRl[kk]->0,{kk,0,3}]/.cR[0]-> R0/.cR[4]->R4]//Simplify]//Normal;
integrandentropySpheressUV[Λ_,R0_,R4_,z_]=%;

(*integrandentropySpheressUV[Λ_,R0_,R4_,z_]=(1/(18 (z^3)))(9 (z^2)+6 (R0^2) (-3+(z^2) (Λ^2))-2 z (-3+(z^2) (Λ^2)) ((R[z])^2) (Sqrt[((9+(9-3 (z^2) (Λ^2)) ((R'[z])^2))/(z^2))]))/.R->Function[z,RUV[Λ,R0,R4,z]];
Assuming[z>0,Series[%,{z,0,9}]]//Simplify;
Integrate[%,z]

SUV[γ_,Λ_][R0_,R4_,z_]=(((-9-48 (R0^3) R4+8 (R0^2) (Λ^2) Log[z]) (z^2))/(48 (R0^2)))+(((-243+10368 (R0^6) (R4^2)-162 (R0^2) (Λ^2)+288 (R0^5) R4 (Λ^2)-10 (R0^4) (Λ^4)-3456 (R0^5) R4 (Λ^2) Log[z]-48 (R0^4) (Λ^4) Log[z]+288 (R0^4) (Λ^4) ((Log[z])^2)) (z^4))/(5184 (R0^4)))+(1/(186624 (R0^6)))(6075+174960 (R0^3) R4+933120 (R0^6) (R4^2)+6075 (R0^2) (Λ^2)+106272 (R0^5) R4 (Λ^2)+165888 (R0^8) (R4^2) (Λ^2)+1818 (R0^4) (Λ^4)+10368 (R0^7) R4 (Λ^4)+104 (R0^6) (Λ^6)-29160 (R0^2) (Λ^2) Log[z]-311040 (R0^5) R4 (Λ^2) Log[z]-17712 (R0^4) (Λ^4) Log[z]-55296 (R0^7) R4 (Λ^4) Log[z]-1728 (R0^6) (Λ^6) Log[z]+25920 (R0^4) (Λ^4) ((Log[z])^2)+4608 (R0^6) (Λ^6) ((Log[z])^2)) (z^6)+(1/(103680000 (R0^8)))(7300125+127008000 (R0^3) R4+381024000 (R0^6) (R4^2)-1161216000 (R0^9) (R4^3)+10507050 (R0^2) (Λ^2)+122644800 (R0^5) R4 (Λ^2)+195955200 (R0^8) (R4^2) (Λ^2)+4551330 (R0^4) (Λ^4)+30291840 (R0^7) R4 (Λ^4)+34560000 (R0^10) (R4^2) (Λ^4)+731364 (R0^6) (Λ^6)+2496000 (R0^9) R4 (Λ^6)+39600 (R0^8) (Λ^8)-21168000 (R0^2) (Λ^2) Log[z]-127008000 (R0^5) R4 (Λ^2) Log[z]+580608000 (R0^8) (R4^2) (Λ^2) Log[z]-20440800 (R0^4) (Λ^4) Log[z]-65318400 (R0^7) R4 (Λ^4) Log[z]-5048640 (R0^6) (Λ^6) Log[z]-11520000 (R0^9) R4 (Λ^6) Log[z]-416000 (R0^8) (Λ^8) Log[z]+10584000 (R0^4) (Λ^4) ((Log[z])^2)-96768000 (R0^7) R4 (Λ^4) ((Log[z])^2)+5443200 (R0^6) (Λ^6) ((Log[z])^2)+960000 (R0^8) (Λ^8) ((Log[z])^2)+5376000 (R0^6) (Λ^6) ((Log[z])^3)) (z^8)+(1/(41990400000 (R0^10)))(274741875-72892710000 (R0^3) R4-712262160000 (R0^6) (R4^2)-356918400000 (R0^9) (R4^3)+403107840000 (R0^12) (R4^4)+917719875 (R0^2) (Λ^2)-85153761000 (R0^5) R4 (Λ^2)-439009632000 (R0^8) (R4^2) (Λ^2)-228427776000 (R0^11) (R4^3) (Λ^2)+1734392250 (R0^4) (Λ^4)-15863914800 (R0^7) R4 (Λ^4)-61366118400 (R0^10) (R4^2) (Λ^4)+1462591080 (R0^6) (Λ^6)+4560782400 (R0^9) R4 (Λ^6)+4976640000 (R0^12) (R4^2) (Λ^6)+444497368 (R0^8) (Λ^8)+1244160000 (R0^11) R4 (Λ^8)+40435200 (R0^10) (Λ^10)+12148785000 (R0^2) (Λ^2) Log[z]+237420720000 (R0^5) R4 (Λ^2) Log[z]+178459200000 (R0^8) (R4^2) (Λ^2) Log[z]-268738560000 (R0^11) (R4^3) (Λ^2) Log[z]+14192293500 (R0^4) (Λ^4) Log[z]+146336544000 (R0^7) R4 (Λ^4) Log[z]+114213888000 (R0^10) (R4^2) (Λ^4) Log[z]+2643985800 (R0^6) (Λ^6) Log[z]+20455372800 (R0^9) R4 (Λ^6) Log[z]-760130400 (R0^8) (Λ^8) Log[z]-1658880000 (R0^11) R4 (Λ^8) Log[z]-207360000 (R0^10) (Λ^10) Log[z]-19785060000 (R0^4) (Λ^4) ((Log[z])^2)-29743200000 (R0^7) R4 (Λ^4) ((Log[z])^2)+67184640000 (R0^10) (R4^2) (Λ^4) ((Log[z])^2)-12194712000 (R0^6) (Λ^6) ((Log[z])^2)-19035648000 (R0^9) R4 (Λ^6) ((Log[z])^2)-1704614400 (R0^8) (Λ^8) ((Log[z])^2)+138240000 (R0^10) (Λ^10) ((Log[z])^2)+1652400000 (R0^6) (Λ^6) ((Log[z])^3)-7464960000 (R0^9) R4 (Λ^6) ((Log[z])^3)+1057536000 (R0^8) (Λ^8) ((Log[z])^3)+311040000 (R0^8) (Λ^8) ((Log[z])^4)) (z^10);

