The band structure of a lattice can contain degeneracies between two or more bands, or singularities, at certain quasimomenta. One prominent example is the Dirac point in honeycomb lattice. The geometric properties of such points have been studied by measuring the associated Berry phase with trajectories that circle the singularity in momentum space. In our work, we instead use trajectories that go right into the singular points, and then turn at varying angles, to probe the population transport properties directly. We also provide to our knowledge the first measurement of the winding number of the quadratic band touching in the excited bands of a honeycomb lattice.
Direct Geometric Probe of Singularities in Band Structure