  We studied electronic band structure and topological property of a
topological insulator thin film under a moir\'e superlattice potential to
search for two-dimensional (2D) $\mathbb Z_2$ non-trivial isolated mini-bands.
To model this system, we assume the Fermi energy inside the bulk band gap and
thus consider an effective model Hamiltonian with only two surface states that
are located at the top and bottom surfaces and strongly hybridized with each
other. The moir\'e potential is generated by another layer of 2D insulating
materials on top of topological insulator films. In this model, the lowest
conduction (highest valence) mini-bands can be $\mathbb Z_2$ non-trivial when
the minima (maxima) of the moir\'e potential approximately forms a hexagonal
lattice with six-fold rotation symmetry. For the nontrivial conduction
mini-band cases, the two lowest Kramers' pairs of conduction mini-bands both
have nontrivial $\mathbb Z_2$ invariant in presence of inversion, while
applying external gate voltages to break inversion leads to only the lowest
Kramers' pair of mini-bands to be topologically non-trivial. The Coulomb
interaction can drive the lowest conduction Kramers' mini-bands into the
quantum anomalous Hall state when they are half-filled, which is further
stabilized by breaking inversion symmetry. We propose the monolayer Sb$_{2}$ on
top of Sb$_2$Te$_3$ thin films to realize our model based on results from the
first principles calculations.
