Ideal type-II Weyl points in twisted one-dimensional dielectric photonic crystals

Weyl points are the degenerate points in three-dimensional momentum space with nontrivial topological phase, which are usually realized in classical system with structure and symmetry designs. Here we proposed a one-dimensional layer-stacked photonic crystal using anisotropic materials to realize ideal type-II Weyl points without structure designs. The topological transition from two Dirac points to four Weyl points can be clearly observed by tuning the twist angle between layers. Besides, on the interface between the photonic type-II Weyl material and air, gappless surface states have also been demonstrated in an incomplete bulk bandgap. By breaking parameter symmetry, these ideal type-II Weyl points at the same frequency would transform into the non-ideal ones, and exhibit topological surface states with single group velocity. Our work may provide a new idea for the realization of photonic Weyl points or other semimetal phases by utilizing naturally anisotropic materials.


Introduction
Three-dimensional(3D) topological semimetal phases have attracted much attentions in recent years, such as Weyl points [1][2][3][4], Dirac points [5,6] and nodal lines [7,8], while Dirac and Weyl points can further be classified to type-I and type-II points depending on their equi-frequency curves at Dirac/Weyl frequency [9][10][11]. As the fundamental state, 3D Dirac points can be seen as a pair of Weyl points with opposite charges, and may be separated in momentum space by breaking time-reversal symmetry or inversion symmetry. Specifially, Dirac/Weyl semimetals have been demonstrated to exhibit several peculiar phenomena, such as Fermi arcs and chiral anomaly [1,11], while for type-II Weyl points, the unique features including tilted cone dispersion and the surface states existed in an incomplete bandgap would bring more exotic effects [10]. (a) Schematic of the unit cell of the twisted layer-stacked photonic crystals, with relative permittivity tensor ε 1 , ε 2 and ε 3 from lower to upper layers, and twist angle θ for the first(clockwisely) and third(counterclockwisely) dielectrics. The period in z direction is h with three layers equally divided to h/3, while in xy plane the dielectrics are continuous.(b) Dependence of the permittivity components on twist angle θ . (c) 3D band structure of the twisted photonic crystal on k x -k y plane with respect to k z = π/h when θ = 0 • , the pair of black dots show the Dirac points (k x = ±1.07π/h, k y = 0) with fourfold degeneracies.(d),(e) and (f) The linear dispersion in three orthogonal directions around the type-II Dirac points (0.52c/h).The blue and red lines in (c) indicate the bands with even and odd M y parities.
In this paper, we will show that the simple layer-stacked one-dimensional(1D) photonic crystal could exhibit ideal type-II Weyl points by introducing the degree of freedom of twist angle. Utilizing nondispersive anisotropic dielectrics combined with twist angles, we are able to realize a topological photonic system without structure design. The evolution from two type-II Dirac points to four type-II Weyl points can be easily achieved by tuning the twist angles, which induces topological phase transitions, and the gapless surface states have also been observed in an incomplete bandgap. Moreover, we further obtain the non-ideal type-II Weyl points and surface states with single group velocity by breaking the parameter symmetry. Our work may pave a new way for the study of Weyl system and the tuning between Dirac and Weyl points.

