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We studied electronic band structure and topological property of atopological insulator thin film under a moir\'e superlattice potential tosearch 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 andthus consider an effective model Hamiltonian with only two surface states thatare located at the top and bottom surfaces and strongly hybridized with eachother. The moir\'e potential is generated by another layer of 2D insulatingmaterials on top of topological insulator films. In this model, the lowestconduction (highest valence) mini-bands can be $\mathbb Z_2$ non-trivial whenthe minima (maxima) of the moir\'e potential approximately forms a hexagonallattice with six-fold rotation symmetry. For the nontrivial conductionmini-band cases, the two lowest Kramers' pairs of conduction mini-bands bothhave nontrivial $\mathbb Z_2$ invariant in presence of inversion, whileapplying external gate voltages to break inversion leads to only the lowestKramers' pair of mini-bands to be topologically non-trivial. The Coulombinteraction can drive the lowest conduction Kramers' mini-bands into thequantum anomalous Hall state when they are half-filled, which is furtherstabilized by breaking inversion symmetry. We propose the monolayer Sb$_{2}$ ontop of Sb$_2$Te$_3$ thin films to realize our model based on results from thefirst principles calculations.

$\mathbb Z_2$-Nontrivial Moir\'e Minibands and Interaction-Driven  Quantum Anomalous Hall Insulators in Topological Insulator Based Moir\'e  Heterostructures