  Realization and design of topological insulators emerging from electron
correlations, called topological Mott insulators (TMIs), is pursued by using
mean-field approximations as well as multi-variable variational Monte Carlo
(MVMC) methods for Dirac electrons on honeycomb lattices. The topological
insulator phases predicted in the previous studies by the mean-field
approximation for an extended Hubbard model on the honeycomb lattice turn out
to disappear, when we consider the possibility of a long-period
charge-density-wave (CDW) order taking over the TMI phase. Nevertheless, we
further show that the TMI phase is still stabilized when we are able to tune
the Fermi velocity of the Dirac point of the electron band. Beyond the
limitation of the mean-field calculation, we apply the newly developed MVMC to
make accurate predictions after including the many-body and quantum
fluctuations. By taking the extrapolation to the thermodynamic and weak
external field limit, we present realistic criteria for the emergence of the
topological insulator caused by the electron correlations. By suppressing the
Fermi velocity to a tenth of that of the original honeycomb lattice, the
topological insulator emerges in an extended region as a spontaneous symmetry
breaking surviving competitions with other orders. We discuss experimental ways
to realize it in a bilayer graphenesystem.
