Spontaneous symmetry and antisymmetry breaking in a ring with two potential barriers

We propose a fundamental setup for the realization of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of the nonlinear Schrödinger equation with the self-attractive or repulsive cubic term, respectively, on a one-dimensional ring split in two mutually symmetric boxes by delta-functional potential barriers, placed at opposite points. The system is relevant to optics and Bose-Einstein condensation. The spectrum of the linearized system is found in analytical and numerical forms. SSB and SASB are predicted by dint of the variational approximation and studied in the numerical form. A particular stable solution, which demonstrates strong asymmetry, is found in an exact form. In the system with the attractive nonlinearity, the SSB of the symmetric ground state is initiated by the modulational instability. It creates stationary asymmetric states through a supercritical bifurcation. In the self-repulsive system, SASB makes the lowest antisymmetric excited state unstable, transforming it into an antisymmetry-breaking oscillatory mode.