Passive topological waveguide controlled by the boundary of the patterned area of external magnetic field with the hybrid quantum Hall and valley Hall effects

Topological waveguides with arbitrary pathway are desirable for many applications. In this paper we construct a triangular compound lattice consisting of magnetic dielectric rods. By breaking the space symmetry and the time-reversal symmetry, the structure generates topological edge states (TESs) from the hybrid quantum Hall effects and valley Hall effects. This topological edge waveguide pathway can be arbitrary arranged just by the external magnetic field. The hybrid topological phase provides a new and ultraflexible way to the reconfiguration of the TESs.


Introduction
In recent years, the analogy of quantum Hall effects (QHEs) [1][2][3], quantum spin Hall effects (QSHEs) [4,5], and quantum valley Hall effects (QVHEs) in optical system [6][7][8] has become a hot research topic, because topological edge states (TESs) provide the unique unidirectional and defect-insensitive transport. Photonic crystals provide more freedom to design band structures, and are a good platform to study and demonstrate the topological phase in a solid electronic system. The main advantage of a topological edge waveguide (TEW) over a traditional waveguide is topological protection. A TEW provides a solid foundation to efficiently guide, switch, and route light in integrated circuits. The topological protection can be combined with the reconfigurability of a TEW. The reconfigurability of a TEW means that the TEW can be dynamically modulated. Most of the reconfigurability literature focuses on the modulation of the modes, such as the transition of topological phase [9], the change of the frequency range of a TEW [10,11], the transformation of the edge states and their hybridization with bulk modes [12], the reversible switching of TESs [13]. On the other hand, the reconfigurability of the TES configuration can enable an arbitrarily shaped pathway of TES. The TESs occur between the trivial and nontrivial topological structures. To reconfigure the TES pathways, the topological phases of the bulky structures have to be changed. There are many ways to change the topological phases, such as sliding dielectric rods [14], rotating the Y-shaped prisms [15], changing the handedness of constituent objects mechanically or electronically [16]. All these ways have severe technological challenges. The bulky structures consist of a large number of unit cells, and the operation of changing the topological phases has to be performed over all the unit cells. Therefore, finding an ultraflexible way and a simple structure to achieve a reconfigurable TES is necessary for practical applications. In [17], a non-Hermitian-controlled topological state can enable the dynamic control of robust transmission, which actively steers topological light on demand by projecting a designed spatial pumping pattern onto the photonic lattice. All the unit cells are fixed and any arbitrary topological pathway inside the bulk of the lattice can be flexibly patterned simply through the boundary of the designed spatial pattern of the external pumping light. However, it is not easy to precisely tune the intensity of the pumping beam to just below the lasing threshold in order to achieve a sufficient gain-loss contrast at the boundary of the pumping area while avoiding nonlinear gain saturation in each element of the photonic lattice in the pump area, where each element acts as an on-chip light source and feeds light into the topological lattice. This method is an active way (the power-consuming pump provides active gain to the shining region) and can hardly work in microwave or millimeter wave (as gain material is common in optical region but not in microwave/millimeter wave region). In the present work we introduce a passive method (using permanent magnets or direct current to generate a static B field, which works in microwave/millimeter wave region). We can set up an arbitrary topological transport pathway inside a bulky lattice through controlling the shape of the projected area of the external magnetic field. The waveguide is determined by the boundary of the projected area of the external magnetic field, which do not require battery, or applied voltage or any energy source. The removal of the external magnetic field will recover the initial structure without any waveguide. Like writing waveguides in glass with a femtosecond laser, we call the method in this study as writing out the TEW by the external magnetic field.

