Hybrid topological photonic crystals

Topologically protected photonic edge states offer unprecedented robust propagation of photons that are promising for waveguiding, lasing, and quantum information processing. Here, we report on the discovery of a class of hybrid topological photonic crystals that host simultaneously quantum anomalous Hall and valley Hall phases in different photonic band gaps. The underlying hybrid topology manifests itself in the edge channels as the coexistence of the dual-band chiral edge states and unbalanced valley Hall edge states. We experimentally realize the hybrid topological photonic crystal, unveil its unique topological transitions, and verify its unconventional dual-band gap topological edge states using pump-probe techniques. Furthermore, we demonstrate that the dual-band photonic topological edge channels can serve as frequency-multiplexing devices that function as both beam splitters and combiners. Our study unveils hybrid topological insulators as an exotic topological state of photons as well as a promising route toward future applications in topological photonics.


Supplementary Note 1: The calculation of band/gap Chern number of HTPC
To demonstrate the nontrivial band induced by the breaking of time-reversal symmetry (i.e., with an external magnetic field), we calculate the band Chern number via the integration of the Berry curvature, which is defined as Note that the integral is over the whole Brillouin zone (BZ) and Ω ( ) is the Berry curvature of the nth band. By further summing the Berry curvature of bands below the gap of interest, we have the gap Berry curvature as Obviously, the integration of Ω ( ) within the whole BZ gives rise to the gap Chern number .
In order to verify the phase transition process of gap III, we adopt the efficient numerical algorithm reported in Ref. 1 where is the Dirac velocity, = ( , ) measures the momentum deviation from the K valley, refers to the mass term induced by the 3 and/or symmetries, and ( = , , ) are Pauli matrices operating on the orbital degree of freedom. The Berry curvature is then given by 2 Henceforth, the valley Chern number can be attained by integrating the Berry curvature over the whole 2D space, which gives by Remarkably, we find that the valley Chern number induced by gapping a quadratic Dirac point is an integer number, in strong contrast to the common perception that the valley Chern number is a half-integer.
As we have pointed out in the maintext, and are the mass terms induced by breaking the 3 (through rotation) and (through external magnetic field) symmetries, separately. Note that in our work the external magnetic field gives < 0, and the rotation operation gives < 0 when ∈ (0°, 60°), and > 0 when ∈ (60°, 120°). Hence, the total mass term induced by breaking both 3 and

Supplementary Note 3: The evolution of the edge states under the perfect electric conductor boundary condition
To manifest the phase transition between QAH and VH, we further examine the evolution of the edge states under the perfect electric conductor boundary condition. As shown in Supplementary Fig. 4a

Supplementary Note 4: Identification of the bulk gap
Before implement the measurement of the edge states, we first identified the full band gap of the HTPC2. As shown in Supplementary Fig. 5a, the HTPC2 composed of in the experiments.

