Needs["MaTeX`"]
SetOptions[MaTeX,"Preamble"->{"\\usepackage{amssymb,amsmath,latexsym,amsfonts,amsthm,xcolor,bm,mhchem}","\\definecolor{darkgreen}{RGB}{0, 180, 0}"}];

RHF-based resummation

Taylor expansion vs Padé approximant

Exact[λ_,t_,U_]=(1/2) (2 U-U λ-(Sqrt[16 (t^2)+(U^2) (λ^2)]));
Taylor2[λ_,t_,U_]=Series[Exact[λ,t,U],{λ,0,2}]//Normal;
Taylor3[λ_,t_,U_]=Series[Exact[λ,t,U],{λ,0,3}]//Normal;
Taylor4[λ_,t_,U_]=Series[Exact[λ,t,U],{λ,0,4}]//Normal;
Taylor5[λ_,t_,U_]=Series[Exact[λ,t,U],{λ,0,5}]//Normal;
Taylor6[λ_,t_,U_]=Series[Exact[λ,t,U],{λ,0,6}]//Normal;
Pade01[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U],{λ,0,{0,1}}],t>0];
Pade11[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U],{λ,0,{1,1}}],t>0];
Pade22[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U],{λ,0,{2,2}}],t>0];
Pade33[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U],{λ,0,{3,3}}],t>0];
Pade44[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U],{λ,0,{4,4}}],t>0];
Pade55[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U],{λ,0,{5,5}}],t>0];

With[{U=3.5,t=1},Plot[{Exact[λ,t,U],Taylor2[λ,t,U],Taylor4[λ,t,U],Pade11[λ,t,U],Pade22[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U]},{λ,0,3/2},PlotRange->{{0,3/2},{-6,2}},PlotLabel->MaTeX["U/t = 3.5",FontSize->30],GridLines->Automatic,
PlotLegends->Placed[{MaTeX["\\text{Exact}",FontSize->20],MaTeX["\\text{2nd-order Taylor}",FontSize->20],MaTeX["\\text{4th-order Taylor}",FontSize->20],MaTeX["\\text{[1/1] Pad\\'e}",FontSize->20],MaTeX["\\text{[2/2] Pad\\'e}",FontSize->20],MaTeX["\\text{[3/3] Pad\\'e}",FontSize->20],MaTeX["\\text{[4/4] Pad\\'e}",FontSize->20]},Right],
PlotStyle->{{Black,Thick},{Gray,Dotted,Thick},{Gray,Dashed,Thick},{Red,Thick},{Blue,Thick},{Darker[Green],Thick},{Magenta,Thick}},
PlotTheme->"Scientific",FrameLabel->{MaTeX["\\lambda",FontSize->20],MaTeX["E_{-}(\\lambda)",FontSize->20]},ImageSize->Large,BaseStyle->20,FrameStyle->{Black,Thick},PlotRangePadding->0]
]
Export["PadeRMP35.pdf",%];

With[{U=4.5,t=1},Plot[{Exact[λ,t,U],Taylor2[λ,t,U],Taylor4[λ,t,U],Pade11[λ,t,U],Pade22[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U]},{λ,0,3/2},PlotRange->{{0,3/2},{-4,3}},PlotLabel->MaTeX["U/t = 4.5",FontSize->30],GridLines->Automatic,
PlotStyle->{{Black,Thick},{Gray,Dotted,Thick},{Gray,Dashed,Thick},{Red,Thick},{Blue,Thick},{Darker[Green],Thick},{Magenta,Thick}},
PlotTheme->"Scientific",FrameLabel->{MaTeX["\\lambda",FontSize->20],MaTeX["E_{-}(\\lambda)",FontSize->20]},ImageSize->Large,BaseStyle->20,FrameStyle->{Black,Thick},PlotRangePadding->0]
]
Export["PadeRMP45.pdf",%];

With[{t=1,U=3.5,λ=1},{Taylor2[λ,t,U],Taylor3[λ,t,U],Taylor4[λ,t,U],Taylor5[λ,t,U],Taylor6[λ,t,U],Exact[λ,t,U]}]
With[{t=1,U=3.5,λ=1},{Pade11[λ,t,U],Pade22[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U],Pade55[λ,t,U],Exact[λ,t,U]}]