Model and topological phases
Our idea comes from the QHEs of the magnetic photonic crystals (MPCs). When time reversal symmetry is broken by the external magnetic field, the degeneracy of the two photonic bands is lifted forming the topologically nontrivial bandgap. The separated bands acquire nonzero Chern numbers, which provides the necessary condition for the TESs appearing at the edges of the MPC. To realize the TESs, a cladding layer (see e.g. [18,19]) has to be added to restrict the TESs and transfer them into guided modes. The cladding layer must have the common bandgap but different topological phase with the MPC. The existence of the cladding layer makes it difficult to reconfigure the pathways of the TESs. Thus, we turn to the QVHEs of ordinary PCs. There are Dirac cones at points K and K' in the Brillouin zone (BZ) in a triangular lattice. The Dirac cones are protected by both the spatial and time inversion symmetries. Breaking the spatial inversion symmetry or the time inversion symmetry will lift the degeneracy at the K valleys and hence open a bandgap. If only the spatial inversion symmetry is broken, the two lifted bands acquire the opposite valley Chern number. By combining two structures with opposite topological phases, i.e. the valley Chern number, the valley topological edge states (VTESs) can be achieved [20][21][22][23]. To change the TES configuration, one has to change the bulky structure. From the single QHEs or QVHEs we cannot find the ultraflexible way to achieve the reconfigurability. Up to now, the QHEs and QVHEs have been always independently studied. One key question is what would take place if the QHEs and QVHEs simultaneously occur in one structure? Studying this question may bring about a new topological phenomena and lead to ultraflexible reconfiguration of the TES pathways.
We begin with a two-dimensional compound triangular lattice in which the unit cell consists of three same triangular yttrium-iron-garnet (YIG) rods with their centers placed at the vertices of an equilateral triangle. The schematic structure is shown in figure 1 in which a 1 and a 2 are the basic vectors and a is the lattice constant. r 1 , r 2 in the inset show the sizes of the equilateral triangle and the rods, respectively. φ shows the orientation of the equilateral triangle. For φ = 0, without external magnetic field (H 0 = 0) the inversion symmetries for both the time and space ensure Dirac degeneracy at K points in the bulky bands. Increasing φ from 0 degree to 30 degree will decrease the symmetry from C 3v to C 3 , which will break the Dirac degeneracy and form a bandgap. The two lifted bands from the Dirac degeneracy have opposite valley Chern number. By combining two structures with the opposite valley Chern number, the VTESs can be achieved [23]. This is just the QVHEs. We choose the compound lattice instead of the single lattice because the added scattering among the rods inside the lattice will broaden the bandgap, and increase the bandwidth and field localization effect.
Without the external magnetic field, the YIG rods have the permittivity ε = 15.26ε 0 and permeability µ = µ 0 [18]. To obtain the larger bandgap from the lifted Dirac cones, we perform the optimization of the structure parameters r 1 and r 2 . We only consider the TM modes (with field components E z , H x and H y ) of the system. The bulky bands are obtained using Comsol. The first hexagonal BZ and the rhombus BZ are plotted in figure 2(a). The fourth and fifth bands forming the degeneracy at the valley point K are plotted in figure 2(b) for φ = 0 and r 2 = 0.12a. With φ = 30 • , we record the frequency positions of two separate valley points with the changing of r 1 in figure 3(a) and find that at r 1 = 0.23a the two points K + and K − have the largest interval. Thus we fix r 1 = 0.23a and record K + and K − with changing of r 2 in figure 3(b). As a result, we find that at r 2 = 0.10a the two points have the largest interval. Hence we set r 1 = 0.23a and r 2 = 0.10a.    If the magnetic field is along the z-axis, the permeability of the YIG rod is taken as the tensor [18,24] with parameters ω 0 = 2πγH 0 , ω m = 2πγM S , γ = 2.8 × 10 6 (rad s −1 G −1 ), M S = 1780 G. H 0 is the external magnetic field. In this study, because we focus on the TESs with limited bandwidth, as indicated in [18], the effect of material dispersion and loss can be neglected. The material parameters for YIG are taken at 4.28 GHz. The topological invariant quantum for a band is Chern number that can be calculated from the Berry connection [23] and the gauge independent Berry curvature where u k is the field distribution corresponding to the mode k on the band. In figure 2(a), the rhombus BZ is divided into a mesh. k 1 and k 2 are the serial number along the two basic vectors of reciprocal lattice, respectively. The Berry curvature can be obtained by the Berry connection loop integration along each side of the mesh. Decreasing the space symmetry, or applying the external magnetic field to break the time-reversal symmetry can open the degeneracy. The two separated bands corresponding to the former and latter cases will acquire the valley Chern number and Chern number, respectively. When the orientation φ of the three rods is changed from 0 to any other value, the symmetry C 3V is decreased to C 3 . Figure 4 shows In figures 4(a) C is equal to zero, but an integration of the Berry curvature over the half BZ around K and K ' will lead to C K = 1/2 and C K' = −1/2 if the open gap from the two valley points is not large. Thus, a valley Chern number C V = C K − C K' = 1 is obtained. In the same way the fifth band acquires the opposite valley Chern number −1. For the gap from a pair of opened Dirac degeneracy points the valley Chern number is denoted as C V = ±1. When the gap becomes wider, the values of C V deviate from ±1, but the fourth and fifth bands have still the opposite valley topological phases. In figure 4(b) with the breaking of time-reversal symmetry, the integration of the Berry curvature over the whole BZ acquire the Chern number C = −1. The fifth and sixth bands acquire the Chern number C = 0 and C = 1, respectively. Therefore, the structures with φ ̸ = 0, H 0 = 0 and φ = 0, H 0 ̸ = 0 will lead to QVHEs and QHEs, respectively.

