Photonic analogues of the Haldane and Kane-Mele models

2.1 Electronic Haldane model in a 2D electron gas

It was recently shown in [23] that the Haldane model can be implemented in artificial graphene, i.e., an electronic platform that mimics the properties of graphene [28], for example, a 2DEG under the influence of a periodic electrostatic potential V(r) with the honeycomb symmetry [29], [30], [31]. As depicted in Figure 1A, a broken IS in artificial graphene can be realized with different potentials V₁ and V₂ in each sublattice, while a broken TRS can be obtained by applying a space-varying static magnetic field B(r)=∇×A with zero spatial average. As proven in [23], a magnetic vector potential A of the form

Figure 1:

Geometry of the system.

Schematic of (A) the electronic and (B) the equivalent photonic structure used to implement the Haldane model. The V_i’s and ω_p,i’s with i=1, 2 are respectively the electrostatic potentials and plasma frequencies of the materials associated with the two different sublattices, V_out and ω_p,b are the electrostatic potential and the plasma frequency of the background material, and A and ξ are the spatially dependent magnetic vector potential and pseudo-Tellegen vector given by Eqs. (1) and (14), respectively.

has the required symmetries to reproduce the Haldane model such that only the phase with dominant broken TRS leads to a nontrivial topology with ν≠0. In the above formula, a is the distance between nearest sites (scattering centers) in the honeycomb lattice, B₀ is the peak magnetic field in Tesla, R=r−r_c, where r_c determines the coordinates of the honeycomb cell’s center, and the b_i’s with i=1, 2 are the reciprocal lattice primitive vectors. The electronic system of Figure 1A is characterized by a microscopic Hamiltonian:

where m_b is the electron effective mass, p^=−iℏ∇ is the momentum operator and e>0 is the elementary charge. It follows that the stationary states ψ with energy E, i.e., the solutions of the time-independent Schrödinger’s equation H^micψ=Eψ, satisfy

Crucially, this microscopic equation with a magnetic vector potential A given by Eq. (1) reproduces the symmetries of the Haldane model [23], and, as shown next, can also be used to develop an analogy with the Maxwell equations.

2.2 Photonic analogue

The strategy adopted by us to obtain a photonic equivalent of the Haldane model (3) is to exploit a formal analogy between the 2D Schrödinger and Maxwell equations. For the sake of clarity, the analysis is divided into two steps: first, it is shown how to create an electromagnetic equivalent of artificial graphene in a 2D photonic crystal. Then it is explained how to break the fundamental symmetries of the system in order to implement the electromagnetic Haldane model.

Using a six-vector formalism for the representation of the electromagnetic fields [32], the Maxwell equations with current sources can be written in a compact manner as

where F=(E H)^T and J=(j_e j_m)^T stand for six-vectors whose components are the electric and magnetic fields and current densities, respectively, and M is the material matrix given by

where ε¯¯,μ¯¯,ξ¯¯ and ζ¯¯ are the relative permittivity, permeability and magnetoelectric tensors, respectively. In addition, it is also assumed that the system is invariant to translations along the z direction (∂/∂z=0).

2.3 Numerical examples

To illustrate the ideas developed so far, next we present the band diagrams and the typical field profiles associated with each topological phase of the photonic Haldane model. All the simulations presented here were numerically obtained with a dedicated finite-difference frequency-domain method (FDFD) whose details can be found in [30].

The electromagnetic system is constructed from a dual electronic system through the equivalence relations (9), (10) and (13). We choose the original electronic platform as the patterned 2DEG of Figure 1A with parameters V₁=V₂=0, Voutmba2ℏ2≈15.83,A=0, R₀/a=0.35 and m_b=0.067m, with m the electron rest mass. With such parameters the electronic system behaves as graphene near the normalized energy Emba2ℏ2≈9.4 where its band diagram consists of two Dirac cones centered at the high-symmetry K and K′ points. For a=150 nm the system reduces to the artificial graphene studied in [23], [30] after adding a constant potential U=0.8 meV to the whole structure to guarantee that V≥0.

