The Origin of the Dirac Linear Spectrum and a Study on the Topological Bilayers

We propose that the linear Dirac spectrum and the Rashba interaction, two important features of the topological insulators, have a common origin rooted on the well-known non-relativistic Schrödinger kinetic energy for fermions subject to the spin-momentum locking condition described here. This approach yields that the quasi-particles with linear Dirac spectrum are skyrmions, namely, three-dimensional vortices with topological charges surrounded by a residual local magnetic field. Within this framework, the spectrum of topological bilayers is obtained and shown for a small wave number to have two branches, one quadratic and the other linear but with twice the velocity of the single layer case.