Entanglement entropy and entanglement spectrum of Bi_1-xSb_x (111) bilayers.

We study topological properties of Bi$_{1-x}$Sb$_{x}$ bilayers in the (111) plane using entanglement measures. Electronic structures are investigated within multi-orbital tight-binding model and structural stability is confirmed through first-principles calculations. Topologically non-trivial nature of bismuth 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 Bi bilayer and strain in Sb bilayer. Composition- and strain-induced phase transitions reveal a finite discontinuity in the entanglement entropy. This quantity remains a continuous function of the electric field strength, but shows a finite discontinuity in the first derivative. We relate the difference in behavior of the entanglement entropy to the breaking of inversion symmetry in the last case.