Design routing and simulations
The main design routine is shown in figure 5. Figure 5(a) shows the bulky bands along the Γ-K-M direction in the first BZ with H 0 = 0 andφ = 0. We notice that the fourth and fifth bands and the fifth and sixth bands form the two Dirac degeneracies at K point and Γ point, respectively. With φ = 30 • an omnidirectional bandgap (grey strip shadow) appears between the lifted two valleys in figure 5(b). The two separated photonic valley states show chirality as they have intrinsic circular-polarized orbital angular momentum, which can be verified by the phase distribution of E z [26]. The topological charge is defined as l =¸L ∇[arg(E z )] · ds/2π. In figure 5(b), the E z phase of the up (down) valley state decreases clockwise (counterclockwise) have the topological charge l = 1 (l = −1). The topological charge can define the valley topological phase in another way. To achieve the general QVHEs, the lattice of φ = 30 • is combined with another lattice of φ = −30 • to form an edge. The lattice of φ = −30 • has the same bandgap with the reversal topological charge l = −1 (l = 1) for the up (down) valley. Because of the different topological phases between the two lattices, the VTESs will be formed within the bandgap. However, the ultraflexible way to make the reconfiguration cannot be obtained from the single QVHEs because changing the edge shape has to change the bulky structures. The advantage of QVHEs for reconfiguration is that the two structures forming the TESs consist of the same lattices but only with different orientation. No cladding layer is needed for QVHEs. The advantage of QHEs in the reconfiguration is that the topological phase of the structure can be changed by the external magnetic field. However, an extra cladding layer is needed for QHEs. We can combine the QHEs and QVHEs, and apply their advantages to form and reconfigure the TESs. Thus, basing on the structure of figure 5(b) with the same φ = 30 • , we apply the external magnetic field with H 0 = 2000 Gauss on the lattice. As shown in figure 5(c), besides the bandgap between the two K points has been enlarged (blue strip shadow), the Dirac degeneracy at Γ point has been also lifted forming a bandgap (green strip shadow). Through the calculations from equation (6), the Chern number for the fourth, fifth and sixth band in figure 5(c) are found to be −1, 0 and 1, respectively. Therefore, such a lattice can achieve the QHEs. The bandgaps with green strip and blue strip formed by two bands with the Chern number overlaps with the bandgap with grey strip formed by two bands with the valley Chern number in figure 5 In order to find the hybrid TESs, we build a rectangular supercell from the compound lattice shown in the inset of figure 6. The supercell has the smallest period in the truncated x direction and is infinite in the y direction, but we take a finite length in the y direction for the calculations. The current supercell is made of the same lattice only with magnetic field applied to a half space. In figure 6 only a single red curve denoting the TES occurs in the up common bandgap with the bandwidth 0.029 (2πc/a), while in the down common bandgap no TES can be excited. Given a frequency ω, a unidirectional transport mode from an ordinary line source will be excited. These TESs are different from those with the QSHEs and QVHEs. For the QSHEs or QVHEs, the TES curves come in pairs so that a source must have a definite spin angular momentum to excite one of the modes on the two curves. Another feature of the current TESs is that the two structures are not only made of the same lattice but also have the same orientation. The boundary is determined by the region of the applied magnetic field. The unique mechanism forming the TESs help us to find the ultra-flexible way to reconfigure the pathway of the TESs because the edge is completely determined by the external magnetic field and can be set as an arbitrary shape. Removing the magnetic field will recover the initial structure without any waveguide so that the designed structure can be used as a reusable platform for optical experiments.    In figure 7, we show how to realize the reconfiguration of the TESs. In the whole two-dimensional plane made of the same lattice in figure 1, a beam of magnetic field with H 0 = 2000 Gauss is incident on the lattice ( figure 7(a)). The edge can be formed directly by the shape of the magnetic beam. The beam shape and position of the magnetic field can be arbitrary changed. Therefore, a dynamic TES waveguide along the boundaries is achieved. The forming process of the waveguide looks like the writing of the magnetic field on a paper. As a magnetic conductor can remove magnetic field in a region with uniform applied magnetic field, the magnetic conductor can be used as a module to define the transport pathway, which is another refiguration way ( figure 7(b)). In figure 8 we show some simulation results with different boundary configurations defined by the magnetic field. The line source has the angular frequency ω = 0.8812(2πc/a) and its position is denoted by the yellow star. Figure 8(a) shows a rectangle loop formed by the transport pathway. Figures 8(b) and (c) show the double 'z' pathways with different directions. Figure 8(d) shows a triangular loop. We can envision arbitrary TES waveguide pathways through the distribution of the magnetic field.