Ideal type-II Weyl points and surface states in the twisted photonic crystals
Here we propose a 1D layer-stacked photonic crystal using constant anisotropic materials, with three types of dielectric layers as unit cell (see FIG. 1(a)). The designed photonic crystal is periodic along z direction with a lattice constant h, while keeps homogeneous in the xy plane. We set the three layers to have the same thickness of h/3, and adopt identical nondispersive anisotropic dielectrics with the relative permittivity ε 0 = diag [ε 11 , ε 22 , ε 33 ], whereas the dielectrics of lower (upper) layer rotate an angle θ counterclockwisely (clockwisely) from the x axis, while the middle layer keep unchanged. In this way, the relative permittivity tensor of the unit cell of such a photonic crystal can be expressed as where ε xx = cos 2 θ ε 11 + sin 2 θ ε 22 , ε yy = sin 2 θ ε 11 + cos 2 θ ε 22 , ε xy = ε yx = (ε 11 − ε 22 )sinθ cosθ , and ε zz = ε 33 are all constant for a fixed θ (see FIG. 1(b) for the dependence of permittivity components on θ ). Thus, this 1D photonic crystal can actually be considered as a threedimensional(3D) structure due to the degree of freedom of the twist angle θ . Considering the initial case, we first numerically calculate the band structure at k z = π/h plane using COMSOL Multiphysics, with the relative permittivity ε 11 = 1, ε 22 = 2 and ε 33 = 14, while the twist angle θ = 0 • . In this case, the twisted photonic crystal is reduced to a homogeneous anisotropic material and the dispersions for lowest four bands are presented in FIG. 1(c), which are doubly degenerate for each two bands. Besides, there is a pair of degenerate points with fourfold degeneracy (the black dots) at (k x = ±1.07π/h, k y = 0). It should be emphasized that the positions of degenerate points can locate outside 0 to π/h due to the non-periodicity in the xy plane. To further demonstrate the properties of these degenerate points, the corresponding 1D band structure in three orthogonal directions are shown in FIG. 1(d)-1(f). We first sweep k x for a fixed (k y , k z ) = (0, 1)π/h and obtain two degenerate points at k x = ±1.07π/h, as shown in FIG. 1(d). For the intrinsic symmetry of electromagnetism in the anisotropic permittivity [23],the M y -even (with only (E x , E z , H y )) and M y -odd (with only (H x , H z , E y )) states with respect to the k y =0 mirror plane are then found in this photonic crystal when θ = 0 • , as denoted by the blue and red bands in FIG. 1(d). Thus the two black dots are exactly the crossing of bands with opposite parities, which indicates the Dirac points. For each point, the two tilted bands with the same sign of group velocity may be further classified as a type-II Dirac point. In FIG. 1(e), by fixing (k x , k z ) = (1.07, 1)π/h and sweeping k y , we get a degenerate point at k y =0. Similarly, there is a fourfold degenerate point situated at k z = π/h when we change k z at (k x , k y ) = (1.07, 0)π/h, as shown in FIG. 1(f). Therefore, the tilted linear dispersions around the two degenerate points verify the existence of type-II Dirac points for θ = 0 • .
Next, we change θ from 0 • to 90 • and study the evolution of degenerate points with twist angle θ at the k z = π/h plane. For the dielectrics in Eq. (1), the period of permittivities ε 1 and ε 3 is θ = 180 • (see FIG. 1(b)), while the band structure for θ = θ 1 is identical to that for θ = (180 • − θ 1 ) when 0 < θ 1 < 180 • , thus we only display the degeneracies of band structure for θ ranging from 0 • to 90 • for simplicity, as shown in FIG. 2(a) and 2(b). Firstly, we study the evolution of these degenerate points from 0 • to 60 • in FIG. 2(a). It is shown that the original two type-II Dirac points (black dots) are split into two pairs of type-II Weyl points for the nonzero twist angle θ , which is formed by the twofold degeneracy of two middle bands (see the 3D band structure in FIG. 2 (c) for θ = 30 • ). Besides, each pair of Weyl points with the same k x will carry +1 and -1 charges, respectively, indicating the nontrivial topology of these type-II Weyl points. The inset for Chern number and topological charges related to the type-II Weyl points at k x = 0.86π/h further verifies this point. During this process, we can see that the k x coordinates of Weyl points gradually decrease with the increase of θ and evatually degenerate at k x = 0 when θ = 60 • . While for the k y coordinates of these Weyl points, they will first undergo an increasing process and then gradually decrease to k y = 0 at θ = 60 • . Thus the evolutionary path of these four Weyl points forms an "eye-glass" shape in momentum space when twist angle θ changes from 0 • to 60 • . Specifically, when the photonic crystal has a twist angle of 60 • , its band structure is the same as that for θ = 120 • , which only presents a degenerate point at (0, 0, π/h)(denoted as A). It can be explained that this layer-stacked crystal has a complete rotation period of 360 • along z direction when θ = 120 • and dispalys a threefold screw symmetryŜ 3z . Together with time-reversal symmety T , the combined symmetry TŜ 3z guarantees that the twofold degeneracy must be at high symmetry A point or Γ point [(0, 0, 0)] in momentum space [46].
The evolution of these type-II Weyl points for θ ranging from 60 • to 90 • is shown in Fig.  2(b). Starting from a degenerate point at (0, 0, π/h) for θ = 60 • , four type-II Weyl points then appear at the high symmetry lines of k x = 0 and k y = 0 when twist angle θ continues to increase. In detail, for a fixed θ the degenerate points located at k x > 0 and k y > 0 regions belong to a pair of Weyl points with opposite charges, while the degenerate points at k x < 0 and k y < 0 regions are another pair. Moreover, the k y coordinates of Weyl points increase faster than those of the k x vector, and finally evolving to (±0.42π/h, 0, π/h) and (0, ±0.98π/h, π/h) at θ = 90 • , as denoted by the red and pink hollow circles in Fig. 2(b). It should be stressed that the frequencies of Weyl points here are not at the same frequency, and the tilted dispersions for each pair of type-II Weyl points are in different directions.
