spontaneous_symmetry_breaking.nb

(*Given Potential*)
V[\[CurlyPhi]_, \[Mu]_] := -\[Mu]^2 \[CurlyPhi]^2/
     2 + \[Lambda]^2 \[CurlyPhi]^4 /2 ;
(*Maxima Minima Routine*)  
	(*Points of Inflection*)
	\[Delta]  = Solve[ V'[\[CurlyPhi]] == 0, \[CurlyPhi]];
	Subscript[\[CurlyPhi], 1] = \[Delta][[1]];
	Subscript[\[CurlyPhi], 2] = \[Delta][[2]];
	Subscript[\[CurlyPhi], 3] = \[Delta][[3]];
V''[\[CurlyPhi]] ;
(*\[Epsilon] =Replace[V''[\[CurlyPhi]] ,\[CurlyPhi] \[Rule] \
Subscript[\[CurlyPhi], 1]]*)
(*If[\[Epsilon] > 0,Print["The minima is Subscript[\[CurlyPhi], \
1]"],Print["The maxima is Subscript[\[CurlyPhi], 2]"]]*)

(* TO DO *)

\[Lambda] = 
  1;(*For the shake of plotting \[Lambda] is taken constant*)
(*2D view*)
Manipulate[
 Plot[y = -\[Mu]^2 \[CurlyPhi]^2/2 + \[Lambda]^2 \[CurlyPhi]^4 /
      2 , {\[CurlyPhi], -8, 8}], {{\[Mu], 0, 
   "Symmetry breaking parameter"}, -6, 6, Appearance -> "Labeled"}]
(* The potential  is symmetric for \[Lambda] = 0 and the symmetry
breaks spontaneously in either directions of \[Lambda] giving the
peculiar Mexican hat*)

(*--------------3D version--------------*)
(*A more realistic 3D view*)
V[\[CurlyPhi]_, \[Mu]_] := -\[Mu] \[CurlyPhi]^2/
    2 + \[Lambda]^2 \[CurlyPhi]^4 /
    2(*Square and 4th power terms \[CurlyPhi]^4+\[Mu] \[CurlyPhi]^2*)
\

Manipulate[
 RevolutionPlot3D[
  V[\[CurlyPhi], \[Mu]], {\[CurlyPhi], 0, 2}, {q, 0, 2 Pi}, 
  PlotPoints -> 30, PlotStyle -> Opacity[.5], Mesh -> 20, 
  MeshStyle -> Opacity[.5], SphericalRegion -> True, 
  BoxRatios -> {1, 1, .5}, ImageSize -> 400 , 
  PlotLabel -> 
   "Spontaneous Symmetring Breaking \nin Electroweak Potential in
Early Universe"], {{\[Mu], 4, "Symmetry breaking parameter"}, -6, 6, 
  Appearance -> "Labeled"}, {{\[CurlyPhi], 10, 
   "Potential in Lagrangian"}, 1, 10, 
  Appearance -> "Labelled"}, {{\[Mu] , 1, "Scalar \[Mu] "}, -5, 5 , 
  Appearance -> "Labelled"}]
(* The potential  is symmetric for \[Lambda] = 0 and the symmetry
breaks spontaneously in either directions of \[Lambda] giving the
peculiar Mexican hat*)