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Spontaneous symmetry breaking in a split potential box.

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This study analyzes spontaneous symmetry breaking (SSB) in a 1D model using the Gross-Pitaevskii equation. The ground state always exhibits supercritical SSB, with excited states showing varied stability and bifurcation behaviors.

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Area of Science:

  • Quantum physics
  • Nonlinear dynamics
  • Mathematical modeling

Background:

  • Spontaneous symmetry breaking (SSB) is a key phenomenon in physics.
  • A 1D model based on the Gross-Pitaevskii-nonlinear Schrödinger equation is utilized.
  • The model features a double-well potential with a tunable barrier strength (ɛ).

Purpose of the Study:

  • To analyze SSB in a simplified 1D setting.
  • To investigate the behavior of ground and excited states under varying barrier strengths.
  • To compare analytical predictions with numerical simulations.

Main Methods:

  • Analytical prediction in limit cases (ɛ≫1 and ɛ≪1).
  • Variational approximation (VA) for generic cases.
  • Numerical simulations and stability analysis via eigenvalue calculations.

Main Results:

  • The ground state consistently undergoes supercritical SSB bifurcation.
  • Variational approximation is accurate for moderate ɛ but fails at small ɛ.
  • Approximation based on a soliton ansatz correctly treats the small ɛ case.
  • The first excited state (antisymmetric mode) destabilizes at a critical norm.
  • The second excited state exhibits SSB bifurcation, yielding an unstable asymmetric mode.

Conclusions:

  • The 1D model effectively demonstrates SSB phenomenology.
  • Supercritical SSB of the ground state is robust across different approximations.
  • Excited states exhibit complex stability and bifurcation dynamics.
  • Unstable modes tend to evolve into the asymmetric ground state.