Supplementary Note 6: Hybrid topological photonic crystals with nonlinear effect
In this section, we discuss the physical phenomenon when adding the nonlinear effect into the hybrid topological systems. To demonstrate the nonlinear optical process, e.g., second-harmonic generation, it is necessary to find two edge states with distinct topologies, which satisfies a double frequency relationship. Hence, we plot the band structures of HTPC1 in the range of 7-13 GHz and 17-21 GHz that consisting complete band gaps (colored by blue and orange) in Supplementary Fig. 7a and 7d, respectively.
For simplicity, we term the band gap within 7-13 GHz (17-21 GHz) as gap L (U). Since we have demonstrate that the gap L is a Chern gap in the maintext, here we utilize the symmetry indicators to characterize the topological property of the gap U.
Following Ref. 6 , the topological crystalline index can be expressed by the full set of the 3 eigenvalues at the high-symmetry points (HSPs). For an HSP denoted by the symbol Π, the 3 eigenvalues can only be Π = 2 ( −1)/3 with = 1,2,3.
Here, the HSPs include Γ, , and ′ points. Hence, for the HTPC with 3 symmetry, the topological indices that describe the band topology is given by where # , # ′ , and #Γ are the numbers of bands below the band gap with the 3 symmetry eigenvalue , ′ , and Γ at the , ′ , and Γ points, respectively.
In this scheme, the point is taken as the reference point to get rid of the redundance.
Any nonzero indicates a topological band gap that is adiabatically disconnected from the trivial atomic insulator. In Supplementary Fig. 8, we present phase distribution of the eigenstates of the lowest ten bands that below gap U at , ′ , and Γ points, where the 3 eigenvalues are also labeled. Therefore, the topological indices of the gap U is given by which indicate the gap U is of nontrivial topological property. Note that in our case, the characterization of the valley topology via the symmetry indicator is equivalent to the Berry curvature calculation, hence, we remark that the gap U is of valley topology.
We then present in Supplementary Figs. 7b and 7e the corresponding photonic band structures of a supercell that consisting of 11 HTPC1 unit-cells terminated by perfect electric conductors. As shown in Supplementary Fig. 7b, two edge branches go across the whole gap II whose typical electric field patterns at frequency of 9.73GHz is shown in Supplementary Fig. 7c. Meanwhile, unbalanced valley Hall edge states ( Supplementary Fig. 7e) emerge in gap IV whose electric field pattern at frequency of 19.46GHz is shown in Supplementary Fig. 7f with different Poynting vector directions. We remark that the gapless edge state in gap II originates from its QAH topology nature while the gapped edge state II is of its valley topology nature.
We now investigate the nonlinear frequency conversion process via the edge modes that indicated in Supplementary Figs. 7c and 7f. The full-wave dynamics of the nonlinear interaction of chiral edge states and valley edge states were determined numerically using the commercial software COMSOL MULTIPHYSICS. To simulate the nonlinear frequency mixing processes in COMSOL, we defined two "Electromagnetic Waves, Frequency Domain" models: one for the fundamental frequency 0 and one for the second harmonic frequency Ω 2 . These two models are coupled using a "Polarization" feature added to each of the models. We assumed that for second harmonic generation, the nonlinear susceptibilities are diagonal tensors with the diagonal elements being 2 . Hence, for second harmonic generation, the nonlinear polarizations at the fundamental frequency and second harmonic are , As an illustration, we consider HTPCs made of homogeneous and isotropic nonlinear material with a scalar nonlinear second-order susceptibility of 2 = 10 −21 C V −2 . The pump electric field 1 is induced by an external point source, where 2 by the nonlinear polarization at the second harmonic, generated by 1 . The main results are summarized in Supplementary Fig. 9.
As shown in Supplementary Fig. 9a, a typical chiral edge mode is excited by a point source at the frequency of 1 = 9.73GHz. As a comparison, at the frequency of 2 = 19.46GHz both edge and bulk modes can be excited (see Supplementary Fig. 9b).
Under the point excitation in the edge, the electrical field distributions mainly at the boundary with strong localization and obvious asymmetry and little into the sample at 2 = 2 1 = 19.4GHz, which matches well with the band structure. After introducing nonlinear polarizations, the electric field 2 in Supplementary Fig. 9c distributes more into the bulk than that in Supplementary Fig. 9b, indicating that these two edge modes are indeed nonlinearly interacting via the second harmonic generation, which demonstrates nonlinear effect generates new physical phenomenon.
Since the phase matching condition plays a key role on the realization of the nonlinear effect, here we also achieve the quasi-phase-matching condition to enhance the nonlinear optical effect on the manipulation of the edge propagation of photons by tailoring the dispersion of the edge states. For this purpose, we tailor the edge dispersions by cutting part of the unit-cells at the edge boundary. This is illustrated in Supplementary Fig. 10b with a geometry parameter = 0.4 where =21mm is the lattice constant. The resultant dispersion of the topological edge states in the quantum anomalous Hall photonic band gap around 10GHz is shown in Supplementary Fig. 10a, while the electric field profile of a specific edge state (labeled the red-letter A) is shown in Supplementary Fig. 10c. The dispersion of the edge states in the valley Hall photonic band gap is shown in Supplementary Fig. 10d.
We consider here the second-harmonic generation between the edge state A in  (Supplementary Fig. 11b), showing bidirectional photonic edge propagation with asymmetric features (i.e., the right-going signal decays faster than the left-going signal).
With second-harmonic nonlinear effects and exciting at the frequency 0 at the same position, the photonic energy flow along the edge channel now switches from unidirectional to bidirectional ( Supplementary Fig. 11c). The simulation results in Supplementary Fig. 11c show comparable left-going and right-going photonic energy flow which is an evidence that the second-harmonic frequency conversion is efficient.
We remark that the above phenomenon has not yet been found before, since in the previous studies, the topological edge states in different band gaps propagate in the same direction 7 .
With the above calculations and discussions, we demonstrate through a concrete example that the phase-matching condition can be satisfied to achieve efficient secondharmonic frequency conversion among the edge states in distinct topological band gaps.
Such frequency conversion can be used to manipulate photonic energy flows in the topological edge channel which is a new degree of freedom that can be promising for future topological photonics.

