Graphene in a real strong magnetic field can generate discrete Landau levels [1,2]. In recent years, the strain engineering of graphene has become a research field attracting much attention [3–12]. Inhomogeneous uniaxial strain can induce the lattice deformation together with the shift of Dirac points, equivalent to modulating the hopping amplitudes between neighboring lattice sites, thereby giving rise to a synthetic gauge field. Similar to a real magnetic field, it further reveals a strong uniform perpendicular pseudomagnetic field as well as quantized Landau levels, which have been extended to acoustic and photonic crystals [13–31]. Unlike the real magnetic field breaking the time-reversal symmetry, the pseudomagnetic field preserves the time-reversal symmetry with opposite signs at two inequivalent valleys. The distinct nature between the real- and pseudo-magnetic fields can bring interesting phenomena.

In the well-known Haldane model exhibiting the quantum Hall effect, a pair of chiral edge states depending on the topological Chern numbers counterpropagates along two parallel edges [32,33]. Recent studies discovered an intriguing modified Haldane model, resulting in the antichiral edge states co-propagating on opposite sides, while the bulk states transport along the opposite direction as required by energy conservation [34,35]. This modification to the Haldane model can be constructed by reversing the next-nearest-neighbor hoppings at different sublattices, equivalent to reverse magnetic fluxes. These models have been realized in a gyromagnetic photonic crystal (GPC) by onsite modulation of magnetization, with such a structure immersed in an external real magnetic field [36–45]. The competitive relationship between the real- and pseudo-magnetic fields has been studied in graphene [46,47]. However, the transition of the topological edge states under their coexistence is unexplored in the classical wave system.

This work studies the edge state transport in a GPC with the coexistence of real- and pseudo-magnetic fields, as displayed in Fig. 1(a). The chiral edge states emerge in the GPC immersed in the same external real magnetic field, breaking the time-reversal symmetry, on different sublattices. The inhomogeneous uniaxial strain is applied to the GPC to construct an effective gauge field. The combined action of the real- and pseudo-magnetic fields modifies the dispersion curves of the chiral edge states and dispersive Landau levels. Secondly, we simultaneously reverse the magnetization directions of two sublattices and apply the strain, thereby inducing the spatially asymmetric propagation of the antichiral edge states in a GPC. The implemented experiments demonstrate the deformed edge state dispersions and their imbalanced transport couplings. Based on the modulation of the intensity and direction of the pseudomagnetic fields, we propose a multichannel waveguide, which paves a new way for the further application of photonic devices.

The Haldane model, with broken time-reversal symmetry in the original Dirac-model-based graphene-like honeycomb lattice, is defined by the Hamiltonian [34,42]: HHaldane=t1∑⟨i,j⟩ci†cj+t2∑⟨⟨i,j⟩⟩e−iνijφci†cj+H.c.,where t1 and t2 are nearest-neighbor hopping and next-nearest-neighbor hopping, respectively. ci† and cj are the creation and annihilation operators at sites i and j. φ represents the hopping phase, and νij=±1 indicates the hopping direction at sublattice A or B. The unequal t2 next-nearest-neighbor hoppings of two sublattices break the time-reversal symmetry, lifting the Dirac degeneracy and opening a completed bandgap. Two edge states counterpropagate along the upper and lower boundaries of the finite structure with the Haldane model. In the continuum theory, the effective Hamiltonian of the Haldane model can be rewritten as [34,48] H=v(σxτzkx+σyτ0ky)+t2aσzτz+t2bσ0τ0,where v is the group velocity around the Dirac point, the Pauli matrices σx,y,z describe the sublattice pseudospin, σ0 is the 2×2 identity matrix, τz=+1(−1) represents the K (K′) valley, and kx,y is the momentum relative to the Dirac points. In addition, t2a=−33t2sinφ and t2b=−3t2cosφ.

In photonics, the Haldane model can be realized in a GPC under uniform external real magnetic fields B0 along the z-direction. Figure 1(b) shows the configuration of the GPC with onsite magnetization modulation [41,42]. The gyromagnetic cylinders, arranged in a honeycomb lattice, consist of yttrium iron garnet (YIG) ferrite materials immersed in air. The honeycomb lattice can be divided into two interpenetrating triangular lattices (sublattices A and B). The lattice constant is a=10mm, the side length is l0=a/3, the radius of the YIG cylinder is r=1.5mm, and the relative permittivity is ε=14.5. Permanent magnets are placed above and below the gyromagnetic cylinders to generate a homogeneous external magnetic field. Both sublattices A and B are subjected to an external real magnetic field, where adjacent sublattices exhibit parallel magnetization bias orientations (as indicated by pink arrows) while maintaining identical magnetic fluxes, as shown in the right panel of Fig. 1(b). This configuration induces strong anisotropy in the YIG cylinders, resulting in a permeability tensor response (see Appendix A for materials characteristics).

