Local manifestations of thickness dependent topology and axion edge state in topological magnet MnBi$_2$Te$_4$

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.