Optical interface states protected by synthetic Weyl points

Weyl fermions have not been found in nature as elementary particles, but they emerge as nodal points in the band structure of electronic and classical wave crystals. Novel phenomena such as Fermi arcs and chiral anomaly have fueled the interest in these topological points which are frequently perceived as monopoles in momentum space. Here we report the experimental observation of generalized optical Weyl points inside the parameter space of a photonic crystal with a specially designed four-layer unit cell. The reflection at the surface of a truncated photonic crystal exhibits phase vortexes due to the synthetic Weyl points, which in turn guarantees the existence of interface states between photonic crystals and any reflecting substrates. The reflection phase vortexes have been confirmed for the first time in our experiments which serve as an experimental signature of the generalized Weyl points. The existence of these interface states is protected by the topological properties of the Weyl points and the trajectories of these states in the parameter space resembles those of Weyl semimetal"Fermi arcs surface states"in momentum space. Tracing the origin of interface states to the topological character of the parameter space paves the way for a rational design of strongly localized states with enhanced local field.

properties. Recent theoretical work suggested that synthetic (man-made) dimensions allow for flexible control over system parameters and can hence facilitate observations of many novel phenomena. Here, we show that Weyl point physics can be explored easily using the concept of synthetic dimensions.
In particular, we consider topological nodal points in a mixed space of momentum and real space structural parameters. Such synthetic nodal points enable us to study Weyl point physics and their topological consequences in simple layer-by-layer structures, which are much easier to fabricate and characterize than ordinary Weyl crystals. As such, it enables the experimental investigation of Weyl physics in the optical regime, which is otherwise very challenging to realize. In addition, the existence of surface states in one-dimensional photonic crystals with complex unit cells can now be understood as topological consequences of nodal points in a higher-dimensional synthetic space.
Looking ahead, our approach not only increases the flexibility of realizing topological physics, but it also provides the possibility of manipulating topological matter in real time.

I. Introduction
Great efforts have been devoted to investigate various intriguing phenomena associated with Weyl points [1][2][3][4][5][6], such as the Fermi arc surface states [7,8] and the chiral anomaly [2] associated with electronic systems. Besides electronic systems, Weyl points have also been found in photonic [9][10][11][12][13][14], acoustic [15,16] and plasmonic [17] systems. Up to now, Weyl points have mostly been identified as momentum space magnetic monopoles, i.e. sources or sinks of Berry curvature defined in the momentum space. As such, Weyl points are usually perceived as topological nodal points in the 3D momentum space defined by Bloch momentum coordinates x k , y k and z k . On the other hand, a few recent works considered the topological singular points in the synthetic dimensions [18][19][20] instead of the momentum space. The interest in considering synthetic dimensions is fueled by the ability of realizing physics in higher dimensions [18,[21][22][23][24] and the possibility of simplifying experimental designs [25]. Moreover, the possible control over the synthetic dimensions enables the experimental verification of the nontrivial topology of any closed surface enclosing the topological singular points [18,19]. Here we experimentally realize generalized Weyl points in the optical frequency regime with one-dimensional (1D) photonic crystals (PCs) utilizing the concept of synthetic dimensions. Different from previous works [18][19][20] that replaced all three dimensions with synthetic dimensions, here we replace two wave vector components with two independent geometric parameters (which form a parameter space) in the Weyl Hamiltonian and keep one dimension as the wave vector. By doing this, we retain the novel bulk-edge correspondence relation between the edge states and the Weyl points. This is not possible for other topological singular points in purely synthetic dimensions [18][19][20]. Meanwhile, our hybridized Weyl points also preserve the advantage of using synthetic dimensions which facilitates the experimental investigation of Weyl physics in the optical region, whose structures are otherwise complex [11,12,15,26,27] and hence difficult to fabricate at such frequencies.
The Weyl Hamiltonian can be written as , , i i j j i j H k v    [28], where , i j v , i k and i  with , , i x y z  represent the group velocities, wave vectors and the Pauli matrices, respectively. Unlike two-dimensional (2D) Dirac points, Weyl points are stable against perturbations when the wave vectors remain as good quantum numbers and the Weyl points do not interact with other bands [9].
Each Weyl point has its associated topological charge given by the Chern number of a closed surface enclosing it [5]. Though Weyl points are usually defined in the 3D momentum space, very recently Weyl physics has been theoretically discussed in synthetic dimensions [25]. Instead of using all three components of the wave vector, here we replace two wave vector components with two independent geometric parameters, and experimentally investigate the generalized Weyl points with simple 1D PCs. Such generalizations preserve the standard Weyl point characteristics such as the associated topological charges [11,13] and robustness against variations in the parameters [9].
Moreover, optical interface states can be found between the PCs possessing synthetic Weyl points and any reflection substrates, whose existence is stable and protected by the topological properties of the synthetic Weyl points. We show that the reflection phase of a truncated PC exhibits vortex structures [29] in the parameter space around a synthetic Weyl point. These vortexes carry the same topological charges as the corresponding Weyl points. The reflection phase along any loop in the parameter space enclosing the center of the vortex (also the position of the Weyl point) varies continuously from   to  . This property guarantees the existence of interface states at the boundary separating the PC and a gapped material such as a reflecting substrate [30,31] independent of the properties of the substrate. The above physical interpretation severs as the bulk-edge correspondence [30,32] for the synthetic Weyl points in our system. These interface states can be regarded as analogues of edge states in Weyl semimetals [3,6,7], and they can be useful in nonlinear optics [33][34][35], quantum optics [36,37], thermal radiation [38] and so on [39][40][41][42]. The winding of the reflection phase has also been discussed in the context of adiabatic charge pumping and Floquet Weyl phases in a three-dimensional network [43][44][45]. Furthermore, we also introduce a third geometric parameter, which extends the three dimensional space to four dimensions. By tuning the third geometric parameter, we also observe the topological transition from Weyl semimetals to nodal line semimetals [9,46].

