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

HsbUMP[λ_,t_,U_]=(({{-((2 (t^2) λ)/U), 0, 0, ((2 (t^2) λ)/U)}, {0, U-((2 (t^2) λ)/U), ((2 (t^2) λ)/U), ((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U)}, {0, ((2 (t^2) λ)/U), U-((2 (t^2) λ)/U), -((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U)}, {((2 (t^2) λ)/U), ((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U), -((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U), -2 U (-1+λ)+((6 (t^2) λ)/U)}}));
HRMP[λ_,t_,U_]=(({{-2 t+U-((U λ)/2), 0, 0, ((U λ)/2)}, {0, U-((U λ)/2), ((U λ)/2), 0}, {0, ((U λ)/2), U-((U λ)/2), 0}, {((U λ)/2), 0, 0, 2 t+U-((U λ)/2)}}));
HsbRMP[λ_,t_,U_]=(({{-U (-2+λ)+((2 (t^2) λ)/U), 0, 0, ((2 (t^2) λ)/U)}, {0, U+((2 (t^2) λ)/U)-U λ, ((2 (t^2) λ)/U), ((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U)}, {0, ((2 (t^2) λ)/U), U+((2 (t^2) λ)/U)-U λ, ((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U)}, {((2 (t^2) λ)/U), ((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U), ((2 t (Sqrt[-4 (t^2)+(U^2)]) λ)/U), -((6 (t^2) λ)/U)+U λ}}));

ϵsbUMP[U_,t_,λ_]=Eigenvalues[HsbUMP[λ,t,U]];
ϵsbRMP[U_,t_,λ_]=Eigenvalues[HsbRMP[λ,t,U]];

Show[
{Plot3D[Evaluate[Re[ϵsbUMP[7,1,rλ+I zλ]]],{rλ,-0,1.5},{zλ,0,1},PlotStyle->Opacity[0.7],BoxRatios->{1, 1, 1},
ImageSize->Large,MaxRecursion->6,Mesh->5,Boxed->False,AxesEdge->{{-1,-1},{1,-1},{-1,-1}},
AxesLabel->{
MaTeX["\\text{Re}(\\lambda)",Magnification->1.5],
MaTeX["\\text{Im}(\\lambda)",Magnification->1.5],
Rotate[MaTeX["\\text{Re}(E_\\text{UMP})/t",Magnification->1.5],(π/2)]},
AxesStyle->Directive[Thick,18,Black]
],
Graphics3D[{Opacity[0.5],Cylinder[{{0,0,-4},{0,0,20}},1]}]},PlotRange->{{0,1.2},{0,1.2},{-2.4,17}},Boxed->False
]

Show[
{Plot3D[Evaluate[Re[ϵsbRMP[7,1,rλ+I zλ]]],{rλ,-0,1.5},{zλ,0,1},PlotStyle->Opacity[0.7],BoxRatios->{1, 1, 1},
ImageSize->Large,MaxRecursion->6,Mesh->5,Boxed->False,AxesEdge->{{-1,-1},{1,-1},{-1,-1}},
AxesLabel->{
MaTeX["\\text{Re}(\\lambda)",Magnification->1.5],
MaTeX["\\text{Im}(\\lambda)",Magnification->1.5],
Rotate[MaTeX["\\text{Re}(E_\\text{RMP})/t",Magnification->1.5],(π/2)]},
AxesStyle->Directive[Thick,18,Black]
],
Graphics3D[{Opacity[0.5],Cylinder[{{0,0,-4},{0,0,20}},1]}]},PlotRange->{{0,1.2},{0,1.2},{-2.4,17}},Boxed->False
]

Solve[
{
Det[HRMP[λ,t,U]-Ε IdentityMatrix[Length[HUMP[λ,t,U]]]],(∂_Ε)Det[HRMP[λ,t,U]-Ε IdentityMatrix[Length[HUMP[λ,t,U]]]]
}==0,{Ε,λ}]

RsbUMP[t_,U_]:=
Sort[
Abs[λ]/.NSolve[
{
Det[HsbUMP[λ,t,U]-Ε IdentityMatrix[Length[HsbUMP[λ,t,U]]]],(∂_Ε)Det[HsbUMP[λ,t,U]-Ε IdentityMatrix[Length[HsbUMP[λ,t,U]]]]
}==0,{Ε,λ},WorkingPrecision->30]
]//DeleteDuplicates

RsbRMP[t_,U_]:=
Sort[
Abs[λ]/.NSolve[
{
Det[HsbRMP[λ,t,U]-Ε IdentityMatrix[Length[HsbRMP[λ,t,U]]]],(∂_Ε)Det[HsbRMP[λ,t,U]-Ε IdentityMatrix[Length[HsbRMP[λ,t,U]]]]
}==0,{Ε,λ}]
]//DeleteDuplicates

RsbUMP[1,3.]

TabRsbUMP=With[{ΔU=(1/100)},Table[{U,RsbUMP[1,U]},{U,2+ΔU,10,ΔU}]];
TabRsbRMP=With[{ΔU=(1/100)},Table[{U,RsbRMP[1,U]},{U,2+ΔU,10,ΔU}]];

TabRRMP=With[{ΔU=(1/100)},Table[{U,(4/U)},{U,ΔU,10,ΔU}]];

ListPlot[{
TabRadConvRMP,
Table[{TabRsbUMP[[k,1]],TabRsbUMP[[k,2,3]]},{k,Length[TabRsbUMP]}],
Table[{TabRsbUMP[[k,1]],TabRsbUMP[[k,2,2]]},{k,Length[TabRsbUMP]}]
},Joined->True,InterpolationOrder->1,PlotTheme->"Scientific",
PlotRange->{{0,10},{0,4}},PlotStyle->Thickness[0.005],
AspectRatio->1/2,
BaseStyle->20,ImageSize->Large,GridLines->Automatic,FrameLabel->{MaTeX["U/t",FontSize->20],MaTeX["\\text{Radius of convergence } r_c",FontSize->20]},
PlotLegends->Placed[{MaTeX["r_c \\text{ for RMP ground state}",FontSize->20],MaTeX["r_c \\text{ for UMP ground state}",FontSize->20],MaTeX["r_c \\text{ for UMP excited state}",FontSize->20]},{Right,Top}]]