Wave-function geometry of band crossing points in two dimensions

Geometry of the wave function is a central pillar modern solid state physics. In this work, we unveil wave-function geometry two-dimensional semimetals with band crossing points (BCPs). We show that Berry phase BCPs are governed by quantum metric describing infinitesimal distance between states. For generic linear BCPs, corresponding determined either an angular integral metric, or equivalently, maximum Bloch This naturally explains origin $\ensuremath{\pi}$-Berry BCP. case quadratic can take arbitrary value 0 and $2\ensuremath{\pi}$. find simple relations phase, distance, in two cases: (i) when one bands flat; (ii) system has rotation and/or time-reversal symmetries. To demonstrate implication continuum model analysis lattice systems, study tight-binding Hamiltonians BCPs. that, curvature absent, BCP always accompanies another so total periodic becomes zero. work demonstrates plays critical role understanding geometric properties topological semimetals.