Boundary conditions: close to the confining IR

orIR=9;
Assuming[(-1+(zs^2) (γ^2) (Λ^2))<0&&0<zs<Λ/γ&&Λ>0&&(2/(Sqrt[6]))>γ>0,eomR/.ruleWarpFactor/.γ-> 1/(Sqrt[3])/.R->Function[z,(∑_(kk=1)^orIR)cR[kk](((zs-z))^(kk/2))+((O[z,zs])^((orIR+1)/2))]/.{cR[1]->(Sqrt[2]) (Sqrt[zs]),cR[2]->0,cR[3]->-((-15+(zs^2) (Λ^2))/(10 (Sqrt[2]) (Sqrt[zs]) (-3+(zs^2) (Λ^2)))),cR[4]->0,cR[5]->-((4725-14550 (zs^2) (Λ^2)+53 (zs^4) (Λ^4))/(8400 (Sqrt[2]) (zs^(3/2)) (((-3+(zs^2) (Λ^2)))^2))),cR[6]->0,cR[7]->((212625+245475 (zs^2) (Λ^2)+964755 (zs^4) (Λ^4)-959 (zs^6) (Λ^6))/(504000 (Sqrt[2]) (zs^(5/2)) (((-3+(zs^2) (Λ^2)))^3))),cR[8]->0,cR[9]->-((1227909375+427045500 (zs^2) (Λ^2)-10410294150 (zs^4) (Λ^4)-6277716612 (zs^6) (Λ^6)+2370175 (zs^8) (Λ^8))/(3104640000 (Sqrt[2]) (zs^(7/2)) (((-3+(zs^2) (Λ^2)))^4))),cR[10]->0}//Simplify]
Solve[%==0,cR[orIR]]//Simplify
Assuming[(-1+(zs^2) (γ^2) (Λ^2))<0&&0<zs<Λ/γ&&Λ>0&&(2/(Sqrt[6]))>γ>0,%//Simplify//FullSimplify]

