Entanglement entropy and entanglement spectrum of Bi${}_{1-x}$Sb${}_x$ (111) bilayers

We investigate topological properties of Bi${}_{1-x}$Sb${}_x$ bilayers in the (111) plane using entanglement measures. Electronic structures are studied using multi-orbital tight-binding model, with structural stability confirmed through first-principles calculations. Topologically non-trivial nature of Bi bilayer is proved by the presence of spectral flow in the entanglement spectrum. We consider topological phase transitions driven by a composition change $x$, an applied external electric field in pure Bi, and strain in pure Sb. Composition- and strain-induced phase transitions reveal a finite discontinuity in the entanglement entropy. However, this quantity remains a continuous function of electric field strength, but shows a discontinuity in the first derivative. We relate the difference in behavior of the entanglement entropy to breaking of the inversion symmetry in the last case.