Geometric properties of band crossing points in two dimensions

The band crossing points (BCPs) in 2D are usually characterized by the Berry phase. In this work, we study the geometric properties of BCPs in 2D. We show that the geometric nature of BCPs are characterized by the quantum distance or quantum metric more appropriately, rather than the Berry phase.

For linear BCPs, the quantum distance is maximal and this result naturally explains the fundamental origin of the π-Berry phase of a Dirac point. Also, we find that the quantum distance fully characterizes quadratic BCPs in the flat band limit where one of the two crossing bands becomes completely flat. To further develop this result, we introduce a construction scheme for flat-band system exhibiting the BCPs and discuss the symmetry properties of the BCPs.