soltip={cR[1]->(Sqrt[2]) (Sqrt[zs]),cR[2]->0,cR[3]->-((-15+(zs^2) (Λ^2))/(10 (Sqrt[2]) (Sqrt[zs]) (-3+(zs^2) (Λ^2)))),cR[4]->0,cR[5]->-((4725-14550 (zs^2) (Λ^2)+53 (zs^4) (Λ^4))/(8400 (Sqrt[2]) (zs^(3/2)) (((-3+(zs^2) (Λ^2)))^2))),cR[6]->0,cR[7]->((212625+245475 (zs^2) (Λ^2)+964755 (zs^4) (Λ^4)-959 (zs^6) (Λ^6))/(504000 (Sqrt[2]) (zs^(5/2)) (((-3+(zs^2) (Λ^2)))^3))),cR[8]->0,cR[9]->-((1227909375+427045500 (zs^2) (Λ^2)-10410294150 (zs^4) (Λ^4)-6277716612 (zs^6) (Λ^6)+2370175 (zs^8) (Λ^8))/(3104640000 (Sqrt[2]) (zs^(7/2)) (((-3+(zs^2) (Λ^2)))^4))),cR[10]->0};

Clear[RIR,dRIR];
RIR[Λ_,zs_,z_]=(∑_(kk=1)^orIR)cR[kk](((zs-z))^(kk/2))/.soltip/.L->1;
dRIR[Λ_,zs_,z_]=D[%,z];

Setup of the numerical integrator

eomNum//Clear
eomNum[Λ_]=eomR/.ruleWarpFactor/.γ-> 1/(Sqrt[3])/.L->1//Simplify

orDeep=10;
Assuming[0< z<(((γ Λ ))^(-1)),(eomNum[Λ])/.R->Function[z,(∑_(kk=0)^orDeep)cR[kk]((( (((γ Λ ))^(-1))- z))^kk)+((O[z,(((γ Λ ))^(-1))])^(orDeep+1))/.γ-> 1/(Sqrt[3])/.solDeepdIRGPPZ]//Simplify//Together//Numerator//Simplify][[3]];
solDeepdIRGPPZ=Solve[%==0,Table[cR[iter],{iter,1,orDeep}]]//Simplify//Flatten

RIRGPPZ[Λ_,Rs_,z_]=(∑_(kk=0)^orDeep)cR[kk]((( (((γ Λ ))^(-1))- z))^kk)/.solDeepdIRGPPZ/.L->1/.γ-> 1/(Sqrt[3])/.cR[0]-> Rs;
dRIRGPPZ[Λ_,Rs_,z_]=D[%,z];

computeSpheres//Clear
computeSpheres[Λ_,zs_]:=
(dog[zs]=NDSolve[{
eomNum[Λ]==0,
R[zs-ϵIR]==RIR[Λ,zs,zs-ϵIR],
R'[zs-ϵIR]==dRIR[Λ,zs,zs-ϵIR]
},R,{z,ϵUV,zs-ϵIR},WorkingPrecision->wpc,Method-> "StiffnessSwitching"][[1]])

plotSpheres//Clear
plotSpheres[zs_]:=Block[{R0},
Plot[{R[ζ],-R[ζ]}/.dog[zs],{ζ,dog[zs][[1,2,1,1,1]],dog[zs][[1,2,1,1,2]]},AxesOrigin->{0,0},PlotRange->All]
]

computeSpheresdeepGPPZ//Clear
computeSpheresdeepGPPZ[Λ_,Rs_]:=
(rcv[Rs]=NDSolve[{
eomNum[Λ]==0,
R[(((1/(Sqrt[3])Λ))^(-1))-ϵIR]==RIRGPPZ[Λ,Rs,(((1/(Sqrt[3])Λ))^(-1))-ϵIR],
R'[(((1/(Sqrt[3])Λ))^(-1))-ϵIR]==dRIRGPPZ[Λ,Rs,(((1/(Sqrt[3])Λ))^(-1))-ϵIR]
},R,{z,ϵUV,(((1/(Sqrt[3])Λ))^(-1))-ϵIR},WorkingPrecision->wpc,Method-> "StiffnessSwitching"][[1]])

