First Principles Calculation of Topological Invariants of non-Hermitian Photonic Crystals

The Chern topological numbers of a material platform are usually written in terms of the Berry curvature, which depends on the normal modes of the system. Here, we use a gauge invariant Green's function method to determine from first principles the topological invariants of photonic crystals. The proposed formalism does not require the calculation of the photonic band-structure, and can be easily implemented using the operators obtained with a standard plane-wave expansion. Furthermore, it is shown that the theory can be readily applied to the classification of topological phases of non-Hermitian photonic crystals with lossy or gainy materials, e.g., parity-time symmetric photonic crystals.

Recently, it was shown that the photonic Chern number can be understood as the quantum of the fluctuation-induced light angular-momentum in a topological material cavity [15,16,17].
The topological properties of a physical system are usually linked to the spectrum of a two-parameter family of linear operators Ĥ k . Typically, the spectrum is formed by the system natural modes (in condensed matter systems Ĥ k is the Hamiltonian of Bloch electronic states). It was recently discovered that the topological classification remains feasible even when the operator Ĥ k is non-Hermitian [18][19][20][21][22][23][24][25]; thereby lossy or gainy photonic systems are characterized by different topological phases.
In this article, we focus on Chern-type topological insulators. A non-trivial Chern phase can only occur when the system has a broken time reversal symmetry. Hence, in optics, non-trivial Chern insulators are necessarily nonreciprocal [26]. Each topological phase is characterized by an integer number (the Chern number), which is a topological invariant insensitive to weak perturbations of the Hamiltonian.
The Chern numbers of a (Hermitian) material system are traditionally obtained from the Berry curvature k [13,14,27]. From the topological band theory, the gap Chern number is 2 gap ..
The integral is over the first Brillouin zone (BZ) and  In this article, we tackle the problem of first principles calculation, i.e., without using tight binding or other approximations, of the Chern number of non-Hermitian photonic systems. Our approach is based on Refs. [24,28], where it was shown that the gap Chern numbers of non-Hermitian systems can be written in terms of the system Green's function. Different from the standard topological band theory, the Green's function approach is gauge invariant and does not require any detailed knowledge of the band structure or of the Bloch eigenstates. The method applies to both fermionic (see also Refs. [30][31][32]) and bosonic platforms (even in case of material dispersion). Different from topological band theory, it is unnecessary to compute the Chern invariants of the individual bands to find the gap topological invariant. The gap Chern number is directly obtained from an integral of the photonic Green's function over a contour in the complex frequency plane that links i − to i + and contained in the relevant band-gap.
The described theory was applied to electromagnetic continua in Refs. [24,28]. Here, we tackle the more challenging and interesting case of photonic crystals. We show how by using the operators obtained from the well-known plane wave method [34] it is possible to find in a relatively simple and computationally inexpensive way the gap Chern number of topological photonic platforms. Furthermore, we study the impact of material loss on the topological invariants.
The article is organized as follows. In Sect. II we present a brief overview of the general Green's function method. In Sect. III we use the Green's function approach to determine the topological phases of lossless, lossy, and lossy-gainy magnetic-gyrotropic photonic crystals. A short summary of the key results is given in Sect. IV.

II. Photonic Green's function formalism
In this Section, we briefly review the general Green's function formalism introduced in Refs. [24,28] to calculate the Chern invariants of photonic platforms. The starting point is the generalized eigenvalue problem with L k a generic differential operator and To this end, we introduce the system Green's function k , defined by The Green's function has poles at the eigenfrequencies n  = k , but otherwise is an analytic function of frequency. Let us first consider, without loss of generality, that L k and g M are Hermitian operators. In that case, the eigenfrequencies n  k are real-valued numbers. Hence, the projection of the system band structure into the complex-frequency consists of line segments contained in the real-frequency axis (see Fig.   1a). The band gaps are the regions of the complex frequency plane that separate disconnected sets of eigenfrequencies. For example, with reference to Fig. 1a the region is a band-gap (vertical strip shaded in yellow in Fig. 1a), as it separates two sets eigenfrequencies, i.e., two bands. This band-gap definition can be readily extended to non-Hermitian systems, with the difference that for non-Hermitian platforms the projected band structure is not restricted to the real-frequency axis. Hence, in the non-Hermitian case the projected band-structure can populate parts of the lower-half (for lossy systems) or upper-half (for gainy systems) complex-frequency plane [24]. In general, the band-gaps are vertical strips in the complex plane, i.e., of the form where the Green's function is analytic (the vertical strip does not need to be rectangular and can have an arbitrary shape provided the initial and end points have  = respectively). Each band-gap is associated with a topological invariant, the gap Chern number, given by [24,28]: where   Tr ... stands for the trace operator, / with 1 x kk = and 2 y kk = . The integral in  is over a contour completely contained in the band-gap that joins the points i − and i +. For simplicity, throughout the article it is assumed that the contour is a straight line of the form   gap Re  = with gap  some (realvalued) frequency in the gap (see Fig. 1a).
The derivatives in frequency and wave vector can be explicitly evaluated as [24,28]. Hence, the gap Chern number can be expressed as: In order to numerically calculate the integral it is convenient to use the coordinates ( ) In practice, the upper-limit of the integral in  needs to be truncated: where max  should be on the order of / ca with c the speed of light and a the lattice constant. Typically, g decays exponentially fast with  and hence the integration in  is quite efficient [24,28]. In practice, the integrals in 12 ,,    are done using numerical quadrature, e.g., using the trapezoidal or the Simpson rules.

