Topological phases of electromagnetic waves in superlattices of negative- and positive- epsilon materials

Recently, a new topological invariant is proposed to classify electromagnetic waves in an isotropic medium with positive or negative permittivity and permeability, and various interface modes such as surface plasmons are interpreted as topological boundary modes [1]. We expect similar topological phases of electromagnetic waves even for metamaterials.
To this end, we consider the electromagnetic waves in a 1D multilayer consisting of positive- and negative- epsilon layers, studied in [2]. Depending on the ratio of layer thicknesses and the values of the dielectric constants of the constituent layers, the spectrum changes, and the dispersion opens a gap at the origin. We associate this with a change in the topological number proposed in [1].
In the work of [3], the topological interface mode is reported due to the Zak phase. In contrast, we focus on the interface mode apart from the Zak phase. We find that interface states appear at an interface between the different topological phases as expected in the longwave limit. [1] K.Y. Bliokh, D. Leykam, M. Lein, F. Nori, Nat. Commun. 10, 580 (2019); [2] A.P. Vinogradov, A.V. Dorofeenko, I.A. Nechepurenko, Metamaterials 4, 181 (2010); [3] F. Yang, S. Ma, K. Ding, S. Zhang, And J. B. Pendry, PNAS 117, 29, 16739-16742 (2020)