plotSpheresdeepGPPZ//Clear
plotSpheresdeepGPPZ[Rs_]:=Block[{R0},
Plot[{R[ζ],-R[ζ]}/.rcv[Rs],{ζ,rcv[Rs][[1,2,1,1,1]],rcv[Rs][[1,2,1,1,2]]},AxesOrigin->{0,0},PlotRange->All]
]

zsmax=(Sqrt[3])

labelStyle=Directive[,FontFamily->"Courier New",FontSize->14];

ϵUV=(10^(-15));
ϵIR=(10^(-6));

Do[
computeSpheres[1,iter zsmax/10]
,{iter,1,9}]

computeSpheres[1,(999/1000) zsmax];

Clear[plotForPaper]
plotForPaper[sol_,plotStyle_:Thick]:=
(If[sol[[0]]=!= List,Print["solution not found"];Return[];];
ParametricPlot[{{-R[ζ]/.sol,-ζ}
,{R[ζ]/.sol,-ζ}},{ζ,sol[[1,2,1,1,1]],sol[[1,2,1,1,2]]},Ticks->{{{-4,4},{-(Sqrt[3]),-1},{(Sqrt[3]),1},{4,4}},None(*{{0,0},{-1,1}}}*)},PlotRange->{(3/2)(Sqrt[3]){-1,1},{0,-1.01zsmax}},AspectRatio->1/3,ImageSize->500,PlotStyle->plotStyle,LabelStyle->labelStyle]
)

Do[
computeSpheresdeepGPPZ[1,(Sqrt[3])iter/10]
,{iter,1,15}]

Show[
Join[
Append[
Table[plotForPaper[dog[zsmax iter/10],{{Darker[Blue],Thick}}],{iter,1,9}],
plotForPaper[dog[zsmax (999/1000)],{{Darker[Blue],Thick}}]
],
Table[plotForPaper[rcv[(Sqrt[3]) iter/10],{{Orange,Thick}}],{iter,1,15}]],
ListLinePlot[(Sqrt[3]){{-2,0},{2,0}},PlotStyle->{Black}],
ListLinePlot[(Sqrt[3]){{-2,-1},{2,-1}},PlotStyle->{Black,Dashed}]
]
Export["GPPZ_embeddings.pdf",%]

Setup of the numerical integrator

Let’s understand first what a good choice for the cutoffs can be, by plotting the integrand of the differential equation:

ϵUV=(10^(-7));
ϵIR=(10^(-6));
computeSpheres[1,1];
{auxR0,auxR4}=relationsR0R4[[2,All,2]]/.z->10ϵUV/.dog[1]/.Λ-> 1
integrandentropySpheress[1,z]/.R0->auxR0/.R4->auxR4/.dog[1]
check1=LogLogPlot[Abs@%,{z,dog[1][[1,2,1,1,1]],dog[1][[1,2,1,1,2]]},WorkingPrecision->wpc,PlotStyle->Red]

ϵUV=(10^(-15));
ϵIR=(10^(-6));
computeSpheres[1,1];
{auxR0,auxR4}=relationsR0R4[[2,All,2]]/.z->ϵUV/.dog[1]/.Λ-> 1
integrandentropySpheress[1,z]/.R0->( R[dog[1][[1,2,1,1,1]]]/.dog[1])/.dog[1];
check2=LogLogPlot[Abs@%,{z,dog[1][[1,2,1,1,1]],dog[1][[1,2,1,1,2]]},WorkingPrecision->40]

check3=LogLogPlot[Abs@integrandentropySpheressUV[1,auxR0,auxR4,z],{z,(10^(-15)),1},WorkingPrecision->40,PlotStyle->{Black,Thick,Dashed}]

Show[check1,check2,check3,PlotRange->Log@{{(10^(-8)),(10^(-3))},{(10^(-9)),10}}]

This suggests that it is better to use a bigger cutoff in the integral than in solving the equation. At the same time, the “analytic” expansion of the integrand does not work so well probably due to the lost in accuracy in computing R4, which appears in the next to leading order in this expression.

