In[]:=
d=1;​​T=Rationalize@0.0001d;​​ϵf0=-Rationalize@1.2000d;​​p=Rationalize@0.8750;​​q0=Rationalize@0.5000;​​v=Rationalize@0.5000d;​​prec=10;​​ds[ϵ_]:=
3
4Sqrt[2]d
Sqrt
(ϵ+d)
d
;​​​​(*wecalculaterenormalizedpositionofthef-level,chemicalpotentialandslavebosonamplitudebysolvingthreenonlinearequations*)​​​​ζ[ϵ_,μ_]:=ϵ-μ;(*ϵ=
2
k
2m
-d*)​​Rk[ϵ_,μ_,ϵf_,a_]:=Sqrt[
2
(ϵf-ζ[ϵ,μ])
+
2
(2va)
];​​Enkp[ϵ_,μ_,ϵf_,a_]:=
1
2
(ϵf+ζ[ϵ,μ]+Rk[ϵ,μ,ϵf,a]);​​Enkm[ϵ_,μ_,ϵf_,a_]:=
1
2
(ϵf+ζ[ϵ,μ]-Rk[ϵ,μ,ϵf,a]);
In[]:=
fwp[ϵ_,μ_,ϵf_,a_]:=
1
2
1-Tanh
Enkp[ϵ,μ,ϵf,a]
2T
;​​fwm[ϵ_,μ_,ϵf_,a_]:=
1
2
1-Tanh
Enkm[ϵ,μ,ϵf,a]
2T
;​​eq1int[ϵ_,μ_,ϵf_,a_]:=
(fwm[ϵ,μ,ϵf,a]-fwp[ϵ,μ,ϵf,a])
Rk[ϵ,μ,ϵf,a]
;​​eq2int[ϵ_,μ_,ϵf_,a_]:=(ζ[ϵ,μ]-ϵf)eq1int[ϵ,μ,ϵf,a];​​​​eq3int[ϵ_,μ_,ϵf_,a_]:=(fwm[ϵ,μ,ϵf,a]+fwp[ϵ,μ,ϵf,a]);​​​​fint1[ϵ_,μ_,ϵf_,a_]:=ds[ϵ]eq1int[ϵ,μ,ϵf,a];​​fint2[ϵ_,μ_,ϵf_,a_]:=ds[ϵ]eq2int[ϵ,μ,ϵf,a];​​fint3[ϵ_,μ_,ϵf_,a_]:=ds[ϵ]eq3int[ϵ,μ,ϵf,a];
eqn1[μ_?NumericQ,ϵf_?NumericQ,a_?NumericQ]:=ϵf-ϵf0-
2
v
NIntegrate[fint1[ϵ,μ,ϵf,a],{ϵ,-d,d},WorkingPrecision->prec,Method->"DoubleExponential",AccuracyGoal->10]​​eqn2[μ_?NumericQ,ϵf_?NumericQ,a_?NumericQ]:=2(q0-
2
a
)-p-NIntegrate[fint2[ϵ,μ,ϵf,a],{ϵ,-d,d},WorkingPrecision->prec,Method->"DoubleExponential",AccuracyGoal->10]​​eqn3[μ_?NumericQ,ϵf_?NumericQ,a_?NumericQ]:=p-NIntegrate[fint3[ϵ,μ,ϵf,a],{ϵ,-d,d},WorkingPrecision->prec,Method->"DoubleExponential",AccuracyGoal->10]
In[]:=
mysol=FindRoot[{eqn1[μ,ϵf,a],eqn2[μ,ϵf,a],eqn3[μ,ϵf,a]},{{μ,-0.2152},{ϵf,1.2474},{a,0.1000}},WorkingPrecision->prec]​​
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in ϵ in the region {{-1.00000000000000000000000000000000000000000000000000000000000,1.000000000}}. NIntegrate obtained 0.000319550801727052935292634882298511852459973227433708543228251 and 0.000319550801727052935292634882298511852459973227433708543228251 for the integral and error estimates.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in ϵ in the region {{-1.00000000000000000000000000000000000000000000000000000000000,1.000000000}}. NIntegrate obtained -0.000492862388363778632505257576779131722316270859439388964733081 and 0.000492862388363778632505257576779131722316270859439388964733081 for the integral and error estimates.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in ϵ in the region {{-1.00000000000000000000000000000000000000000000000000000000000,1.000000000}}. NIntegrate obtained 0.000493910552453103274830219161257980780261571940085876483050803 and 0.000493910552453103274830219161257980780261571940085876483050803 for the integral and error estimates.
General
:Further output of NIntegrate::ncvi will be suppressed during this calculation.
FindRoot
:Encountered a singular Jacobian at the point {μ,ϵf,a} = {-0.2152,1.2474,0.1}. Try perturbing the initial point(s).
Out[]=
{μ-0.2152,ϵf1.2474,a0.1}
In[]:=
{eqn1[μ,ϵf,a],eqn2[μ,ϵf,a],eqn3[μ,ϵf,a]}/.mysol
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in ϵ in the region {{-1.00000000000000000000000000000000000000000000000000000000000,1.000000000}}. NIntegrate obtained 0.000319550801727052935292634882298511852459973227433708543228251 and 0.000319550801727052935292634882298511852459973227433708543228251 for the integral and error estimates.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in ϵ in the region {{-1.00000000000000000000000000000000000000000000000000000000000,1.000000000}}. NIntegrate obtained -0.000492862388363778632505257576779131722316270859439388964733081 and 0.000492862388363778632505257576779131722316270859439388964733081 for the integral and error estimates.
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in ϵ in the region {{-1.00000000000000000000000000000000000000000000000000000000000,1.000000000}}. NIntegrate obtained 0.000493910552453103274830219161257980780261571940085876483050803 and 0.000493910552453103274830219161257980780261571940085876483050803 for the integral and error estimates.
General
:Further output of NIntegrate::ncvi will be suppressed during this calculation.
Out[]=
{2.44732,0.105493,0.8745060894475}