Second-order topological modes in two-dimensional continuous media

We present a symmetry-based scheme to create zero-dimensional (0D) second-order topological modes in continuous two-dimensional (2D) systems. We show that a metamaterial with a p6m-symmetric pattern exhibits two Dirac cones, which can be gapped in two distinct ways by deforming the pattern. Combining the deformations in a single system then emulates the 2D Jackiw-Rossi model of a topological vortex, where 0D in-gap bound modes are guaranteed to exist. We exemplify our approach with the simple hexagonal, kagome, and honeycomb lattices. We furthermore formulate a quantitative method to extract the topological properties from finite-element simulations, which facilitates further optimization of the bound mode characteristics. Our scheme enables the realization of second-order topology in a wide range of experimental systems.