Spatial Chern-Simons Interactions and Complex Magnetic Penetration Depth

This paper examines the Landau-Ginzburg theory in the presence of spatial Chern-Simons interactions, which typically emerge in Weyl semimetals due to domain-wall motion. It is demonstrated that the incorporation of a purely spatial Chern-Simons term, which violates parity, into the Landau-Ginzburg free energy leads to a complex magnetic penetration depth. This characteristic indicates that, aside from possessing an effective penetration depth, the magnetic field on the surface of the superconductor experiences periodic spatial oscillations, which significantly deviates from the conventional Meissner effect. In particular, it is observed that as the degree of parity breaking, quantified by the strength of the spatial Chern-Simons term, increases, a vortex solution with magnetic field inversion may emerge. With the discovery of superconductivity in certain Weyl semimetals, the possibility of experimentally observing this phenomenon in these materials is anticipated.