First Principles Calculation of the Topological Phases of the Photonic Haldane Model

Photonic topological materials with a broken time reversal symmetry are characterized by nontrivial topological phases, such that they do not support propagation in the bulk region but forcibly support a nontrivial net number of unidirectional edge states when enclosed by an opaque type boundary, e.g., an electric wall. The Haldane model played a central role in the development of topological methods in condensed matter systems, as it unveiled that a broken time reversal is the essential ingredient to have a quantized electronic Hall phase. Recently, it was proved that the magnetic field of the Haldane model can be imitated in photonics with a spatially varying pseudo Tellegen coupling. Here, we use a Greens function method to determine from first principles the band diagram and the topological invariants of the photonic Haldane model, implemented as a Tellegen photonic crystal. Furthermore, the topological phase diagram of the system is found, and it is shown with first principles calculations that the granular structure of the photonic crystal can create nontrivial phase transitions controlled by the amplitude of the pseudo Tellegen parameter.


I. Introduction
The study of topological properties of physical systems and of how the topology influences the physical responses and phenomena has been a very active field of research in recent years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The topology of physical system is typically determined by the global properties of the operator that describes the time evolution of the system state (in quantum systems, this operator is the Hamiltonian). Here, we focus on a particular class of topological systems known as Chern insulators [8]. Such systems have a broken time reversal symmetry and are characterized by a topological invariant known as the Chern number. The key fingerprint of a nontrivial topological phase is the emergence of gapless scattering immune unidirectional edge-states at the boundary of the material. This property makes the response of topological systems rather insensitive to fabrication imperfections, disorder and other perturbations [10].
In the 1980s, Haldane discovered that electronic systems with a broken time-reversal symmetry (e.g., an electron gas biased with a magnetic field) may have a quantized Hall conductivity, even if the spatial-average of the magnetic field vanishes [2]. The Haldane model is essentially a tight-binding description of the propagation of an electron wave in a honeycomb lattice of scattering centers (electric potential interaction), subject also to the influence of a periodic magnetic potential. The periodicity of the magnetic potential ensures that the net magnetic flux in a unit cell vanishes. Sometime ago, it was shown that the model theoretically envisioned by Haldane may be implemented in practice using an "artificial graphene" superlattice biased with a spatially varying magnetic field [21][22]. More recently, Ref. [23] developed a photonic analogue for the Haldane model, with the magnetic field of the original electronic model imitated by a spatially varying pseudo-Tellegen coupling. In Ref. [23], the topological phases of the photonic system were determined relying on a tightbinding approximation. Here, building on these previous works, we use an exact Green's -3-function method [20,[24][25][26] to calculate from "first principles" the topological invariants of the Haldane photonic crystal.
The article is organized as follows. In Section II we present a brief overview of the electronic Haldane model and of its electromagnetic analogue. In Section III the Green's function formalism is applied to the electronic (tight-binding) Haldane model and to the Haldane photonic crystal formed by materials with a pseudo-Tellegen response. In Section IV the topological phases of the electronic and photonic models are calculated based on a Green's function approach. A short summary of the key results is given in Section V.

II. The Haldane model
In this Section we briefly review the original Haldane model and its electromagnetic analogue [2,22,23].

A. The electronic Haldane model
In the above, i σ are the Pauli matrices and the relevant coefficients are: Here, 1 t and 2 t are the nearest-neighbors and next-nearest neighbors hopping energies, respectively, M is the so-called "mass term" and  is the phase factor determined by the coupling between next-nearest neighbors due to the applied magnetic potential.
. The wave vector is defined over the first Brillouin zone. The primitive vectors of the reciprocal lattice are: The energy bands The only points of the Brillouin zone that may satisfy are the two high symmetry (Dirac) points: -5-Thus, in the Haldane model the two bands are typically separated by a complete band gap unless they touch at one of the Dirac points. Below, we show that for large values of 2 t the bandgap may be closed, even if the two bands do not intersect. Reference [22] introduced a possible physical realization of the Haldane model. It relies on a 2D electron gas superlattice patterned with scattering centers, whose effect is modelled by a periodic electric potential   V r . The electric potential is equal to b V in the background region and is equal to 1 V or 2 V in each scattering center (disk) sublattice. A periodic spatially varying magnetic field      B r A determined by the vector potential is also applied to the system. Here a is the distance between the nearest scattering centers in the hexagonal lattice, 0 B is the peak magnetic field in Tesla, 1 b and 2 b are the reciprocal lattice primitive vectors and   R r r c where determines the coordinates of the honeycomb cell's center [ Fig. 1a]. The two scatterers are centered at 0,1â   r x and 0,2  r 0 , respectively, and have radii 1 R and 2 R .
The stationary states of the (spinless) electronic system are the solutions of the timeindependent scalar Schrödinger equation: where  is the wavefunction, b m is the electron effective mass, e is the elementary charge and  is the Planck constant.