The direct application of the equivalence relations (9) and (10) to the artificial graphene leads to a simple photonic crystal made of air cylinders (ω_p,1=ω_p,2=0) embedded in a metal with a Drude dispersion. The band diagram of the photonic crystal close to the Dirac point K is depicted in Figure 2A as a function of the wavevector q=|k−K| taken relatively to the K point. As expected, the frequency dispersion is approximately a linear function of q forming a Dirac cone at K and K′ (not shown), thus validating that the proposed structure is a photonic equivalent of graphene. As seen in Figure 2B and C, by breaking either the IS (ω_p,1≠ω_p,2) or the TRS (ξ≠0) it is possible to open a gap in the band diagram. Moreover, similar to the electronic Haldane model, the phases induced by each of the broken symmetries are topologically distinct, as expressed by the different values of the photonic Chern number 𝒞. Specifically, 𝒞=0 for the phase with broken IS, whereas 𝒞=±1 for the phase with broken TRS. It should be noted that with a frequency independent ξ₀ and with ω_pa/c=const. the properties of the photonic system are fully scalable with frequency, and the exact gap spectral range is determined only by a.

Figure 2:

Normalized dispersion diagrams of (Haldane) photonic graphene.

The geometry of the photonic crystal is as in Figure 1B with ωp,b a/c≈5.63. (A) “Photonic graphene” with ω_p,1=ω_p,2=0 and ξ=0. (B) “Photonic graphene” with a broken IS such that ξ=0, ω_p,1=0 and ωp,2 a/c≈1.09. (C) “Photonic graphene” with a broken TRS such that ξ₀≈0.677 and ω_p,1=ω_p,2=0. The photonic Chern numbers 𝒞 associated with the two bands are given in insets.

According to the bulk-edge correspondence [1], [14], [48], the difference in the Chern numbers should manifest itself directly on the propagation of the edge states at the interface with a trivial photonic insulator. In particular, it is expected that the phase with broken TRS supports unidirectional edge states protected against backscattering with a dispersion that spans the entire band gap. To confirm this property the solutions E_z of the wave equation in the closed cavity of Figure 3A,

Figure 3:

Edge states of a photonic graphene cavity.

(A) Schematic of the cavity: the photonic crystal of Figure 1B with an absorber located at the right-bottom region and surrounded by PEC walls. The pseudo-Tellegen response ξ is not represented in the figure. (B) Time snapshot (t=0) of E_z for the material with a broken IS of Figure 2B. (C)–(E) Time snapshot (t=0) of E_z for the material with a broken TRS of Figure 2C. The pseudo-Tellegen parameter is ξ₀≈0.677 in (C) and (D) and ξ₀≈−0.677 in (E).

were computed with the FDFD method [30] with the wave excited by a point-like electric current distribution j_e. The cavity walls were chosen to be perfect electric conductors (PEC), that is, the Haldane photonic crystal is surrounded by a trivial insulator. The oscillation frequency of the current source is centered in the band gap. To ease the field visualization in the closed cavity, an absorber was placed at the right-bottom part of the structure. The absorption is stronger near the center (darker colors in Figure 3A imply a stronger absorption).

The excited field profiles for the distinct topological phases of photonic Haldane graphene are represented in Figure 3B–E. As seen in Figure 3B, the phase characterized by a broken IS does not support any bulk or unidirectional edge mode at this frequency, confirming that this phase is topologically trivial. The absence of bulk modes for the phase with a broken TRS is also verified in Figure 3C, which reveals that for an excitation far from the edges, the fields decay rapidly with the distance to the source.

The situation is dramatically different for sources positioned near the boundaries. Figures 3D and E show that in such a case a unidirectional edge state with propagation direction locked to the sign of ξ₀ is excited near the PEC walls. Crucially, the topologically protected modes do not experience any backscattering at the corners of the cavity. The edge mode dispersion in the band gap region was found by numerically fitting the spatial variation of the edge waves to that of a Bloch wave. As depicted in Figure 4, the edge mode dispersion is approximately linear, and for the edge wave propagating attached to the top interface it is centered about the K point and spans the entire band gap. The gap Chern number – given by the sum of the Chern numbers of the bands below the gap, including the negative frequency bands – is 𝒞_gap=sgn(ξ₀) for the phase with a dominant broken TRS. In agreement with the results of [48], [49], the gap Chern number is positive (negative) for an energy flow in the clockwise (anticlockwise) direction. The electric field time animations of the examples of Figure 3D and E are available in the Supplementary Material.

