Counterintuitive Emergence of the π Zak Phase on the (111) Surface of Bismuth

Bismuth (Bi) has emerged as a crucial element in the realm of topological materials due to its strong spin-orbit coupling, leading to band inversions and various topologically nontrivial phases. This study unveils an additional nontrivial π Zak phase in Bi, inducing metallic in-gap states on the (111) surface. The Zak phase, one of the fundamental topological invariants, is shown to depend on surface orientation and termination (i.e., the sign of dimerization). In sharp contrast to the Su-Schrieffer-Heeger model, the one-dimensional counterpart of Bi, where nontrivial edge modes emerge at ends cutting strong bonds, the nontrivial surface states of Bi appear on the (111) surface where weak bonds are cut, which is counterintuitive. These findings elucidate the underlying origins of the intriguing (111) surface states of Bi, often interpreted as an analog of antimony, exhibiting a similar but topologically distinct electronic structure.