To show that the type-II Weyl points indeed exist in such a twisted layer-stacked photonic crystal, the 3D band structure for θ = 30 • at k z = π/h plane is choosen to display the degenerate points and tilted cone dispersion, as presented in Fig. 2(c). We can see that the doubly degenerate bands are splitted when the twisting between different layers happens, with two nondegenerate bands separated into the high and low frequencies, respectively. Meanwhile, two middle bands degenerate at four Weyl points (W 1, W 2, W 3 and W 4) in momentum space with their locations in (k x , k y , k z ) = (±0.86, ±0.23, 1)π/h, and having the same frequency of 0.48c/h. Besides, it is clearly shown that the cone spectrum near these Weyl points is strongly tilted for k x direction. On the right side of Fig. 2(c), we further plot the linear degeneracy in the vicinity of these type-II Weyl points along k x , k y and k z directions for fixed k y = 0.23π/h, k x = 0.86π/h and (k x , k y ) = (0.86, 0.23)π/h, respectively. For the k x direction, it is shown that the sign of group velocities for the two intersecting bands are the same around each degenerate point, which coincides with the features of type-II Weyl points. While for the other two directions, linear dispersions are also clearly presented. Therefore, combining with the characteristic of iso-frequency (at 0.48c/h) for these Weyl points, this twisted layer-stacked photonic system can actually exist four ideal type-II Wely points by tuning the twist angle θ within an appropriate range.
According to the bulk-edge correspondence, topological surface states should exist at the interface between a photonic type-II Weyl semimetal and the trivial material. To investigate the gapless surface states, we establish a supercell with 20 unit cells along z direction and keep it infinite along the x and y directions, thus forming two truncated surfaces in a direction perpendicular to the z axis, as shown in Fig. 3(c). Here we use air as the topological trivial material, and study the projected band structures corresponding to the case (Fig. 2(c)) at θ = 30 • . A full wave numerical calculation is adopted to calculate the band structures and eigenfields. For k x = 0.86π/h (see Fig. 3(a)), the W 2 and W 4 Weyl points of opposite charges are projected onto the xy surface, and there are two surface states (the red and blue curves) existed in the regions when k y ≥ 0.23π/h (for the other pair of W 1 and W 3 Weyl points of k x = −0.86π/h, similar results can also be found), indicating the nontrivial topology of these type-II Weyl points. To further verify the topological properties when k y ≥ 0.23π/h, we show the projected bands for k y = π/h and the eigenfields of surface states in Fig. 3(b) and 3(c). We find that two gapless surface states are located at an imcomplete bandgap, and they have the same sign of group velocity. This phenomenan is actually resulted from the tilted dispersions around type-II Weyl point. The -E-of these two surface states shows that the eigenfields for the blue band are localized on the boundary in −z direction, while the eigenfields for the red band are localized on the boundary in +z direction, demonstrating the ±1 charges of W 2 and W 4 Weyl points and the existence of topological surface states. The projected spectra at k y =-0.5π/h, k y =0 and k y =0.5π/h, respectively, where the gapless surface state( the blue lines) with single group velocity is observed when k y = ±0.5π/h.

