Curvature induced quantum phase transitions in an electron-hole system

Zhuo Bin Siu,Ching-Ray Chang,Mansoor B A Jalil,Jian-Yuan Chang,Seng Ghee Tan

doi:10.1038/s41598-018-34903-9

Abstract

In this work, we study the effect of introducing a periodic curvature on nanostructures, and demonstrate that the curvature can lead to a transition from a topologically trivial state to a non-trivial state. We first present the Hamiltonian for an arbitrarily curved nanostructure, and introduce a numerical scheme for calculating the bandstructure of a periodically curved nanostructure. Using this scheme, we calculate the bandstructure for a sinusoidally curved two-dimensional electron gas. We show that the curvature can lead to a partner switching reminiscent of a topological phase transition at the time reversal invariant momenta. We then study the Bernevig-Hughes-Zhang (BHZ) Hamiltonian for a two-dimensional quantum well. We show that introducing a curvature can lead to the emergence of topological surface states.

Highlights

Quantum cosmology is intimately linked to space-time curvature1–3
We show that the curvature can substantially change the band-structure in a manner that is reminiscent of the transition between a topologically trivial insulator to a non-trivial state
In Section (IV), we demonstrate that for the Bernevig-Hughes-Zhang Hamiltonian24 describing a two-dimensional topological insulator, introducing a curvature does lead to a topological phase transition from a topologically trivial insulator to a non-trivial insulator with topological surface states

Summary

Introduction

Quantum cosmology is intimately linked to space-time curvature. It is difficult to obtain experimental evidence of the curvature effect in the universe. We show that the curvature can substantially change the band-structure in a manner that is reminiscent of the transition between a topologically trivial insulator to a non-trivial state. This is, not yet a true topological phase transition. In Section (IV), we demonstrate that for the Bernevig-Hughes-Zhang Hamiltonian describing a two-dimensional topological insulator, introducing a curvature does lead to a topological phase transition from a topologically trivial insulator to a non-trivial insulator with topological surface states. Where the metric tensor matrix elements gij ≡ ei ⋅ ej, and the momentum operator pi acting to the right on a spatial wavefunction ψ, piψ ≡ −igij∂jψ These transformation rules allow us to, for instance, convert a system written in terms of the Cartesian coordinates (x, y, z) into the curvilinear qi coordinates

Objectives

Methods

Conclusion