ϵUV=(10^(-15));
ϵUVintegral=(10^(-5));

Clear[Sct]
ℒct
Integrate[ℒct,z]//Simplify
Sct[γ_,Λ_,z_,R0_]=%/.L->1

Clear[computeESpheres]
computeESpheres[zs_]:=

Block[{R0,R4},

{R0,R4}=relationsR0R4[[2,All,2]]/.z->ϵUV/.dog[zs]/.Λ-> 1;

NIntegrate[integrandentropySpheressUV[1,R0,R4,z]/.dog[zs],{z,0,ϵUVintegral},WorkingPrecision->40]+
NIntegrate[integrandentropySpheress[1,z]/.dog[zs],{z,ϵUVintegral,dog[zs][[1,2,1,1,2]]},WorkingPrecision->40]+Sct[γ,1,zs,R0]

]


Clear[computeESpheresdeepGPPZ]
computeESpheresdeepGPPZ[Rs_]:=
Block[{R0,R4},

{R0,R4}=relationsR0R4[[2,All,2]]/.z->ϵUV/.rcv[Rs]/.Λ-> 1;


NIntegrate[integrandentropySpheressUV[1,R0,R4,z]/.rcv[Rs],{z,0,ϵUVintegral},WorkingPrecision->40]+NIntegrate[integrandentropySpheress[1,z]/.rcv[Rs],{z,ϵUVintegral,rcv[Rs][[1,2,1,1,2]]},WorkingPrecision->40]+Sct[1/(Sqrt[3]),1,(((1/(Sqrt[3])))^(-1)),R0]

]

Actual computation

zsmax=1/(1/(Sqrt[3]));

gridzs1=(zsmax/101) Join[
Table[k/5,{k,1,500}]
];

LaunchKernels[8]

tabledisk1=ParallelTable[
$MinPrecision=wpc;
computeSpheres[1,zs];
Block[{R0now,R4now,EEnow},

{R0now,R4now}=relationsR0R4[[2,All,2]]/.z->ϵUV/.dog[zs]/.Λ-> 1;
EEnow=computeESpheres[zs];

{zs,R0now,EEnow}]
,{zs,gridzs1}];

gridzs2= Table[k,{k,(100/101)zsmax,zsmax-(10^(-10)),(10^(-5))}][[2;;-1]];

tabledisk2=ParallelTable[
$MinPrecision=wpc;
computeSpheres[1,zs];
Block[{R0now,R4now,EEnow},

{R0now,R4now}=relationsR0R4[[2,All,2]]/.z->ϵUV/.dog[zs]/.Λ-> 1;
EEnow=computeESpheres[zs];

{zs,R0now,EEnow}]
,{zs,gridzs2}];

tabledisk=Join[tabledisk1,tabledisk2];

(*DeleteFile["EE_GPPZ_connected.mx"];
Save["EE_GPPZ_connected.mx",{tabledisk}];

Now we compute the embeddings that do reach the IR.

gridIR=Join[
Table[k,{k,(10^(-4)),(10^(-1)), (10^(-3))}],
Table[k,{k,(10^(-1)),1,(1/100)}],
Table[k,{k,1,10,(1/10)}]
]//DeleteDuplicates;

tablediskdeep=ParallelTable[

computeSpheresdeepGPPZ[1,Rs];

Block[{R0now,EEnow},

{R0now,R4now}=relationsR0R4[[2,All,2]]/.z->ϵUV/.rcv[Rs]/.Λ-> 1;
EEnow=computeESpheresdeepGPPZ[Rs];

{Rs,R0now,EEnow}]
,{Rs,gridIR[[2;;-1]]}];

(*Save["EE_GPPZ_disconnected.mx",{tablediskdeep}];

Results

Plot

You may load the following file, which contains our results.
In particular, it contains tabledisk and tablediskdeep.

