Continuum Theory of Edge States of Topological Insulators: Variational Principle and Boundary Conditions

Abstract

We develop a continuum theory to model low energy excitations of a generic four-band time reversal invariant electronic system with boundaries. We propose a variational energy functional for the wavefunctions which allows us derive natural boundary conditions valid for such systems. Our formulation is particularly suited to develop a continuum theory of the protected edge/surface excitations of topological insulators both in two and three dimensions. By a detailed comparison of our analytical formulation with tight binding calculations of ribbons of topological insulators modeled by the Bernevig-Hughes-Zhang (BHZ) hamiltonian, we show that the continuum theory with the natural boundary condition provides an appropriate description of the low energy physics. As a spin-off, we find that in a certain parameter regime, the gap that arises in topological insulator ribbons of finite width due to the hybridization of edges states from opposite edges, depends non-monotonically on the ribbon width and can nearly vanish at certain "magic widths".

8 pages, 5 figures

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Cited by in corpus (9)

• Correlation effects in two-dimensional topological insulators
• Unidirectional Maxwellian Spin Waves
• Helical edge states in multiple topological mass domains
• Modeling Near-Surface Bound Electron States in Three-Dimensional Topological Insulator: Analytical and Numerical Approaches
• Confining electrons on a topological insulator surface using potentials and a magnetic field
• Topological viewpoint of two-dimensional group III-V and IV-IV compounds in the presence of electric field and spin-orbit coupling by density functional theory and tight-binding model
• Low-energy modeling of three-dimensional topological insulator nanostructures
• Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
• Hard wall edge confinement in 2D topological insulators and the energy of the Dirac Point