III. Magnetic-gyrotropic photonic crystal A. Physical model
To illustrate the ideas, we consider a photonic crystal formed by a hexagonal array of cylindrical rods with radius R embedded in air as illustrated in Fig. 1b. The periodic structure contains two rods per unit cell, i.e., it is formed by two sub-lattices (honeycomb lattice). The direct lattice primitive vectors are taken equal to: where a is the distance between nearest neighbors (circles with different colors in Fig.   1b). The relative permittivity and permeability tensors of the photonic crystal are of the form: 33 , gyrotropic. This type of material response occurs in natural ferrimagnetic materials (e.g., ferrites) biased with a magnetic field directed along the z-direction [33].
As seen, g M is a multiplication operator (multiplication by the material permittivity) and ( )L L i = −  is a differential operator.
The operator L k is obtained from L with the substitutions

B. Band structure
The band structure of the photonic crystal can be found with the plane wave method [34].   Fig. 1c. The radius of the scattering centers is 0.346 Ra = and the gyrotropic material constitutive parameters are taken equal to 12 12  == and 12 1  == . As seen in Fig. 2a, for the reciprocal case, the bands touch at the Dirac point (K) due the symmetry of the hexagonal lattice ( 12  = ). Indeed, the reciprocal photonic crystal is a photonic analogue of graphene [35]. When a static magnetic field is applied to the system, so that 0   , the degeneracy around the Dirac points is lifted, leading to a complete photonic band-gap.

D. Non-Hermitian systems
The formalism can be applied with no modifications to take into account the effect of material dissipation in the cylindrical rods. Non-energy conserving (non-Hermitian) platforms have recently raised a lot of interest due to the exotic physics of systems with exceptional points [42]. For simplicity, here we model the material loss by considering . As in the previous subsection, we take 1  = .    The formalism can also be applied with no modifications to gainy systems. To illustrate this we consider a parity-time ( ) symmetric [43,44] gyrotropic photonic crystal.
-symmetric systems have rather unique features and can be implemented at optics through a judicious inclusion of gain/loss regions [45][46][47][48] or with moving media [49,50]. The spectrum of -symmetric systems is real-valued when the eigenfunctions simultaneously diagonalize the system Hamiltonian and the operator. Otherwise, the spectrum can be complex-valued, which corresponds to a spontaneously broken -symmetry.
In our case, the -symmetry can be enforced by assuming that 1

IV. Summary
We used a Green's function method to calculate from "first principles" the topological invariants of Hermitian and non-Hermitian photonic crystals with a broken time-reversal symmetry. The main advantage of our formalism is that it does not require any detailed knowledge of the photonic band structure or of the Bloch modes. In particular, different from the topological band theory, the Green's function approach can be applied with no modifications when the different photonic bands cross at one or more points of the Brillouin zone. The computational effort for the non-Hermitian case is essentially the same as for the Hermitian case. The Green's function is numerically calculated using the standard plane-wave method. We applied the formalism to magnetic-gyrotropic photonic crystals. It was shown that the topological phases of a photonic crystal are strongly robust to non-Hermitian perturbations (dissipation and/or gain). We expect that our work will find widespread application in the characterization of emergent topological photonic platforms.