Three-dimensional non-Abelian Bloch oscillations and higher-order topological states

Recently, higher-order topological insulators (HOTIs) have been introduced, and were shown to host topological corner states under the theoretical framework of Benalcazar-Bernevig-Hughes. Here we unveil some topological effects in HOTIs by studying the three-dimensional (3D) non-Abelian Bloch oscillations (BOs). In HOTIs, BOs with a multiplied period occur when a force with a special direction is applied due to the effect of the non-Abelian Berry curvature. Along the direction of the oscillations we find a higher-order topological state that goes beyond the theoretical framework of multipole moments. The emergence of such a higher-order topological state coincides with the appearance of the 3D non-Abelian BOs. That is, the 3D non-Abelian BOs can be used as a tool to probe higher-order topological states. These phenomena are observed experimentally with designed electric circuit networks. Our work opens up a way to detect topological phases theoretically and experimentally.