With[{t=1,U=4.5,λ=1},{Taylor2[λ,t,U],Taylor3[λ,t,U],Taylor4[λ,t,U],Taylor5[λ,t,U],Taylor6[λ,t,U],Exact[λ,t,U]}]
With[{t=1,U=4.5,λ=1},{Pade11[λ,t,U],Pade22[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U],Pade55[λ,t,U],Exact[λ,t,U]}]

((4t)/U)/.{t->1,U->3.5}
Abs[λ]/.Solve[((Pade11[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.5}
Abs[λ]/.Solve[((Pade22[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.5}
Abs[λ]/.Solve[((Pade33[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.5}
Abs[λ]/.Solve[((Pade44[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.5}
Abs[λ]/.Solve[((Pade55[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.5}

((4t)/U)/.{t->1,U->4.5}
Abs[λ]/.Solve[((Pade11[λ,t,U])^(-1))==0,λ]/.{t->1,U->4.5}//Sort
Abs[λ]/.Solve[((Pade22[λ,t,U])^(-1))==0,λ]/.{t->1,U->4.5}//Sort
Abs[λ]/.Solve[((Pade33[λ,t,U])^(-1))==0,λ]/.{t->1,U->4.5}//Sort
Abs[λ]/.Solve[((Pade44[λ,t,U])^(-1))==0,λ]/.{t->1,U->4.5}//Sort
Abs[λ]/.Solve[((Pade55[λ,t,U])^(-1))==0,λ]/.{t->1,U->4.5}//Sort

TableForm[{{With[{U=4,t=1},ComplexPlot3D[Exact[λ,t,U],{λ,-2I-2,2I+2},Mesh->{Range[0,5],Range[-Pi,Pi,Pi/4]},MeshFunctions->{Log[Abs[#2]]&,Arg[#2]&},PlotPoints->75]],
With[{U=4,t=1},ComplexPlot3D[Pade12[λ,t,U],{λ,-2I-2,2I+2},Mesh->{Range[0,5],Range[-Pi,Pi,Pi/4]},MeshFunctions->{Log[Abs[#2]]&,Arg[#2]&},PlotPoints->75]],
With[{U=4,t=1},ComplexPlot3D[Pade23[λ,t,U],{λ,-2I-2,2I+2},Mesh->{Range[0,5],Range[-Pi,Pi,Pi/4]},MeshFunctions->{Log[Abs[#2]]&,Arg[#2]&},PlotPoints->75]],
With[{U=4,t=1},ComplexPlot3D[Pade34[λ,t,U],{λ,-2I-2,2I+2},Mesh->{Range[0,5],Range[-Pi,Pi,Pi/4]},MeshFunctions->{Log[Abs[#2]]&,Arg[#2]&},PlotPoints->75]]
}}]

Quadratic approximants

QuadraticApproximant[dP_,dQ_,dR_]:=Module[{dE,c,P,Q,R,E},
dE=dP+dQ+dR+1;
If[dR≥0,
(P=(∑_(k=0)^dP)(Subscript[p, k])(λ^k);	Q=(∑_(k=0)^dQ)(Subscript[q, k])(λ^k)/.(Subscript[q, 0])->1;	R=(∑_(k=0)^dR)(Subscript[r, k])(λ^k);
E=Series[Exact[λ,t,U],{λ,0,dE}];
c=Solve[CoefficientList[Series[Q (E^2)-P E+R,{λ,0,dE}],λ]==0,Flatten[{Table[(Subscript[p, k]),{k,0,dP}],Table[(Subscript[q, k]),{k,1,dQ}],Table[(Subscript[r, k]),{k,0,dR}]}]][[1]];
Return[FullSimplify[{"["<>ToString[dP]<>"/"<>ToString[dQ]<>","<>ToString[dR]<>"]",
"(Subscript[n, bp]) = "<>ToString[Max[2dP,dQ+dR]],
(1/(2Q))(P-(Sqrt[(P^2)-4Q R])),(1/(2Q))(P+(Sqrt[(P^2)-4Q R]))}/.c,t>0&&U>0]];
)
];
];

QA000[λ_,t_,U_]=QuadraticApproximant[0,0,0];
QA100[λ_,t_,U_]=QuadraticApproximant[1,0,0];
QA101[λ_,t_,U_]=QuadraticApproximant[1,0,1];
QA111[λ_,t_,U_]=QuadraticApproximant[1,1,1];

