  We investigate the ground-state phase diagram of the spinful extended
Haldane-Hubbard model on the honeycomb lattice using an exact-diagonalization,
mean-field variational approach, and further complement it with the infinite
density matrix renormalization group, applied to an infinite honeycomb
cylinder. This model, governed by both on-site and nearest-neighbor
interactions, can result in two types of insulators with finite local order
parameters, either with spin or charge ordering. Moreover, a third one, a
topologically nontrivial insulator with nonlocal order, is also manifest. We
test expectations of previous analyses in spinless versions asserting that once
a local order parameter is formed, the topological characteristics of the
ground state, associated with a finite Chern number, are no longer present,
resulting in a topologically trivial wave function. Our study confirms this
overall picture, and highlights how finite-size effects may result in
misleading conclusions on the coexistence of finite local order parameters and
nontrivial topology in this model.
