Topological photonic Tamm states and the Su-Schrieffer-Heeger model

In this paper we study the formation of topological Tamm states at the interface between a semi-infinite one-dimensional (1D) photonic crystal and a metal. We show that when the system is topologically nontrivial there is a single Tamm state in each of the band gaps, whereas if it is topologically trivial the band gaps host no Tamm states. We connect the disappearance of the Tamm states with a topological transition from a topologically nontrivial system to a topologically trivial one. This topological transition is driven by the modification of the dielectric functions in the unit cell. Our interpretation is further supported by an exact mapping between the solutions of Maxwell's equations and the existence of a tight-binding representation of those solutions. We show that the tight-binding representation of the 1D photonic crystal, based on Maxwell's equations, corresponds to a Su-Schrieffer-Heeger-type model (SSH model) for each set of pairs of bands. By expanding this representation near the band edge we show that the system can be described by a Dirac-like Hamiltonian. It allows one to characterize the topology associated with the solution of Maxwell's equations via the winding number. In addition, for the infinite system, we provide an analytical expression for the photonic bands from which the band gaps can be computed. N.M.R.P., M.I.V., and Y.V.B. acknowledge support from the European Commission through the project GrapheneDriven Revolutions in ICT and Beyond (Ref. No. 785219) and the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Financing UID/FIS/04650/2019. N.M.R.P., T.G.R., and Y.V.B. acknowledge COMPETE2020, PORTUGAL2020, FEDER, and the Portuguese Foundation for Science and Technology (FCT) through Project No. POCI-01-0145-FEDER-028114. The authors acknowledge Andre Chaves for suggesting the starting point of the analytical approach to the photonic bands. N.M.R.P. acknowledges stimulating discussions with Joaquin Fernandez-Rossier on the topic of the paper. J.C.G.H. acknowledges the hospitality of the physics department of SDU, Denmark, where this work was completed. The authors are thankful to Asger Mortensen and Mario Silveirinha for their careful and critical reading of the manuscript.