With[{U=4,t=1},Plot[{Exact[λ,t,U],QA000[λ,t,U][[3]],QA100[λ,t,U][[4]],QA101[λ,t,U][[3]]},{λ,0,2},PlotLegends->Automatic]]

UHF-based resummation

Taylor expansion vs Padé approximant

Exact[λ_,t_,U_]=Table[((Root[4 (t^2) (U^4) λ-4 (t^2) (U^4) (λ^2)+(2 (U^4)-8 (t^2) (U^2) λ-2 (U^4) λ+4 (t^2) (U^2) (λ^2)) #1+(-3 (U^2)+2 (U^2) λ) ((#1)^2)+((#1)^3)&,k])/U),{k,3}];
Taylor2[λ_,t_,U_]=Series[Exact[λ,t,U][[1]],{λ,0,2}]//Normal;
Taylor4[λ_,t_,U_]=Series[Exact[λ,t,U][[1]],{λ,0,4}]//Normal;
Pade11[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U][[1]],{λ,0,{1,1}}],t>0];
Pade22[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U][[1]],{λ,0,{2,2}}],t>0];
Pade33[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U][[1]],{λ,0,{3,3}}],t>0];
Pade44[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U][[1]],{λ,0,{4,4}}],t>0];
Pade55[λ_,t_,U_]=FullSimplify[PadeApproximant[Exact[λ,t,U][[1]],{λ,0,{5,5}}],t>0];

With[{U=7,t=1},Plot[{Exact[λ,t,U],Taylor4[λ,t,U],Pade11[λ,t,U],Pade22[λ,t,U]},{λ,0,2},
PlotLegends->Placed[{MaTeX["\\text{Exact}",FontSize->12],MaTeX["\\text{4th-order Taylor}",FontSize->12],MaTeX["\\text{[1/1] Pad\\'e}",FontSize->12],MaTeX["\\text{[2/2] Pad\\'e}",FontSize->12]},{Left,Bottom}],
PlotStyle->{{Black,Thick},{Gray,Dashed,Thick},{Red,Thick},{Blue,Thick},{Darker[Green],Thick},{Magenta,Thick}},
PlotTheme->"Scientific",FrameLabel->{MaTeX["\\lambda",FontSize->20],MaTeX["E(\\lambda)",FontSize->20]},ImageSize->Large,BaseStyle->16,PlotRange->{All,{-3/4,1/4}}]
]

Abs[λ]/.Solve[((Pade22[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.}//Sort
Abs[λ]/.Solve[((Pade33[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.}//Sort
λ/.Solve[((Pade44[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.}//Sort
λ/.Solve[((Pade55[λ,t,U])^(-1))==0,λ]/.{t->1,U->3.}//Sort

Abs[λ]/.Solve[((Pade22[λ,t,U])^(-1))==0,λ]/.{t->1,U->7.}//Sort
Abs[λ]/.Solve[((Pade33[λ,t,U])^(-1))==0,λ]/.{t->1,U->7.}//Sort
Abs[λ]/.Solve[((Pade44[λ,t,U])^(-1))==0,λ]/.{t->1,U->7.}//Sort
Abs[λ]/.Solve[((Pade55[λ,t,U])^(-1))==0,λ]/.{t->1,U->7.}//Sort

{Pade22[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U],Pade55[λ,t,U]}/.{λ->1,t->1,U->3.}
{Pade22[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U],Pade55[λ,t,U]}/.{λ->1,t->1,U->7.}

Quadratic approximants

QuadraticApproximant[dP_,dQ_,dR_]:=Module[{dE,c,P,Q,R,E},
dE=dP+dQ+dR+1;
If[dR≥0,
(P=(∑_(k=0)^dP)(Subscript[p, k])(λ^k);	Q=(∑_(k=0)^dQ)(Subscript[q, k])(λ^k)/.(Subscript[q, 0])->1;	R=(∑_(k=0)^dR)(Subscript[r, k])(λ^k);
E=Series[Exact[λ,t,U][[1]],{λ,0,dE}];
c=Solve[CoefficientList[Series[Q (E^2)-P E+R,{λ,0,dE}],λ]==0,Flatten[{Table[(Subscript[p, k]),{k,0,dP}],Table[(Subscript[q, k]),{k,1,dQ}],Table[(Subscript[r, k]),{k,0,dR}]}]][[1]];
Return[{"["<>ToString[dP]<>"/"<>ToString[dQ]<>","<>ToString[dR]<>"]",
"(Subscript[n, bp]) = "<>ToString[Max[2dP,dQ+dR]],
(1/(2Q))(P-(Sqrt[(P^2)-4Q R])),(1/(2Q))(P+(Sqrt[(P^2)-4Q R]))}/.c];
)
];
];

