Topological systems have exciting properties that may lead to new physics. The Chern numbers of a material platform are usually written in terms of the Berry curvature which depends on the normal modes of the system. From a computational point of view the calculation of the Chern number can be a rather formidable problem.
Here, we use a gauge invariant Green’s function method (Phys. Rev. B, 99, 125155, 2019) to determine the topological invariants. We apply the method to a lossy photonic crystal formed by a hexagonal array of ferrite cylinders embedded in air with radius r=0.35a [Fig. 1a]. The Chern numbers are calculated from first principles, i.e., without a tight-binding approximation.
The ferrite is characterized by a standard gyrotropic model with losses included. The band structure of the photonic crystal (with E=(wa/c)2 where w is the oscillation frequency) is plotted in Fig. 2a for a lossless system. The band-gap is shaded in blue. Figures 2bi-iii show the projection of the band structure on the complex plane, E=E’+iE’’ as the real-valued wave vector is swept along the Brillouin zone for non-zero and increasing values of the material loss. In the non-Hermitian case the projected band structure is formed by two non-intersecting regions separated by a band-gap (vertical strips shaded in blue).
The gap Chern number is given by an integral of the photonic Green function over the first Brillouin zone and over a line parallel to the imaginary frequency axis in the band-gap. The Green function is found using a standard plane-wave expansion. The first Brillouin zone [Fig.1b] is sampled with N points along each direction of space. Figures 3a and 3b show (for the lossless case and for the lossy case corresponding to Fig. 2bii, respectively) that for moderately large N the numerically calculated Chern number C quickly approaches unity. The computation time is a few minutes in a standard personal computer. Our formalism not only allows for a rigorous characterization of the topological phases of Hermitian and non-Hermitian systems, but may also enable theoretic developments that shed light on the origin of the topological numbers.