Bose-Einstein condensates in a ring-shaped trap with a nonlinear double-well potential

We develop the mean-field theory for Bose-Einstein condensates in a one-dimensional ring with two types of nonlinear double-well potentials, based on a pair of δ functions or Gaussian of a finite width, placed at diametrically opposite points. By analyzing the ground states (GSs) in these cases, we find a qualitative difference between them. With the Gaussian profile, the GS always undergoes the phase transition from the symmetric shape to an asymmetric one at a critical value of the norm. In contrast, the symmetry-breaking transition does not happen with the δ-functional profile of the nonlinearity, and the GS always remains symmetric. In addition, the numerical analysis for the Gaussian profile demonstrates that the type of symmetry-breaking transition depends on the width of the nonlinear-potential well.