A. Synthetic Weyl points in parameter space
To illustrate the idea of synthetic Weyl points in a generalized parameter space, we consider a 1D PC consisting of four layers per unit cell, as shown in the inset in Fig. 1(a). In our experiments, the first and third layers (blue) are made of HfO2 with refractive index a n =2.00, and the second and fourth layers (red) are made of SiO2 with refractive index b n =1.45. The thickness of each layer is given by: Since the thickness of each layer cannot be a negative value, so p and q are both fall in [-1,1], which makes the p-q space is an closed parameter space. The total optical length L inside the unit cell is a constant   2 a a b b n d n d  for the whole p q  space. As illustrated in Fig. 1(a), the structural parameters p and q , together with one Bloch wave vector k , form a 3D parameter space in which Weyl physics can be studied.
We start with the PCs with only two layers inside each unit cell, a layer of HfO2 with thickness a d and a layer of SiO2 with thickness b d . The band dispersion is plotted in Fig. 1(b) in red. A fourlayer PC with parameters 0 p  and 0 q  simply doubles the length of each unit cell and folds the Brillouin zone. The dispersion of this four-layer PC is shown in Fig. 1(b) in blue. Such artificial band folding gives a linear crossing along the wavevector direction. Away from the point where 0 p  and 0 q  , the degeneracy introduced by the band folding is lifted and a band gap emerges.
Two bands form a conical intersection indicating that band dispersion is linear in all directions. To characterize this degenerate point, we derive an effective Hamiltonian for parameters around it (see In addition to the Weyl point constructed above, we also find Weyl points on higher bands and at different positions in the parameter space. In Fig. 2, we show all the Weyl points on the lower five bands at either , and these Weyl points will all possess the same topological charge. Note that while the total charges of Weyl points must vanish in periodic systems [47] as the reciprocal space is periodic, such a constraint does not apply here as the parameter space is not periodic. This features one of the main difference between Weyl points in synthetic dimensions and those in momentum space.

B. Reflection phases around the Weyl points
We then consider the reflection phase of a normal incident plane wave when the PC is semi-infinite.
The working frequency of the incident wave is chosen to be the frequency of the Weyl point between band 1 and band 2. Except for the 0 p q   point, the working frequency is inside the band gap for all other p and q values. Hence the reflection coefficient can be written as  being a function of p and q . In Fig. 3(a), we show the reflection phase in the whole p q  space, where the truncation boundary is at the center of the first layer. The reflection phase distribution shows a vortex structure, with the Weyl point at the vortex center. The topological charge of this vortex is given by the winding number of the phase gradient [48], which is the same as that of the Weyl point. For illustration purposes, we choose a circle centered at this Weyl point as marked by the gray dashed circle in Fig. 3(a). The reflection phase along the circular loop decreases with the polar angle  , and picks up a total change of 2  after each circling. We note that the same circle does not enclose any Weyl points between band 2 and band 3. The reflection phase at the frequency of the Weyl points between band 2 and band 3 is shown in Fig. 3(c) (the circular loop is also marked). Now the reflection phase along the loop in Fig. 3(c) covers a range much less than 2π.
The above conclusion can also be experimentally verified. We choose five PC configurations whose locations in the p q During the evaporation, the pressure in the chamber was kept below Ref. [45] and the discussions in Appendix C.

