  We study the effect of stacking faults on the topological properties of the
magnetic topological insulator MnBi$_{2}$Te$_{4}$ (MBT) using density
functional theory calculations and the Hubbard $U$ being tuned with many-body
diffusion Monte Carlo techniques. We show that a modest deviation from the
equilibrium interlayer distance leads to a topological phase transition from a
non-trivial to a trivial topology, suggesting that tuning the interlayer
coupling by adjusting the interlayer distance alone can lead to different
topological phases. Interestingly, due to the locally increased interlayer
distance of the top layer, a metastable stacking fault in MBT leads to a nearly
gapless state at the topmost layer due to charge redistribution as the topmost
layer recedes. We further find evidence of spin-momentum locking in the surface
state along with a weak preservation of the band inversion in the near gapless
state, which is indicative of the non-trivial topological surface states for
the metastable stacking fault. Our findings provide a possible explanation for
reconciling the long-standing puzzle of gapped and gapless states on MBT
surfaces.