Figure 4:

Dispersion diagram of the bulk and edge modes (top interface) as a function of the normalized wavevector.

The structure parameters are as in Figure 2C. The slight asymmetry of the bulk bands with respect to q=0 and the frequency shift with respect to Figure 2C are numerical artifacts caused by a relatively coarse mesh in the simulations of the electrically large topological cavity of Figure 3.

It is relevant to mention that because of computational restrictions the maximum number of mesh cells allowed by our numerical code is limited. For this reason the results of Figures 3 and 4 – for a computational domain formed by many cells – were obtained with a relatively low mesh density. For consistency, the band structure shown in Figure 4 was calculated using the same mesh as the one used to obtain the edge state dispersion. The coarser mesh leads to a slight numerical shift with respect to the more exact band structure results of Figure 2.

Also because of computational limitations, the calculation of the edge modes dispersion is challenging when they are weakly confined to the cavity walls. To obtain the most accurate results, we focused on a phase with a broken TRS, but with a preserved IS (ω_p,1=ω_p,2=0) to maximize the band gap width. Indeed, the edge state confinement is stronger for larger gap energy. We numerically verified that as long as TRS remains the dominant broken symmetry, the introduction of a broken IS (ω_p,1≠ω_p,2) does not affect the topological properties of the system, and the presence of gapless edge states.

In conclusion, we numerically demonstrated that the structure of Figure 1B described by Eq. (12) is a photonic equivalent of the Haldane model, with topological properties determined by the dominant broken symmetry exactly as its electronic counterpart. The proposed platform is especially interesting from a theoretical perspective, as materials with a strong Tellegen response are not readily available in nature [50]. In the next section, we introduce a practical path for the realization of two copies of the photonic Haldane model simply relying on anisotropic dielectrics.

3.1 Kane-Mele model in a nonreciprocal system

Consider the photonic crystal of Figure 1B with the same relative permittivity (6) and magnetoelectric tensors (11) as in Section 2, but with matched permittivity and permeability tensors:

As in Section 2, it is assumed that ε_||=μ_|| are space independent and that the zz components of the permittivity and permeability tensors are given by a Drude model εzz(r)=μzz(r)=1−ωp2(r)/ω2 in the frequency range of interest. From Eqs. (A.8) and (A.9) of Appendix A, the wave equations for TE (E=Ezz^) and TM (H=Hzz^) polarized waves are

Remarkably, even though the two polarizations are uncoupled, the wave propagation of TE and TM waves is ruled essentially by the same equation, except for the pseudo-Tellegen vector which has an opposite orientation for TE and TM waves. Here, ε_|| only plays the role of a normalization factor and can without loss of generality be taken equal to unity. In this case and for ξ given by Eq. (14), the wave equations (19) reduce to two copies of the photonic Haldane model (12) with an opposite orientation of the pseudo-Tellegen vector. By virtue of Eq. (13), this situation corresponds in the electronic case to two copies of the Haldane model with opposite magnetic fields, which is precisely the Kane-Mele model [19], [20]. Hence, Eq. (19) yields a strict photonic analogue of the Kane-Mele model with the spin-orbit coupling mimicked by the nonreciprocal pseudo-Tellegen response. From Section 2, it is clear that the pseudo-Tellegen coupling opens a topologically nontrivial band gap wherein TE- and TM-polarized edge states can propagate in opposite directions without back-reflections at the interface with a trivial photonic insulator.

