  Time-reversal symmetric topological insulator is a novel state of matter that
a bulk-insulating state carries dissipationless spin transport along the
surfaces, embedded by the $Z_2$ topological invariant. In the noninteracting
limit, this exotic state has been intensively studied and explored with
realistic systems, such as HgTe/(Hg,Cd)Te quantum wells. Yet an interacting
topological insulator is still an elusive subject, and most related analyses
rely on the mean-field approximation and numerical simulations. Among the
approaches, the mean-field approximation fails to predict the topological phase
transition, in particular at intermediate interaction strength without
spontaneously breaking symmetry. In this review, we develop an analytical
approach based on a combined perturbative and self-consistent mean-field
treatment of interactions that is capable of capturing topological phase
transitions beyond either method when used independently. As an illustration of
the method, we study the effects of short-ranged interactions on the $Z_2$
topological insulator phase, also known as the quantum spin Hall phase, in
three generalized versions of the Kane-Mele model at half-filling on the
honeycomb lattice. The results are in excellent agreement with quantum Monte
Carlo calculations on the same model, and cannot be reproduced by either a
perturbative treatment or a self-consistent mean-field treatment of the
interactions. Our analytical approach helps to clarify how the symmetries of
the one-body terms of the Hamiltonian determine whether interactions tend to
stabilize or destabilize a topological phase. Moreover, our method should be
applicable to a wide class of models where topological transitions due to
interactions are in principle possible, but are not correctly predicted by
either perturbative or self-consistent treatments.