SetDirectory[NotebookDirectory[]];
Get["EE_GPPZ_.mx"];

labelStyle=Directive[,FontFamily->"Courier New",FontSize->14];

zsmax=(Sqrt[3]);

pEE=Show[
ListPlot[tabledisk[[All,{2,3}]],PlotRange->{{0,4},{2,-6}},
Joined->True,Frame->True,AxesOrigin->{0,0},FrameStyle->Black,PlotStyle->{Darker[Blue],Thick}],
ListPlot[tablediskdeep[[All,{2,3}]],
Joined->True,Frame->True,AxesOrigin->{0,0},FrameStyle->Black,PlotStyle->{Orange,Thick}
],
ImageSize->350,
LabelStyle->labelStyle,Frame->True,AspectRatio->(GoldenRatio^(-1)),
PlotStyle->{{Darker[Blue],Thick},{Orange,Thick}},
PlotRangePadding->0,
ImagePadding->{{36,15},{15,5}}
]
Export["GPPZ_EE.pdf",%]

Check the nature of the phase transition

Note there is a tiny region where both configurations appear to be present. This is probably a numerical artifact, and thus we remove this points.

Get["EE_GPPZ_.mx"];

DeleteFile["EE_GPPZ_.mx"]
Save["EE_GPPZ_.mx",{tabledisk,tablediskdeep}];

Show[
ListLinePlot[tabledisk[[-500;;-1,{1,2}]],PlotMarkers->{Automatic, 4}],
ListLinePlot[{{zsmax,0},{zsmax,10}},PlotStyle->{Black,Thin}]
]

Show[
ListLinePlot[tabledisk[[-500;;-1,{1,3}]],PlotMarkers->{Automatic, 4}],
ListLinePlot[{{zsmax,0},{zsmax,10}},PlotStyle->{Black,Thin}]
]

Show[
ListLinePlot[tablediskdeep[[6;;80,{1,2}]],PlotMarkers->{Automatic, 4},PlotStyle->Orange,PlotRange->All]]

Show[
ListLinePlot[tablediskdeep[[6;;80,{1,3}]],PlotMarkers->{Automatic, 4},PlotStyle->Orange,PlotRange->All]]

Indeed this region is tiny, both in radii and entropy,

Show[pEE,PlotRange-> {{1.50704, 1.50715},{-0.97885,-0.97900}},
ImageSize->350,Ticks->{{1.507, 1.50715},{-1.9576/2,-1.958/2}},
LabelStyle->labelStyle,Frame->True,AspectRatio->(GoldenRatio^(-1)),
PlotStyle->{{Darker[Blue],Thick},{Orange,Thick}},
PlotRangePadding->0,FrameTicks->{{1.50704, 1.50715},{-0.97885,-0.97895},None,None},
ImagePadding->2{{36,15},{15,5}}]

(1/2)(1.50704-1.50715)
(1/2)(-0.97885+0.97900)

I will remove the points that overlap, consequently

tablediskremove=Select[tabledisk,#[[2]]<1.50705&];
tablediskdeepremove=Select[tablediskdeep,#[[2]]>1.50714&];
tabledisk=tablediskremove;
tablediskdeep=tablediskdeepremove;

pEE=Show[
ListPlot[tabledisk[[All,{2,3}]],PlotRange->{{0,4},{2,-6}},
Joined->True,Frame->True,AxesOrigin->{0,0},FrameStyle->Black,PlotStyle->{Darker[Blue],Thick}],
ListPlot[tablediskdeep[[All,{2,3}]],
Joined->True,Frame->True,AxesOrigin->{0,0},FrameStyle->Black,PlotStyle->{Orange,Thick}
],
ImageSize->350,
LabelStyle->labelStyle,Frame->True,AspectRatio->(GoldenRatio^(-1)),
PlotStyle->{{Darker[Blue],Thick},{Orange,Thick}},
PlotRangePadding->0,
ImagePadding->{{36,15},{15,5}}
]