In the above simulations, at the edge the magnetic fied changes abruptly from 2000 Gauss to 0. In real experimental condition, the magnetic field at the interface should be gradually varied. In order to match the real case, we construct a supercell interface at which the values of H 0 change linearly from 2000 Gauss to 0 with different slope widths h. The values of h are based on the possible range of the gradually changing of the magnetic field. The obtained bands for different h and and the supercell are shown in figure 9. We find that in spite of the gradually changing of the magnetic field, the TESs still occur in the band gap. However, with the increasing of h, more and more the TES curve has entered into the bulky bands. In figure 10 we plot the E z field distributions excited from the point source of ω = 0.8812(2πc/a) at the interface y = 0. For h smaller than a, the transports keep the property of the TES, but the transport path in below the source. For h = 1.2a, the transport path becomes vague. However, when we move the source at y = 1.2a, the transport path is recovered, as shown in figure 10(e). The simulations demonstrate that the TESs in this study are robust to the magnetic field boundary condition and can be achieved in some designed magnetic fields [27].

Comparison of the hybrid and single TESs
The most important feature of TESs is their topological protection, i.e. the ability to overcome the structure defect in transport. In figures 5(b) and (c), we have plotted the bands of the infinite periodic lattices with the QVHE and QHE, respectively. However, in order to find the TESs, we have to perform the eigen-frequency calculations based on the supercell. The results are shown in figures 11(a) and (b), corresponding to the QVHE and QHE, respectively. In the supercell of figure 10(a) the edge is formed between the two lattices with φ = 30 • and φ = −30 • , respectively. A pair of TES curves with opposite group velocities occur in a bandgap with bandwidth 0.12 (2πc/a). According to the QVHE, a spin source can excite a mode on one of the two curves through the spin-locked property, which leads to the unidirectional transport. In the supercell of figure 11(b) the edge is formed between the two lattices with φ = 0 with and without magnetic field, respectively. A monotonic TES curve closes the whole narrow bandgap with bandwidth 0.015 (2πc/a). According to the QHE, an ordinary line source can excite the mode on the curve and achieve the unidirectional transport.