Interestingly, it can be readily checked that a photonic platform with effective parameters that satisfy (17) and (18) is PTD invariant [24]. The scattering phenomena in PTD-invariant systems are characterized by an anti-symmetric scattering matrix, and this property guarantees that a generic PTD-invariant microwave network is matched at all ports [24], [52]. In particular, any PTD-invariant waveguide that supports a single propagating mode (for fixed direction of propagation) is completely immune to back-reflections [24]. Note that PTD-invariant systems are always bidirectional. Thus, the absence of backscattering in the proposed photonic Kane-Mele model can also be explained as a consequence of PTD invariance.

3.2 Kane-Mele model in a reciprocal system

While the ideas developed in the last section are interesting from a theoretical standpoint, their impact on more practical grounds is admittedly limited, mainly because of the difficulty to obtain a pseudo-Tellegen response [50]. Next, we show that the nonreciprocal PTD-invariant platform studied in Section 3.1 can be transformed into an equivalent fully reciprocal non-bianisotropic system by means of a duality transformation [24].

The key observation is that a duality transformation acts exclusively on the fields, leaving the space and time unaffected [24], [53], [54]. Thereby, the wave phenomena and topological properties of two systems linked by a duality transformation are fundamentally the same. In particular, a duality transformation preserves the band diagram dispersion and the immunity to backscattering [24], [53], [55]. The origin of this unexpected link between the scattering properties of reciprocal and nonreciprocal systems is thoroughly covered in [24].

As in [24], we consider a duality transformation D of the form

where η0=μ0/ε0 is the free space impedance. The duality transformation changes the material parameters as M(r)→DM′(r)≡det(D)⋅(D−1)T⋅M(r)⋅D−1 [24]. For a bianisotropic material with matched permittivity and permeability μ¯¯=ε¯¯ and matched magnetoelectric couplings ξ¯¯=ζ¯¯, it leads to a new material described by

Notably, the new material has no magnetoelectric coupling. If the parameters of the original material are given by (17) and (18), then the permittivity ε¯¯′ and permeability μ¯¯′ tensors of the new material are explicitly

Interestingly, it can be readily verified that a system described by the tensors ε¯¯′ and μ¯¯′ satisfies Eq. (12) of [24], implying that it is PTD invariant. This further confirms that the proposed platform can support scattering-immune edge modes. Furthermore, the tensors ε¯¯′ and μ¯¯′ are symmetric, indicating that the transformed system is also reciprocal.

The tensors ε¯¯′ and μ¯¯′ have the same diagonal part, whereas their out-of-diagonal elements have opposite signs. Clearly, the duality transformation preserves the diagonal elements of the permittivity and permeability, whereas the pseudo-Tellegen response of the original material becomes the out-of-diagonal part of ε¯¯′ and μ¯¯′. Evidently, the out-of-diagonal parameters play the role of an effective magnetic field for photons and are responsible for the bidirectional and reflectionless edge wave propagation in this structure. Interestingly, Liu and Li have previously shown using totally different physical arguments that spatially dependent anisotropic media with a structure analogous to (22) can be used to create a pseudomagnetic field for photons [25], [56], [57]. Thus, our theory merges the concepts of pseudomagnetic fields and time-reversal invariant topological matter as particular cases of the broader class of PTD-invariant systems [24].

It is highlighted that the wave propagation in a structure described by the parameters (22) is fully determined by the propagation in the two associated copies of the photonic Haldane model. Thereby, similar to Section 2.3, the material characterized by (22) supports topologically protected (but bidirectional) edge states. The cavity walls should enforce the (PTD-invariant) mixed boundary conditions E_z=0 and H_z=0. These “soft” boundary conditions were originally studied by Kildal and can be implemented with corrugated surfaces [58]. The field profiles of the cavity edge modes can be found from the numerical simulations of Section 2.3 using the duality transformation (20).

In conclusion, we proposed a fully reciprocal PTD-invariant photonic platform based on non-uniform anisotropic dielectrics that supports bidirectional edge mode propagation protected against back-reflections. Importantly, the concept of PTD invariance is a single-frequency condition, and thereby it is sufficient that the material parameters satisfy Eq. (22) in some frequency in the band gap to observe the scattering-immune bidirectional edge state propagation. This property let us hope that even though challenging a practical implementation of the proposed metamaterials with spatially dependent parameters may be feasible in the future.