Spontaneous Lorentz Symmetry Breaking and Topological Defects for the Chern-Simons Matter Vector Field

The study of topological defects occurring in vector and tensor fields is an intriguing subject and little explored in the literature. In this article, we analyze the topological defects arising from the spontaneous violation of Lorentz symmetry for a vector matter field with Chern-Simons term in a Minkowski spacetime in $(1+2)D$. As a consequence, the resulting nonlinear equations include a topological mass via the Chern-Simons term, which leads to a vector version of a soliton state. We show that the topological defects arising from the vector field can be categorized as either vortices, with topology $S^{1}\times\mathbb{R}$, or domain-walls, with topology $S^{0} \times\mathbb{R}^2$. The vortex solutions were analyzed using a procedure similar to the Nielsen-Olesen one, though extended to the vector case to account for a spontaneous violation of Lorentz symmetry. We also analyze the influence of the topological mass and verify the stability of the model as well as the magnetic vortex in $(1+2)D$. We show that domain-wall solutions also emerge as an effect of the violation of the Lorentz symmetry expressed by the vacuum of the vector field. We obtain general equations for this new class of domain walls involving the Lorentz violation parameter and the topological mass. By means of the energy-momentum tensor, we verify the instability of the formation of these domain walls in $(1+2)D$.