Based on all the TESs discussed above, we construct three kinds of straight TES waveguides with a defect (removing a medium rod at the boundary) to understand the robustness of the TESs. The frequency-domain simulations of transport along the straight waveguides have been performed through Comsol software. The structure size is 18a × 16a and the absorbing boundaries are used. The waveguide is at the center that devides the structure into two half spaces with different topological phases. The 2D E z amplitude field patterns are shown in figures 12-14, in which the two insets give the 1D field distributions along the horizontal waveguide and the vertical line across the propagation path, respectively. The structure model of figure 12 is based on the supercell shown in the inset of figure 10(a). A clockwise spin source with ω 1 = 0.8975 (2πc/a) is placed at the edge. The frequency line in figure 11(a) intersects the TES curves at two points A and B. The 2D E z amplitude field paterns show that only mode A can be excited with clear transport from the left to the right direction. However, a large reflection occurs at the defect. The horizontal and vertical 1D fields show the clear backward scattering (which generates high oscillation due to the standing wave between the excitation wave and backward scattering wave) and the good field localization of the mode. The structure model of figure 13 is based on the supercell shown in the inset of figure 11(b). An ordinary line source with ω 2 = 0.8312 (2πc/a) at the edge excites only the unidirectional transport (i.e. both clockwise and anticlockwise spin sources will excite the transport along the same direction). The 2D E z amplitude field patterns show the clear left transport and a small reflection at the defect. The horizontal 1D fields show the small backward scattering. However, the field localization from the vertical 1D fields is not good because the bandgap in figure 11(a) is very narrow so that the field has leaked into the side with magnetic field. The structure model of figure 14 is based on the hybrid QVHE and QHE. A ordinary line source with ω 2 = 0.8812 (2πc/a) at the edge excites only the left transport. The 2D E z amplitude field paterns show little reflection at the defect. The horizontal 1D fields shows no backward scattering. Furthermore, the vertical 1D fields shows a good field localization. Therefore, the QVHEs have the best field localization and worst ability to resist the defect. The QHEs in this structure have a good feature of robustness against a defect but a drawback of weak localization of the field. The hybrid TESs have inherited both the field localization feature of QVHE and the robustness feature of QHE against a defect. These are two good properties of the hybrid TESs. In this study, the two isolated TESs can not simutaneously have the two good properties, while the hybrid TESs can. The filed localization feature can localize the energy and enhance the transport efficiency. The unidirectional transport and robustness feature against a defect are useful in constructing a slow-light waveguide. A slow-light system typically has some problem from fabrication imperfections: as the group velocity of the light decreases, it becomes increasingly sensitive to the disorder, leading to significant backscattering, loss, and Anderson localization [28]. A topological protection can resist backscattering and localization in the presence of a disorder or defect.

Conclusion
In this paper we have proposed a novel model that can produce the hybrid QHES and QVHES. The hybrid QHES and QVHES enable an ultraflexible way to reconfigure a TES waveguide. The configuration of the TES waveguide is totally defined by the external magnetic field boundaries. The hybrid QVHE and QHE have inherited both the field localization feature of QVHE and the robustness feature of QHE to a defect. We envision that the new topological phase may occur in an electronic system or condensed matter. We also hope that this theoretical scheme will excite some relevant experimental study.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix. The role of valley topological phase in the hybrid TESs
The current TESs result from the hybrid QHE and QVHE. As is known, the TESs can be also formed between the lattice with QHE and the ordinary lattice if they have common band gaps. One may ask what is the role of QVHE in the hybrid TESs. In this appendix, we make a supplementary study through constructing a supercell composed of two combined lattices: The down one is the same as the half bottom part as figure 6, the other is a square lattice in which the lattice site composed of three triangular rods with r 1 = 0.23a and r 2 = 0.12a. The vertex angle of the three triangular rods is along the horizontal direction and without the magnetic field. The schematic structure is show in the inset of figure A1. The two lattices have the common bandgaps so that they can form a TES which is shown in figure A1. The TESs take on a typical QHE with the monotonous curve in the band gap. Because of the square lattice, the rod rotation cannot lead to the change of the topological phase.
In order to find the effect of QVHE on the hybrid TES, basing on the supercell of figure 6, we invert the directions of the triangular rods in the up-half supercell, as shown in the inset of figure A2. The eigenfrequency bands are shown in figure A2. Besides the monotonous TES curve occurs in the up-band gap, an approximate symmetric TES curve is formed in the down-band gap. As is seen in figure 6, no TES has been excited in the down-band gap. The inversion of the triangular rods will change the valley topological phase, i.e. the phase reversal at the valley points, so that a new TES curve has been formed. It is interesting that the TESs in the two band gaps take on different topological properties. The up one is the chiral edge state from which a unidirectional transport can be excited from an ordinary point source, while the down one is the helical edge state from which a unidirectional transport can be excited from a spin point source and locked by the spin direction of the source. Therefore, the valley topological phase takes a clear effect on the hybrid TESs.
In figure A3, we plot the E z field distributions with the point source excitation from the structure based on the supercell in figure A2. In figure A3(a), an ordinary point source with ω 4 excites only right-transport. In figures A3(b) and (c), two spin point sources with ω 3 for opposite spin directions excites only right-and left-transport, respectively.