Quantum Band Structure and Topology in One Dimensional Modulated Plasmonic Crystal

Band structures of electrons in a periodic potential are well-known to host topologies that impact their behaviors at edges and interfaces. In this work, we consider topology of plasmons in a two-dimensional metal subject to a unidirectional periodicity. Plasmon modes and wavefunctions may be computed for these in terms of quantized photon degrees of freedom, with spectra consisting of bands and gaps. When there is inversion symmetry, we show that each band hosts a Zak phase which may only take the values 0 or π. Each gap has a topological index that is determined by the sum of the Zak phases below it. When the system has an interface with the vacuum, topological spectral gaps develop in-gap modes, which are confined to the interface. Interfaces between systems that are the same spectra but different topological classes -- which can be created by a defect in the lattice in which half a unit cell has been removed -- host in-gap, confined states when the topology of the relevant gaps are different. We demonstrate these properties by analyzing a Kronig-Penney-type model in graphene, in which the electron density is piecewise constant. In addition, we consider a plasmon system analogous to the SSH model, with results highly analogous to those of the SSH tight-binding system.