Topological Transitions of Evanescent Waves

Abstract

We study the band structure of evanescent waves, that is, the decaying or growing solutions that live within the gaps of tight-binding Hamiltonians. Although the underlying Hamiltonian is Hermitian, these modes carry complex wavevectors and therefore exhibit a non-Hermitian dispersion. Using the transfer matrix formalism in one dimension, we show that, in addition to the standard exceptional points that appear at the band edges, a different type of exceptional point can arise inside the gap, protected by spinless time-reversal symmetry. Tuning model parameters drives a topological transition where such in-gap exceptional points are nucleated in pairs, mediated by a crossing between purely imaginary momentum branches. We show that this spectral structure is diagnosed by a $\mathbb{Z}_2$ index assigned to every energy inside the gap, with the in-gap exceptional points arising as its domain walls. A generic bound state in the system has a tail that is purely exponential when its energy lies between a band edge and the nearest exceptional point, but becomes oscillatory - with a wavevector differing from the band-edge momentum - when its energy lies between the pair of in-gap exceptional points. We illustrate these results using continuum and tight-binding models, as well as an anisotropic photonic crystal.