Abstract
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 $\ensuremath{\delta}$ 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 $\ensuremath{\delta}$-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.
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