How can I sort the eigenvalues of a matrix when passing to Plot3D?

these eigenvalues should be separated when sorted in the correct way but it is not the case when passing to Plot3D

Plot3D[Evaluate[Sort[Eigenvalues[{{2 Sin[\[Pi]/6-y],-1,-E^(3 I x)},{-1,2 Sin[\[Pi]/6+y],-1},{-E^(-3 I x),-1,-2 Cos[y]}}]]],{x,-\[Pi]/3,\[Pi]/3},{y,-\[Pi],\[Pi]},BoxRatios->{1,1,1},Mesh->None,PlotRange->{-4,4},PlotTheme->"Detailed",PlotLegends->False,ClippingStyle->Automatic]


I tried

dat3D = ParallelTable[
Flatten@{x, y,
Sort[Eigenvalues[
N@{{2 Sin[\[Pi]/6 - y], -1, -E^(3 I x)}, {-1,
2 Sin[\[Pi]/6 + y], -1}, {-E^(-3 I x), -1, -2 Cos[
y]}}]]}, {x, -\[Pi]/3, \[Pi]/
3, \[Pi]/(60.)}, {y, -\[Pi], \[Pi], \[Pi]/(20.)}];
ListPlot3D[{Flatten[dat3D, 1]}, BoxRatios -> {1, 1, 1},
PlotRange -> {-4, 4}, PlotTheme -> "Detailed", PlotLegends -> False]


but does not give the correct results

• This looks like a bug of "Evaluate" to me. Please report it to "support@wolfram.com" Jan 9 at 11:15

Removing Evaluate command does give me separated eigenvalues.Is this correct?

Plot3D[Sort[Eigenvalues[{{2*Sin[Pi/6 - y], -1, -E^(3*I*x)}, {-1,
2*Sin[Pi/6 + y], -1},
{-E^(-3*I*x), -1, -2*Cos[y]}}]], {x, -Pi/3, Pi/3}, {y, -Pi, Pi},
BoxRatios -> {1, 1, 1}, Mesh -> None,
PlotRange -> {-4, 4}, PlotTheme -> "Detailed", PlotLegends ->
False, ClippingStyle -> Automatic]


Also,attempting your approach of sorting individual points gave me this

   Graphics3D[Point[Sort[Flatten[Table[({#1[[1]], #1[[2]], #1[[3]][[n]]} & ) /@
Flatten[Table[{x, y, Eigenvalues[N[{{2*Sin[Pi/6 - y], -1, -
E^(3*I*x)}, {-1, 2*Sin[Pi/6 + y], -1},
{-E^(-3*I*x), -1, -2*Cos[y]}}]]}, {x, -Pi/3, Pi/3, Pi/60.},
{y, -Pi, Pi, Pi/20.}], 1], {n, 1, 3}], 1]]]