Type-II Weyl points in twisted photonic crystals with bilayer unit cell
At last, we continue to explore the evolutions of location and shaping of the type-II Weyl points in momentum space by varying the permittivity distribution of unit cell structure to reduce its symmetry. As displayed in the inset of Fig. 4(a), the unit cell of twisted photonic crystal is replaced by the bilayer anisotropic materials ε 1 and ε 2 with different thicknesses of h ε1 =2h/3 and h ε2 =h/3, where the lower layer (dielectric ε 1 ) has a twist angle β respect to the x axis. Here, ε 1 and ε 2 are still taken from Eq. 1, in addition to using β to distinguish the twist angle. We can see that the parameter symmetry is thus broken in such a unit cell constructed of bilayer anisotropic materials. In Fig. 4(a), we show the evolution of Weyl points formed by the degenerate bands (see the band structure in Fig. 4(b) for β = 60 • ) at k z = π/h plane when β changes from 0 • to 90 • . Starting from the initial case of β = 0 • , it is shown that the type-II Dirac points (i.e., the black solid dots in FIG. 1(c)) are splitted into two pairs of type-II Weyl points, with each pair situated in the same quadrant, which is totally different from the locations in Fig.2. Besides, due to the fewer symmetry of parameters, the type-II Weyl points of each pair do not locate at the same frequency, implying the non-ideal type-II Weyl points. The evolutionary paths fitted in Fig. 4(a) further reveal that there are two paths for each pair of Weyl points with different features. Taking the Weyl points in the first quadrant as an example. One path has a shorter distance, with the k x coordinates of Weyl point decreases gradually to 0.42π/h, while k y increases first and then decreases to 0 (the red hollow circle). Another path has a longer evolutionary distance in momentum space, where the k x coordinate declines from 1.07π/h to 0 and the k y coordinate increase from 0 to 0.98π/h (the pink hollow circle), respectively. It is shown that the final state corresponds exactly to the positions of Weyl points in Fig. 2(b) when θ = 90 • , which can be easily understood that the twisted structure in Fig. 1(a) is reduced to the asymmetrical case in Fig. 4(a) when θ = 90 • . The results are echoed back and forth.
In Fig. 4(b), we show the 3D band structure for β = 60 • at k z = π/h plane and mark the locations of four non-ideal type-II Weyl point by the orange and magenta stars. It is intuitive to see that each pair of Weyl points are located in the same quadrant with different frequencies, and the locations of these Weyl points in momentum space are (−0.66, −0.24, 1)π/h and (0.66, 0.24, 1)π/h [the orange stars], (−0.53, −0.86, 1)π/h and (0.53, 0.86, 1)π/h [the magenta stars], respectively. In addition, linear dispersions around these Weyl points can also be observed, including the tilted cone dispersion in k x direction for the two Weyl points marked as orange stars, while for the other two Weyl points (magenta stars), the linear dispersion is tilted in k y direction, as denoted by the inset. This unique tilting structure of each pair of Weyl points is different from that in Fig. 2. Next we will study the projected band structures of such twisted photonic crystal at different k y to investigate the existence of surface states. The topological charges for Weyl points are ±1 as shown in Fig. 4(c), where the topological transitions take place, and the positions and charges of Weyl points thus induce the k y dependence of Chern number [16]. Three representative k y values lying between the Weyl points with different charges are taken to study the band structure projected along the z direction (i.e., the Weyl photonic crystal is truncated in z direction ). When k y = −0.5π/h, the calculated dispersions near the frequencies of Weyl points are displayed in Fig. 4(d), where the surface state is plotted in blue curve and the bulk states are shown in grey. It can be seen that there exists a gapless surface state with negative group velocity, which is consistent with the Chern number C = −1 in Fig. 4(c) and is resulted from the opposite topological charges of type-II Weyl points. As k y increases to 0 (see Fig. 4(e)), there is no surface state located in the projected bandgap due to the trivial phase between two Weyl points with the same charge. Next, as k y further increases to 0.5π/h, the projected band structure can be found in Fig. 4(f), where the surface state reappears with positive group velocity, corresponding to the +1 Chern number in Fig. 4(c). The surface states thus verify the topology of these four non-ideal type-II Weyl points.
Finally, we propose that the naturally anisotropic materials with elliptic in-plane dispersion may be used to realize the ideal model of such a layer-stacked 1D photonic crystals, though losses can be introduced due to the dispersion of practical material [47,48]. By controlling the twist angle between layers, the type-II Weyl points residing in different positions can be found in momentum space.

Conclusion and outlook
To summarize, we theoretically designed the layer-stacked twisted photonic crystals using anisotropic dielectrics, and proved the existence of photonic type-II Weyl points in such a system. By symmetrically tuning the twist angle in a unit cell consisting of trilayer dielectrics, we split two type-II Dirac points into the four ideal type-II Weyl points, and analyzed the linear dispersions around them. The evolutions of these Weyl points with respect to twist angle is detailly studied in a rotation period. In addition, gapless surface states are found at the truncated surface of such photonic crystal slab. We further explore a twisted structure with fewer symmetry and obtain the non-ideal type-II Weyl points, verifying the existence of surface states with single group velocity. In the twisted photonic system, naturally anisotropic materials are expected to be used as a platform for studying the topological physics of Weyl points, or other semimetal phases, which do not require the complex structural design as in 3D photonic crystals and metamaterials, and may open up new ideas for exploring topological states in systems composed of naturally anisotropic materials.