Non-Abelian effective mechanism from layer pseudospin and lattice pseudospin in twisted bilayer graphene

In the work by Tarnopolsky [Phys. Rev. Lett. 122, 106405 (2019)]PRLTAO0031-900710.1103/PhysRevLett.122.106405 they obtained the magic angles for superconductivity in twisted bilayer graphene (TBG) with a fundamental continuum model for TBG which captures the physics of the flat band. In this Letter we study the continuum model in detail and demonstrate that the system of TBG described by the continuum model is equivalent to a dynamical system that is composed of three parts of effective interactions: the coupling between two effective non-Abelian gauge fields originating from the periodic hopping potential and two effective massless fermion fields which have 1/2 lattice pseudospin and a twist-angle-dependent effective charge, the coupling between a pseudomagnetic field created by gauge fields and a layer pseudospin, and an effective pseudo-Rashba effect in an equivalent unrotated single layer with twist-angle-dependent Rashba coupling strength. The effective Rashba effect is shown to be also the sum of the pseudo-Rashba effects in two effective twisted single layers with a relative twist angle π-θ. The longitudinal component τz of the layer pseudospin vector is revealed to have the physical meaning of an interlayer electric field. Due to the relative twist between the two layers it geometrically leads to the nonvanishing curvature on a vector bundle in a two-dimensional effective hexagonal lattice. The geometric curvature on the vector bundle creates a pseudomagnetic field physically and interacts with the layer pseudospin.