Topological non-Hermitian origin of surface Maxwell waves

Maxwell electromagnetism, describing the wave properties of light, was formulated 150 years ago. More than 60 years ago it was shown that interfaces between optical media (including dielectrics, metals, negative-index materials) can support surface electromagnetic waves, which now play crucial roles in plasmonics, metamaterials, and nano-photonics. Here we show that surface Maxwell waves at interfaces between homogeneous, isotropic media described by real permittivities and permeabilities have a purely topological origin explained by the bulk-boundary correspondence. Importantly, the topological classification is determined by the helicity operator, which is generically non-Hermitian even in lossless optical media. The corresponding topological invariant, which determines the number of surface modes, is a Z4 number (or a pair of Z2 numbers) describing the winding of the complex helicity spectrum across the interface. Moreover, there is an additional pair of non-topological Z2 indices, which describe zones of the TE and TM polarizations at the phase diagram of surface modes. Our theory provides a new twist and insights for several areas of wave physics: Maxwell electromagnetism, topological quantum states, non-Hermitian wave physics, and metamaterials.