We consider an inhomogeneous uniaxial strain applied to a GPC along the y-direction, as shown in Fig. 1(c). The translation invariance is preserved along the x-direction, indicating that the hopping amplitude in the other two directions remains constant. Stretching and compressing the original hexagonal lattice induces the lattice deformation and shifts the degenerate Dirac points along the K-K′ direction, as displayed in Fig. 1(d). This is mathematically equivalent to modulating the hopping amplitudes between neighboring lattice sites. Here, we introduce a distortion quantity β to quantify the lattice deformation. The displacement of the Dirac points exhibits a linear relationship with mβ for the mth layer, as illustrated in Fig. 1(e). Consequently, the inhomogeneous strain-induced hopping of the mth stretched and compressed honeycomb layer along the y-direction can be expressed as lm=l0+(−1)αmβ, where α=0 (1) for stretched (compressed) layers. The hopping amplitudes are set as fixed values: lN=l0+Nβ for the top layer and l−M=l0−Mβ for the bottom layer, where M and N denote the number of stretched and compressed layers, respectively. Both the distortion quantity and the associated number of layers can be modulated. In the GPC, the distortion β is modulated as β=(lN−l−M)/(N+M+1). The displacement of Dirac cones gives rise to an effective vector potential A=(δkx,0). According to B=∇×A, the linear variation of Ax(y) induces the emergence of the pseudomagnetic field Bs=Ax(y)/y perpendicular to the x–y plane. The pseudomagnetic fields Bs are oriented along the +z and −z directions at the K and K′ valleys, respectively. The number of compressed and stretched layers and the distortion quantity determine the pseudomagnetic field intensity. The effective Hamiltonian at the valleys becomes [46,49] H=v[σxτz(kx+Ax)+σyτ0(ky+Ay)]+t2aσzτz+t2bσ0τ0,where Ax,y is the vector potential. Due to the uniform pseudomagnetic field, the energy spectrum in the strained GPC is quantized into discrete Landau levels: En={−Δsign(Bs),n=0±ωcΔ2+ωc2n,n≠0,where n is the Landau level index, the cyclotron frequency is ωc=v2|Bs|, and the introduced parameter is Δ=t2aτz−t2b.

We consider the GPC under the coexistence of the real magnetic field B0 and pseudomagnetic field Bs. The presence of the B0 (Bs) breaks (preserves) the time-reversal symmetry in the system. While the real magnetic field is the same for the K and K′ valleys, the pseudomagnetic fields exhibit opposite directions for these two valleys. When both B0 and Bs coexist, the total effective magnetic fields at the two valleys become unbalanced. In the Haldane model, the chiral edge states propagate in opposite directions along two parallel zigzag boundaries. The introduction of the pseudomagnetic field further induces an energy transport imbalance between these two boundaries.

To demonstrate the properties of GPCs under the coexistence of the real- and pseudo- magnetic fields, we conduct an experiment to investigate imbalanced chiral transport. The experimental configuration is shown in Fig. 2(a). The gyromagnetic cylinders, with a radius of r=1.5mm and height of 5 mm, are precisely positioned between two parallel aluminum plates. A pair of permanent magnets [samarium-cobalt (SmCo)], possessing the same biasing directions, are placed directly above and below the gyromagnetic cylinders to provide a uniform external real magnetic field along the z-direction. Permanent magnets have a radius of 1.5 mm and height of 0.5 mm. To fix their positions, the magnets are embedded in two metallic plates with heights of h2=h3=4mm. Two additional metallic plates, with heights of h1=h4=1mm, are placed on the top and bottom to cover the magnets, preventing the microwave leakage in the experimental sample. To probe the internal electric field within the sample, the first and second metallic plates are drilled with 2 mm diameter holes. Due to experimental platform size constraints, the number of layers of the GPC along the y-direction is limited to 7, corresponding to the distortion β=0.4mm. A larger number of layers result in a weaker pseudomagnetic field while fixing the lN=l0+1.6 and l−M=l0−0.8. Perfectly electrically conducting boundaries are placed around the gyromagnetic cylinders to confine the edge states, while absorbing materials surround the left and right boundaries to minimize edge scattering. Figure 2(b) shows the top view of the first aluminum plate, featuring two detection channels along the upper and lower boundaries for measurement. Microwave excitation is provided by dipole antennas positioned at the left and right ends of the upper and lower boundaries of the GPC sample.

