Electromagnetic response in a quantum spin Hall insulator with strong correlation

The quantum spin Hall system is a time-reversal invariant band insulator with a non-trivial topological electronic structure caused by the spin-orbit coupling. We investigate such a system with strong electron correlation. The on-site correlation can be expressed by the spin gauge field coupled to the electron spin $(s_x,s_y,s_z)$ in the strong coupling limit. Electromagnetic response of this system can be derived to integrate out Fermions and also spin gauge field. We find that the system shows superconducting response when $s_z$ is conserved, and becomes insulating when $s_z$-conservation is broken by a perturbation like the Rashba term. In our discussions an induced BF-term, which is the topological term with mixing of electromagnetic gauge field and spin gauge field coupled to $s_z$, plays an important role.