Analytical approach for the Mott transition in the Kane-Mele-Hubbard model

The description of interactions in strongly-correlated topological phases of matter remains a challenge. Here, we develop a stochastic functional approach for interacting topological insulators including both charge and spin channels. We find that the Mott transition of the Kane-Mele-Hubbard model may be described by the variational principle with one equation. We present different views of this equation from the electron Green's function, the free-energy and the Hellmann-Feynman theorem. The band gap remains finite at the transition and the Mott phase is characterized by antiferromagnetism in the $x-y$ plane. The interacting topological phase is described through a $\mathbb{Z}_2$ number related to helical edge modes. Our results then show that improving stochastic approaches can give further insight on the understanding of interacting phases of matter.