Show[pEE,PlotRange-> {{1.50704, 1.50715},{-0.97885,-0.97900}},
ImageSize->350,Ticks->{{1.507, 1.50715},{-1.9576/2,-1.958/2}},
LabelStyle->labelStyle,Frame->True,AspectRatio->(GoldenRatio^(-1)),
PlotStyle->{{Darker[Blue],Thick},{Orange,Thick}},
PlotRangePadding->0,FrameTicks->{{1.50704, 1.50715},{-0.97885,-0.97895},None,None},
ImagePadding->2{{36,15},{15,5}}]

EEofLdisk=Interpolation[tabledisk[[1;;-1,{2,3}]],InterpolationOrder->8];

interpolORder=4;

Roftaus=Interpolation[tabledisk[[All,{1,2}]],InterpolationOrder->interpolORder];
EEoftaus=Interpolation[tabledisk[[All,{1,3}]],InterpolationOrder->interpolORder];

EprimeOfTaus[τs_]=EEoftaus'[τs]/Roftaus'[τs];
EppOfTaus[τs_]=(Roftaus'[τs] EEoftaus''[τs]-EEoftaus'[τs] Roftaus''[τs])/((Roftaus'[τs])^3);

Roftausdisc=Interpolation[tablediskdeep[[All,{1,2}]],InterpolationOrder->interpolORder];
EEoftausdisc=Interpolation[tablediskdeep[[All,{1,3}]],InterpolationOrder->interpolORder];

EprimeOfTausDisc[τs_]=EEoftausdisc'[τs]/Roftausdisc'[τs];
EppOfTausDisc[τs_]=(Roftausdisc'[τs] EEoftausdisc''[τs]-EEoftausdisc'[τs] Roftausdisc''[τs])/((Roftausdisc'[τs])^3);

We check that the derivative is continuous.

{Roftaus[zs],EprimeOfTaus[zs]}/.zs->tabledisk[[All,1]]//Transpose;
p1aux=ListLinePlot[%,PlotStyle->{Darker[Blue],Thick},PlotRange->{All,{-.6,-1.5}}];

{Roftausdisc[τs],EprimeOfTausDisc[τs]}/.τs->tablediskdeep[[All,1]]//Transpose;
p2aux=ListLinePlot[%,PlotStyle->{Orange,Thick},PlotRange->{{1.4,3},{-.6,-2}}];

Show[p1aux,p2aux,PlotRange->All,LabelStyle->labelStyle,Frame-> True,ImageSize->400,ImagePadding->{{100,1},{50,10}},FrameLabel-> {"R","E'(R)"}]

plotRange={{0.5,2.5},Automatic};
{Roftaus[zs],EppOfTaus[zs]}/.zs->tabledisk[[All,1]]//Transpose;
p1aux=ListLinePlot[%,PlotStyle->{Darker[Blue],Thick},PlotRange->plotRange];

{Roftausdisc[τs],EppOfTausDisc[τs]}/.τs->tablediskdeep[[All,1]]//Transpose;
p2aux=ListLinePlot[%,PlotStyle->{Orange,Thick},PlotRange->plotRange];

Show[p1aux,p2aux,PlotRange->plotRange]

list2={Roftaus[zs],(1/2)Roftaus[zs] (-EprimeOfTaus[zs]+Roftaus[zs] EppOfTaus[zs])}/.zs->tabledisk[[6;;-5,1]]//Transpose;
p2=ListLinePlot[list2[[1;;-500]],PlotStyle->{Darker[Blue],Thick},PlotRange->{{0,1.7},{-1,1/2}}];

{Roftausdisc[τs],cDiconnectedOfTau[τs]}/.τs->tablediskdeep[[All,1]]//Transpose;
p1=ListLinePlot[%,AspectRatio->(GoldenRatio^(-1)),PlotStyle->{Orange,Thick},PlotRange->All];

Show[p1,p2,PlotRange->{{0,4},{-.2,0.6}},AxesOrigin->{0,0},
ImageSize->350,
LabelStyle->labelStyle,Frame->True,AspectRatio->(GoldenRatio^(-1)),
PlotStyle->{{Darker[Blue],Thick},{Orange,Thick}},
PlotRangePadding->0,
ImagePadding->{{36,15},{15,5}}]
Export["GPPZ_c_function.pdf",%]