WKB Theory for \(\alpha -\mathcal {T}_3\) Materials and a Dice Lattice

This chapter is concerned with developing complete Wentzel–Kramers–Brillouin theory for a dice lattice and for the \(\alpha \) - \(\mathcal {T}_3\) model, which describes a wide class of existing pseudospin-1 Dirac materials. Creating a semiclassical model for any new material traditionally builds upon an expansion of the sought wave function in a power series over the Planck’s constant \(\hbar \) and includes deriving and solving the transport equations connecting different terms of this series expansion and finding the semiclassical action. Most importantly, we are going to obtain a set of analytical expressions for the leading-order WKB wave function which is the key prerequisite of studying the unusual electronic phenomena in each new material. We will also derive and analyze the applicability conditions of the WKB approximation for the \(\alpha -\mathcal {T}_3\) model.