B. Photonic analogue of the Haldane model
It was shown in Refs. [23,27] that a pseudo-Tellegen coupling is the counterpart for photons of the magnetic field coupling for electrons. Specifically, consider a bianisotropic material described by constitutive relations of the type: where  ,  ,  and  are the relative permittivity, permeability and magnetoelectric tensors, respectively. The pseudo-Tellegen response is determined by traceless symmetric magneto-electric tensors [28]. In this article, we assume that the relevant tensors are of the form: where ˆx x y is a generic vector lying in the xoy plane. We will refer to ξ as the pseudo-Tellegen vector. Some anti-ferromagnets such as Cr2O3 have a Tellegen-type response, albeit it is typically very weak [29][30][31]. Furthermore, it has been recently shown that some electronic topological insulators may be characterized by a Tellegen type (axion) response [32][33][34][35]. Materials with a Tellegen response are nonreciprocal and enable peculiar effects and exotic physics [36][37][38][39]. It is interesting to point out that the Tellegen coupling is real-valued. This means that any homogeneous Tellegen material is certainly topologically trivial. Furthermore, it is worth noting that most of the solutions that yield nontrivial photonic topologies rely on gyrotropic materials with a complex-valued material response, very different from the Tellegen case.
Suppose that the relativity permittivity and permeability tensors are of the form Then, it can be shown that the wave propagation of transverse electric (TE) waves (with 0 z E  and 0 z H  ) in a (possibly inhomogeneous) photonic system described by the constitutive relations (7) is ruled by the following wave equation [23,27]: It is implicit that the system is invariant to translations along the z-direction and that / 0 z    . As discussed in Ref. [23], the solutions of Eq. (10) can be transformed into the solutions of Eq. (6) using the mapping: where 0  is the (dimensionless) peak amplitude of the pseudo-Tellegen vector [23]. Note that the pseudo-Tellegen coupling varies continuously in space, which implies that the dielectric axes that diagonalize the magneto-electric tensors must be space dependent. Furthermore, the spatially varying electric potential can be mimicked by tailoring the electric response ( zz  ), i.e., by tailoring the plasma frequency p  . As illustrated in Fig. 1b, the plasma frequency associated with the scattering centers ( ,

III. Topological classification with the Green's function
In a recent series of works [20,24,25], we introduced a general Green's function formalism to calculate the gap Chern numbers of non-Hermitian and possibly dispersive photonic crystals. In its most general form, the spectrum of the system under study is determined by a generic differential operator L k (which is not required to be Hermitian) and by a multiplication (matrix) operator g M , which determine a generalized eigenvalue problem ˆn n g n L    where   basis. The integrations can be performed using standard numerical quadrature rules [25]. For further details on the numerical aspects of the method the reader is referred to Ref. [25].
Different first principles methods for the Chern number calculation are reported in Ref. [40,41].

A. Gap Chern number for the electronic Haldane model
It is instructive and pedagogical to apply the Green's function method to the electronic  In this manner, we recovered the well-known phase diagram represented in Fig. 2bi).  The Brillouin zone is parameterized as  To begin with, we rewrite secular equation (10) as where       Thus, we obtain a standard eigenvalue problem with a trivial g  M 1 operator.
In order to find the spectrum of L k and the topological phases, next we obtain a representation of L k in a plane wave basis [42]. First, we expand the periodic functions ξ and  into a Fourier series as .
From here it is evident that the operator L k is represented by the matrix The operators î L  k are represented by matrices with generic elements where 1 u x and 2 u y are unit vectors along the coordinates axes. In the numerical calculations the plane wave expansion is truncated imposing that To conclude, we note that the spectral problem [Eq. (17)] is equivalent to the

C. Topological phases of the photonic Haldane model: Numerical results
The band structures of two Haldane photonic crystals are plotted in Fig. 4.  The reciprocal case characterized by 0 0   is shown in Fig. 4a, and corresponds to a photonic analogue of graphene [23]. Due to the plasmonic response of the host medium there is a bandgap for low frequencies (  the gap Chern number is +1, but for values of 0  slightly larger than 1.36, the sign of the gap Chern number changes due to a phase transition. As a consequence the topological phase diagram is formed by a sequence of bubble-type regions with alternating sign of the gap Chern number. This example illustrates the richness of topological phenomena in photonic crystals, which arises due to periodicity and granular nature of the system, and which can only be captured with a "first principles" approach. It is relevant to note that in a tight-binding approximation it is possible to map the parameters   (1)], see Ref. [22]. Comparing Figs. 2b and 5b, one sees that, for a sufficiently small 0  , the sign of the equivalent  is opposite to the sign of 0  [22,23].

IV. Summary
In this article, we determined the topological phases of a photonic analogue of the Haldane model using a first principles Green's function approach. The proposed system consists of a hexagonal array of dielectric cylinders embedded in a metallic host, with a spatially varying pseudo-Tellegen coupling playing the role a pseudo-magnetic field. The Tellegen nonreciprocal response is at origin of the nontrivial topology of the photonic crystal.
Interestingly, the results of the first principles calculations show that even though a bulk pseudo-Tellegen medium has a trivial topology, a nonuniform pseudo-Tellegen structure (photonic crystal) can have topological bandgaps. Furthermore, it was found that due to the complex wave interactions arising from the scattering by the potential-well centers, the phase diagram of the photonic crystal can have nontrivial features and a bubble-type structure different from what is predicted by Haldane's tight-binding model. We expect that the Green's function method can find many other applications in the characterization of the topology of emerging photonic systems.