  The interplay of non-trivial band topology and magnetism gives rise to a
series of exotic quantum phenomena, such as the emergent quantum anomalous Hall
(QAH) effect and topological magnetoelectric effect. Many of these quantum
phenomena have local manifestations when the global symmetry is broken. Here,
we report local signatures of the thickness dependent topology in intrinsic
magnetic topological insulator MnBi$_2$Te$_4$(MBT), using scanning tunneling
microscopy and spectroscopy on molecular beam epitaxy grown MBT thin films. A
thickness-dependent band gap with an oscillatory feature is revealed, which we
reproduce with theoretical calculations. Our theoretical results indicate a
topological quantum phase transition beyond a film thickness of one monolayer,
with alternating QAH and axion insulating states for even and odd layers,
respectively. At an even-odd layer step, a localized gapped electronic state is
observed, in agreement with an axion insulator edge state that results from a
phase transition across the step. The demonstration of thickness-dependent
topological properties highlights the role of nanoscale control over novel
quantum states, reinforcing the necessity of thin film technology in quantum
information science applications.
