Inhomogeneous abelian Chern-Simons Higgs model with new inhomogeneous BPS vacuum and solitons

We study an inhomogeneous U(1) Chern-Simons Higgs model with a magnetic impurity in the BPS limit. The potential is sextic with both broken and unbroken phases, but its minimum varies spatially depending on the strength of the impurity. While the system lacks translation symmetry, it admits a supersymmetric extension. Depending on the sign of the impurity term, it has either a BPS sector or an anti-BPS sector (but not both), which satisfies the Bogomolny equations. The vacuum configuration of the broken phase is not simply determined by the minimum of the potential since it is no longer constant, but it becomes a nontrivial function satisfying the Bogomolny equations. Thus, the energy and angular momentum densities of the vacuum locally have nonzero distributions, although the total energy and angular momentum remain zero. As in the homogeneous case, the theory supports various BPS soliton solutions, including topological and nontopological vortices and Q-balls. The vorticities as well as the U(1) charges are exclusively positive or negative. For a Gaussian type impurity as a specific example, we obtain rotationally symmetric numerical solutions and analyze their detailed properties. We also discuss the case of a delta-function impurity as the infinitely thin limit of the Gaussian impurity which shows some nontrivial feature of BPS Chern-Simons Higgs theory.