Supplementary Note 7: Photonic Floquet hybrid topological insulators
In the maintext, the hybrid topological system is demonstrated by using gyromagnetic materials, which have weak magneto-optical responsive in the optical frequencies. However, we remark that the results still can be extended to optical frequencies since the design principle of the hybrid topological system merely requires for the breaking of both and symmetries. To this end, we implement the Floquet engineering to realize the hybrid topology in coupled waveguide arrays, which offers a possible scheme to extend to optical frequencies.
The time varying model is shown in Supplementary Fig. 12a, which contains six steps. The time-dependent Hamiltonian of the system can be described by, where ( ) is set to be at odd steps, and zero at even steps; ∆( ) equals ∆ at even steps, and zero at odd steps; ± = � ± �/2, where , , are Pauli matrices; the vectors 2 +1 are given by 1 is the lattice constant. The spectrum of the system can be obtained by solving eigen equation of Floquet operator, here the Floquet operator is defined as where is the time-ordering operator. Different from the static system, here we study the quasienergy , which has a periodicity of 2 / .
From the model, we notice the time reversal symmetry of the system is broken due to the periodically driving and the parity symmetry is also broken when ∆ is nonzero.
Besides, the lattice sites are arranged in a honeycomb lattice so that the model is suitable for studying valley topology and quantum anomalous Hall physics, and may support hybrid topological states. The phase diagram of the system is shown in Supplementary   Fig. 13d, which contains topological trivial state with vanishing Chern number, Chern insulator with Chern number ±1 and anomalous Floquet topological insulator also with vanishing Chern number but supports chiral edge state. We study the strip structure shown in Supplementary Fig. 12b which contains the interface between topological trivial state (upper) and hybrid topological state (lower) with different valley topology.
The Chern number is −1 for the first bulk band of HTPC which can be obtained from Berry phase calculation in Supplementary Fig. 13b, we can see the winding of Berry phase as function of 1 is -1. The Chern number is 0 for topological trivial state whose Berry phase winding as function of 1 is 0 as shown in Supplementary Fig. 13c. The quasi-energy spectrum of the strip structure is shown in Supplementary Fig. 12c Supplementary Fig. 12d. We notice the field of the chiral edge state is localized at the lower edge which can be treated as the interface between HTPC and vacuum, the fields of the chiral interface state and valley interface state are localized at interface. It is worthy to mention that, although here we propose a time varying system, in the real optical experiment we can use the coupled helical waveguides arrays which treat one spatial direction as time. Such system has been used to study photonic Floquet topological insulator 8,9 , photonic valley topological insulator 10 , photonic anomalous Floquet topological insulator [11][12][13][14] and photonic anomalous Flouqet higher-order topological insulator 15 in optical region. Our proposal here provides a new member to the family of topological insulator in optical region.