Conical intersections in condensed matter physics

So far we have seen how conical intersections (CIs) appear in molecular systems, and how they may influence physical properties. These CIs showed up in the Born–Oppenheimer treatment and could be analysed in terms of synthetic gauge fields. Even though the adiabatic potential surfaces (APSs) often are given in normal vibrational coordinates, they still represent real space coordinates, i.e., the CIs appear in the real space. In this chapter, we see how CIs can occur in momentum space instead as so called Dirac cones or, as we will call them, Dirac CIs. The paradigmatic example of such Dirac CIs is graphene where carbon atoms form a two-dimensional hexagonal lattice. The periodicity of the lattice implies that the spectrum consists of energy bands restricted to the first Brillouin zone. Dirac CIs are point degeneracies of pairs of energy bands. We show, analogously to what we saw in the previous chapter, that there is a gauge structure connected to the physics of the bands. From this, one can define topological invariances like the Chern number. We discuss how the topological features are manifest in physical observables such as conductivity and edge states. While much of the chapter is devoted to periodic systems, we also give two examples where Dirac CIs emerge in continuum systems: spin–orbit coupled systems and superconductors. Superconductors differ from the other examples since the Dirac CIs appear at the mean-field level and require interaction between the electrons.