Higher-order topology in two-dimensional crystals

We study two-dimensional spinful topological phases of matter protected by time-reversal and crystalline symmetries. To define the topology we employ the concept of corner charge fractionalization: We show that corners in a higher-order topological phase can carry charges that are fractions of even multiples of the electric charge. These charges are quantized and topologically stable as long as all symmetries are preserved. We classify the topologies corresponding to different corner charge configurations for all 80 layer groups, and present their bulk topological indices. These can be calculated from the symmetry data and Brillouin zone Wilson loops. To corroborate our findings, we present tight-binding models and density functional theory calculations for various material realizations.