A repulsive core + attractive shell (Mexican hat) gives an ordered ground state — qualitative SSB / Higgs-mode geometry in GPE.

Purpose

Model **symmetry breaking** in the same GPE toolbox: an isotropic Mexican-hat external potential $V_{\rm ext}\propto (r^2/R^2-1)^2$ drives the condensate into an ordered, **ring-shaped** ground state rather than a uniform VEV.

What it proves

Imaginary-time relaxation places the density peak off-centre on the scale $R$ , depresses the core below a fraction of the peak, and concentrates a majority of the norm in an annulus around $R$ — matching the qualitative SSB / 'Higgs mode' geometry checklist used in-code.

Relation to current theory

Electroweak physics is not solved here; this is the vortex-condensate analogue of 'flat directions lifted → ordered vacuum', parallel in spirit to sim_25 domain walls and sim_56 holonomy toys.

Key equation

V_{\rm ext}(r) \propto \bigl(r^2/R^2 - 1\bigr)^2

Interactive visualization

Illustrative live GPE field for this sim's physics

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Plots

Scalar metrics

mass in annulus100

stdout tail

=================================================================
 SVT Simulation #60: double-well / Mexican-hat GPE ground state
=================================================================
  r_peak / R        : 0.977  (expect ~1, not 0)
  rho_center / peak : 0.000  (expect < 0.65)
  mass in annulus   : 100.0%

=================== VALIDATION ===================
  ring peak off-centre         : PASS
  central depression           : PASS
  annulus mass fraction      : PASS
  Overall                      : PASS

  SVT Prediction: A repulsive core + attractive shell (Mexican hat) gives an ordered ground state — qualitative SSB / Higgs-mode geometry in GPE.
==================================================

  Matches data:   YES — Validated