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

TL;DR

This paper studies the topological properties of Bi$_{1-x}$Sb$_{x}$ (111) bilayers using entanglement measures, revealing phase transitions and symmetry effects through theoretical and computational methods.

Contribution

It demonstrates the use of entanglement spectrum and entropy to identify topological phases and phase transitions in Bi-Sb bilayers, including effects of composition, electric field, and strain.

Findings

Spectral flow confirms topologically non-trivial Bi bilayer.

Discontinuities in entanglement entropy indicate phase transitions due to composition and strain.

Electric field induces continuous entanglement entropy, but with a discontinuous first derivative.

Abstract

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.