The dispersion of chiral edge states is experimentally measured and numerically calculated using COMSOL Multiphysics finite element analysis software, as shown in Fig. 2(c). Due to the interplay between the B0 and Bs, the band dispersion deformation of chiral edge states emerges. The crossing point of two dispersion curves is no longer symmetric about the center of the Brillouin zone and shifts toward the K′ valley compared to the Bs=0 case. The group velocities of two chiral edge states are unequal at the K and K′ valleys. The upper and lower edges of the sample support right-propagating and left-propagating edge states, respectively. After performing Fourier transforms on the edge Ez electric field mappings, we obtain the edge state dispersions for both the upper and lower boundaries, as shown in the insets of Fig. 2(c). The measured edge state dispersions are in excellent agreement with the simulation results. The varying number of layers in GPCs leads to distinct pseudomagnetic field intensities, which manipulates the edge state dispersion curves, as displayed in Fig. 2(d). The pseudomagnetic field exhibits a significant impact on the upper edge state dispersion compared to that on the lower edge. As the number of structural layers increases, the intensity of the pseudomagnetic field decreases. In the band structure, while the slope direction of the edge state dispersion curves remains unchanged, its magnitude decreases, indicating that the group velocity direction of the upper edge remains constant while its magnitude decreases.

The pseudomagnetic field Bs is larger than the real magnetic field B0, causing the signs of the total effective magnetic fields for two valleys (K and K′) to be opposite. In our GPC structure, the B0 is 0.08 T, while the Bs is 1.3 T under the distortion. β is 0.4 mm, corresponding to the number of layers of 7. The coexistence of Bs and B0 results in deformed edge state dispersions and imbalanced bulk state distributions of Landau levels at the K and K′ valleys [45,46]. Furthermore, the chiral edge states do not cross the zeroth Landau level without hybridizing with it. The Landau levels within the bulk bands, induced by strong pseudomagnetic fields, exhibit dispersive rather than flat characteristics. The electric fields of n=0, −1, and −2 Landau levels exhibit distinct spatial distributions for the K′ and K valleys in Fig. 2(e), which are attributed to the unequal total effective magnetic fields for two valleys.

When the excitation source is positioned at two boundaries of the GPC experimental sample, indicated by two yellow stars in Figs. 3(a) and 3(b), the edge states propagate rightward (leftward) along the upper (lower) edge. The measured electric field distributions, obtained by inserting a probe antenna into the holes within the experimental setup, reveal stronger electric field intensity at the upper edge compared to the lower edge. Subsequently, the probe antennas are placed at P1, P2, and P3 points marked by blue stars to measure the transmission coefficients, respectively. The transmission of the chiral edge states is controlled and influenced by the pseudomagnetic field intensity. The pseudomagnetic field leads to an uneven energy distribution of the upper and lower edges during the chiral transmission. This transmission property can be leveraged in the design of photonic devices.

Here, we investigate the interplay between the pseudomagnetic field Bs applied to the modified Haldane model and the real magnetic field B0, which results in spatially asymmetric edge state distributions, as shown in Fig. 4(a). The modified Haldane model exhibits antichiral edge states, where two edge states propagate in the same direction while the bulk states propagate in the opposite direction. The next-nearest-neighbor hopping directions of sublattices A and B are identical, while their hopping phases have opposite signs, as displayed in Fig. 4(b). The Dirac points still remain degenerate, but their energies shift in opposite directions. The most intuitive manifestation in the energy band is the emergence of degenerate edge state dispersions with identical dispersion slopes. The modified Haldane model can be realized in the GPC structure by applying real magnetic fields B0 in opposite directions to different sublattices. The related parameters of the GPC are the same as those in Fig. 2. The pseudomagnetic field Bs parallel to the +z-direction at the K valley is generated by lattice deformation, similar to that in the original Haldane model. In Fig. 4(c), the pseudomagnetic field leads to the band deformation and lifts two degenerate dispersion curves, with the upper and lower edge state dispersions represented by red and blue lines, respectively. In the experiment configuration, we reverse the biasing directions of adjacent permanent magnets to uniformly apply opposite magnetic fluxes to sublattices A and B. After performing a Fourier transform on the mapped electric fields of the upper and lower edges, we obtain deformed band dispersions that agree well with simulated results. Figure 4(d) shows that the intensity of the pseudomagnetic field Bs significantly influences the upper branch of the antichiral edge state dispersions; its direction of the group velocity remains unchanged while its magnitude increases with the enhancement of the pseudomagnetic field intensity.

