s ={Cos[\[Pi]/8] E^(-I \[Pi]/8), Sin[\[Pi]/8] E^(I \[Pi]/8)};
s = s/Norm[s];
evox = Reap[
    For[\[Theta] = 0, \[Theta] <= 4 \[Pi], \[Theta] = \[Theta] + 0.1,
      tmp = MatrixExp[I/2 PauliMatrix[1]*\[Theta]].s;
      Sow[Chop[tmp]];
      ];][[2, 1]];
(*The "circle3D" function definition was taken from https://mathematica.stackexchange.com/questions/79524/how-to-draw-a-circle-in-3d-on-a-sphere*)
circle3D[centre_: {0, 0, 0}, radius_: 1, normal_: {0, 0, 1}, 
  angle_: {0, 2 Pi}] := 
 Composition[Line, Map[RotationTransform[{{0, 0, 1}, normal}, centre], #] &, Map[Append[#, Last@centre] &, #] &, Append[DeleteDuplicates[Most@#], Last@#] &, Level[#, {-2}] &, MeshPrimitives[#, 1] &, DiscretizeRegion, If][First@Differences@angle >= 2 Pi, Circle[Most@centre, radius], Circle[Most@centre, radius, angle]]
(*The "Unwrap" function definition was taken from https://community.wolfram.com/groups/-/m/t/1340126*)
ClearAll[Unwrap]
Unwrap[lst_List] := Unwrap[lst, 2 Pi] (*phase jumps of 2Pi is the default because of trigonometric funtions*)
Unwrap[lst_List, \[CapitalDelta]_] := Unwrap[lst, \[CapitalDelta], Scaled[0.5]] (*default tolerance is half the phase jump \[CapitalDelta]*)
Unwrap[lst_List, \[CapitalDelta]_, tolerance_] := Module[{tol, jumps}, tol = If[Head[tolerance] === Scaled, \[CapitalDelta] tolerance[[1]], tolerance];
  jumps = Differences[lst];
  jumps = -Sign[jumps] Unitize[Chop[Abs[jumps], tol]];
  jumps = \[CapitalDelta] Prepend[Accumulate[jumps], 0];
  jumps + lst]
(**)
pole[s_] := {Conjugate[s].PauliMatrix[1].s, Conjugate[s].PauliMatrix[2].s, Conjugate[s].PauliMatrix[3].s};
flag[s_] := (Arg[s[[2]]] + Arg[s[[1]]])/2;
\[Alpha] = Unwrap[flag /@ evox, \[Pi]];
\[Alpha] = \[Alpha] - Min[\[Alpha]];
\[Alpha] = \[Alpha]/Max[\[Alpha]];
psx = Table[
   Graphics3D[{Text[Style["x", Bold, Large], {1.4, 0, 0}], Text[Style["y", Bold, Large], {0, 1.4, 0}], Text[Style["z", Bold, Large], {0, 0, 1.4}], Thick, 
     Arrow[{{-1.2, 0, 0}, {1.3, 0, 0}}], Arrow[{{0, -1.2, 0}, {0, 1.3, 0}}], Arrow[{{0, 0, -1.2}, {0, 0, 1.3}}], Hue[\[Alpha][[j]]], PointSize[0.05], Point[pole[evox[[j]]]], Line[{{0, 0, 0}, pole[evox[[j]]]}], Opacity[0.8], White, Sphere[{0, 0, 0}, 1], Opacity[1], Gray, circle3D[{0, 0, 0}, 1, {0, 0, 1}], 
     circle3D[{0, 0, 0}, 1, {1, 0, 0}], circle3D[{0, 0, 0}, 1, {0, 1, 0}]}, Boxed -> False, Lighting -> "Neutral", ViewPoint -> {2, 2, 2},
    Epilog -> Inset[BarLegend[{Hue[#/(2 \[Pi])] &, {0, 2 \[Pi]}}, Ticks -> Table[j, {j, 0, 2 \[Pi], \[Pi]/2}], LegendLabel -> "Spinor's\nglobal phase"], {0.9, 0.5}]
    ]
   , {j, 1, dim}];
ListAnimate[psx]

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