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. The concept however is more general than the single-electron setting. In this work, we consider topology of plasmons in a two-dimensional metal, subject to a unidirectional periodicity. We show how the plasmon modes and wavefunctions may be computed for such a periodic system, by focusing on the confined, quantized photon degrees of freedom associated with the plasmon modes. At low frequencies the plasmons disperse with wavevector as $\sqrt{q}$; however at higher frequencies one finds a series of bands and gaps in the spectrum. For a unidirectional periodic electron density with inversion symmetry, we show that each band hosts a Zak phase $\gamma_n$ which may only take the values $0$ or $\pi$. Each gap has a topological index $\nu$ that is determined by the sum of the Zak phases below it. When the system has an interface with the vacuum, one finds in-gap modes for gaps with non-trivial topologies, which are confined to the interface. In addition, interfaces between systems that are the same except for their topologies -- 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 topological indices of the relevant gaps are different. We demonstrate these properties numerically by analyzing a Kronig-Penney-type model in graphene, in which the electron density is piecewise constant, modulating between two different densities. In addition, we consider a related plasmon system modeled after the Su-Schrieffer-Heeger (SSH) model. We show that the topological phase diagram of the lowest energy bands is highly analogous to that of the SSH tight-binding system.