  Topological insulators present a bulk gap, but allow for dissipationless spin
transport along the edges. These exotic states are characterized by the $Z_2$
topological invariant and are protected by time-reversal symmetry. The
Kane-Mele model is one model to realize this topological class in two
dimensions, also called the quantum spin Hall state. In this review, we provide
a pedagogical introduction to the influence of correlation effects in the
quantum spin Hall states, with special focus on the half-filled
Kane-Mele-Hubbard model, solved by means of unbiased determinant quantum Monte
Carlo (QMC) simulations. We explain the idea of identifying the topological
insulator via $\pi$-flux insertion, the $Z_2$ invariant and the associated
behavior of the zero-frequency Green's function, as well as the spin Chern
number in parameter-driven topological phase transitions. The examples
considered are two descendants of the Kane-Mele-Hubbard model, the generalized
and dimerized Kane-Mele-Hubbard model. From the $Z_2$ index, spin Chern numbers
and the Green's function behavior, one can observe that correlation effects
induce shifts of the topological phase boundaries. Although the implementation
of these topological quantities has been successfully employed in QMC
simulations to describe the topological phase transition, we also point out
their limitations as well as suggest possible future directions in using
numerical methods to characterize topological properties of strongly correlated
condensed matter systems.