Figures 5(a) and 5(b) illustrate the transport properties of the antichiral edge state in simulations and experiments. Both the upper and lower edge states propagate toward the right. Since the antichiral propagation occurs in a gapless system, the reverse transmission of the bulk states is essential to ensure the energy conservation of the system; thus, there is a weak reverse transmission in the bulk states accompanying the edge state that propagates to the left. The antichiral transmission is controlled by the pseudomagnetic field, which causes the intensity of the upper edge to exceed that of the lower edge. As shown in Figs. 5(c) and 5(d), the transmission at port P1 within the frequency range of 9.16–9.4 GHz is higher than that at port P3. Meanwhile, the transmission at port P2 always remains at a relatively low level compared to that at ports P1 and P3, indicating that the bulk state transmission is relatively weak. Additionally, the uneven energy distribution of the edge state transmission can be attributed to the presence of the pseudomagnetic field. Figures 5(e) and 5(f) illustrate that, when the excitation source is placed on the left side and incident from the left port, energy transmission in the bulk is prohibited. However, when incident from the right port, it exhibits unidirectional bulk transport, indicating that leftward bulk transport shows strong non-reciprocity.

The energy division on the edges of the GPCs depends on the directions and intensities of the real magnetic field B0 and pseudomagnetic field Bs. Here, we construct a five-channel unidirectional waveguide based on the characteristic of the counterpropagating chiral edge states and co-propagating antichiral edge states, as shown in Fig. 6(a). We combine two GPCs with opposite B0, indicated by red and blue cylinders, to construct a three-channel waveguide with an air gap width of 0.3a. The inverted uniaxial strain is applied to these two GPCs, resulting in their possession of +Bs and −Bs parallel to the z-direction, indicated by the color map in the inset. Three types of edge state dispersion curves emerge within the GPC bandgap. The group velocities of the chiral edge states on the top and bottom edges are positive, in contrast to the negative group velocity of the chiral edge state in the middle channel, as shown in Fig. 6(c). Therefore, the excited waves propagate toward the right along the top and bottom channels and toward the left along the middle channel. The opposite Bs are also generated in two GPCs with the same modified Haldane modes, as shown in Fig. 6(d). The antichiral edge states propagate toward the left along the three channels. The manipulation of B0 and Bs directions enables control over the energy distributions and transmission directions in different channels, thereby facilitating the design of the beam splitter, as illustrated by the electric field distribution in Fig. 6(b). Since the edge states are strongly confined to the edges and the bulk states are spatially separated from the edge states, they can propagate robustly through a shape 60° angle without backscattering. When we further change B0 and Bs in the four domains of the beam splitter, the transmission ratio of the five channels can be manipulated (see Appendix C for simulation results), thus allowing additional control over the on–off states of electromagnetic waves at specific output ports. Even though the multichannel waveguide is not experimentally realized owing to the size limit of the experimental platform, the abundant simulated results demonstrate remarkable tunability in the designed multichannel waveguide, therefore providing new methods to design more photonic devices.

By applying the uniaxial strain to GPCs with Haldane model and modified Haldane model, a synthetic pseudomagnetic field parallel to the z-direction can be obtained. The induced pseudomagnetic field does not change the transmission direction of the chiral and antichiral edge states, while leading to an imbalance in the edge energy distribution. This phenomenon of uneven energy distribution between the upper and lower edges is verified in the measured edge state dispersions and electric fields in the performed experiments. Subsequently, with the joint regulation of the real- and pseudo-magnetic fields, a multichannel electromagnetic wave waveguide is proposed. By manipulating the intensity and direction of the pseudomagnetic field, the transmission energy ratio of each channel can be controlled, which improves the space utilization rate.