QA000[λ_,t_,U_]=QuadraticApproximant[0,0,0];
QA100[λ_,t_,U_]=QuadraticApproximant[1,0,0];
QA101[λ_,t_,U_]=QuadraticApproximant[1,0,1];
QA111[λ_,t_,U_]=QuadraticApproximant[1,1,1];
QA211[λ_,t_,U_]=QuadraticApproximant[2,1,1];
QA212[λ_,t_,U_]=QuadraticApproximant[2,1,2];
QA222[λ_,t_,U_]=QuadraticApproximant[2,2,2];
QA322[λ_,t_,U_]=QuadraticApproximant[3,2,2];
QA323[λ_,t_,U_]=QuadraticApproximant[3,2,3];
QA333[λ_,t_,U_]=QuadraticApproximant[3,3,3];
(*QA433[λ_,t_,U_]=QuadraticApproximant[4,3,3][[3]];
QA443[λ_,t_,U_]=QuadraticApproximant[4,4,3][[3]];
QA444[λ_,t_,U_]=QuadraticApproximant[4,4,4][[3]];*)

{QA212[λ,t,U][[3;;4]],QA222[λ,t,U][[3;;4]],QA322[λ,t,U][[3;;4]],QA323[λ,t,U][[3;;4]],QA333[λ,t,U][[3;;4]],Exact[λ,t,U]}/.{t->1,U->3.,λ->1}
{QA212[λ,t,U][[3;;4]],QA222[λ,t,U][[3;;4]],QA322[λ,t,U][[3;;4]],QA323[λ,t,U][[3;;4]],QA333[λ,t,U][[3;;4]],Exact[λ,t,U]}/.{t->1,U->7.,λ->1}

With[{U=3,t=1},Plot[{Exact[λ,t,U],Pade33[λ,t,U],Pade44[λ,t,U],QA222[λ,t,U][[3;;4]],QA333[λ,t,U][[3;;4]]},{λ,0,2},
PlotLegends->Placed[{MaTeX["\\text{Exact}",FontSize->12],MaTeX["\\text{[3/3] Pad\\'e}",FontSize->12],MaTeX["\\text{[4/4] Pad\\'e}",FontSize->12],MaTeX["\\text{[2/2,2] Quadratic}",FontSize->12],MaTeX["\\text{[3/3,3] Quadratic}",FontSize->12]},{Left,Bottom}],GridLines->Automatic,
PlotStyle->{{Black,Thick},{Red,Dashed,Thick},{Blue,Dashed,Thick},{Darker[Green],Thick},{Magenta,Thick}},AspectRatio->1,
PlotTheme->"Scientific",FrameLabel->{MaTeX["\\lambda",FontSize->20],MaTeX["E(\\lambda)",FontSize->20]},ImageSize->Medium,BaseStyle->16,PlotRange->{All,{-5,5}},FrameStyle->{Black,Thick},PlotRangePadding->0]
]
Export["QuadUMP.pdf",%];

Abs[λ]/.NSolve[QA101[λ,t,U][[3]]==QA101[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA111[λ,t,U][[3]]==QA111[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA211[λ,t,U][[3]]==QA211[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA212[λ,t,U][[3]]==QA212[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA222[λ,t,U][[3]]==QA222[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA322[λ,t,U][[3]]==QA322[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA323[λ,t,U][[3]]==QA323[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort
Abs[λ]/.NSolve[QA333[λ,t,U][[3]]==QA333[λ,t,U][[4]]/.{U->7,t->1.},λ]//Sort

With[{U=4,t=1},ListPlot[{Table[{λ,Exact[λ,t,U]},{λ,0,2,0.1}],Table[{λ,QA444[λ,t,U]},{λ,0,2,0.1}]},,PlotLegends->Automatic]]