C. Fermi-arc-like interface states
The vortex structure and the associated topological charge guarantee the existence of interface states [30,31], between the PCs with Weyl points and the reflecting substrates, regardless of the substrate properties. The existence of interface states [30] is given by where PC  and S  represent the reflection phases of the PC and the reflecting substrate, space. The behavior of these interface states connecting Weyl points with opposite charges in the parameter space has the same mathematical origin as that of the Fermi arc [1,2,4,7] in Weyl semimetals. There is however a crucial difference: the Fermi arc starts and ends with Weyl points in a periodic system, while the interface states in our system can connect Weyl points to the boundary of the parameter space because the total charge of the Weyl points does not vanish inside the parameter space.
As an example, we consider the Weyl points on the fourth and fifth bands as shown in Fig. 2 There are a total of eight Weyl points: six with charge -1 and the remaining two with charge +1. As

D. "Nodal lines" in higher dimensional space
Compared with Bloch momentum space, synthetic dimensions provide a flexible way to construct topological system in higher dimensional space, which enables the study of phenomena that occur only in higher dimensional spaces [23]. As an example, we can define another parameter R as the ratio    [49]. When R passes through these transition points, Weyl points will immerse as node line and reappear. In the process, a pair of Weyl point with positive and negative charge will appear or disappear simultaneously. Although the number of Weyl points will changes, the total charge remains constant.

III. Conclusion
We have shown the existence of Weyl points in the parameter space and their topological consequences. In the specific example of dielectric superlattices, the reflection phase of the semi-infinite multilayered PC shows a vortex structure with the same topological charge as the synthetic Weyl points defined in the parameter space. The vortex structure guarantees the existence of interface states, which can be used in various systems [33][34][35][36][37][38][39][40][41][42] (See also one example given in the Supplementary Material Sec. IV). In general, interface states may or may not exist at the boundary between a 1D PC and a reflecting substrate [30]. The Weyl points here provide a deterministic scheme to construct interface states between multi-layered PCs and reflecting substrates of arbitrary reflection phases [31]. In the past, numerical simulation is the only way to predict the optical properties of photonic crystals with complex unit cells defined by many parameters and there is no easy way to connect the bulk properties to the surface properties such as reflectance and existence of interface states. Here, we see that the topological character of the nodal points in the higher dimension space defined jointly by momentum and structural parameters actually connects the properties of the bulk to those of the surface. We emphasize that the topological notions apply to all frequencies, including high frequency gaps where the PCs cannot be described by the effective medium theory.
The geometric parameters p and q are fixed for each PC in this work. However, there are techniques [50][51][52][53] that can be employed to control the geometric parameters as well as the refractive index in real time. Combined with the understanding of the topological origin of the interface states, such tunability allows for the control of the interface states which may facilitate various applications. In addition, the reflection phase vortex offers a flexible way to manipulate the electromagnetic wave such as generating vortex beams and controlling the reflection direction. This work also opens up a new direction for experimentally exploring the physics in topological theory in higher dimensions [23]. With more parameters involved, we can also construct Weyl points with a higher topological charge [12,13,54].

Appendix A: Effective Hamiltonian around the Weyl points
Here we adapt the transfer matrix method to derive the effective Hamilton around the Weyl points.
Each unit cell is composed of four layers, the transfer matrix can be written as And the wave equation can be written as where c  and c  represent the coefficients of the forward propagating and backward propagating waves inside the first layer respectively. Define the following dimensionless , , , , c c c   . Keeping only the first order, Eq.
(A6) can be written as where   are Hermitian matrixes. The matrix on the right hand side of Eq. (A9) can be decomposed as 3 3 After some simple mathematics, Eq. (A9) can be written as matrix. Hence the matrix on the left hand side of Eq. (A12) works as an effective Hamiltonian of the system. This effective Hamiltonian processes a Weyl form.
We choose the Weyl point between band 1 and band 2 as an example. Here  and N has been chosen to be large enough such that n  converges. In Fig. 7(b), we plot Berry phases on band 1 (blue) and band 2 (red) as a function of  . In the calculation, the radius of the sphere in Fig. 7 is set to be r=0.001.
We can see that the variation range of the Berry phase is 2 for the upper band (the red line) and 2  for the lower band (the blue line), which means the charge of this Weyl point is -1.

Appendix C: Reflection phases around Weyl points
In this appendix, we show how to get the reflection phase from the effective Hamiltonian obtained in Appendix A. We choose the Weyl point between the first and second bands as an example. The Hamiltonian of our system near this Weyl point is given by: As expected, the wave vector k  becomes purely imaginary. We assume that the direction of the incident wave is along the positive direction, and hence we require   Im 0 k   . The corresponding eigenstate is given by: and Now Eq. (C7) can be simplified as Then the corresponding electronic field at the boundary of the PC is: where the subscript "1y" means the electric field is along the y direction and inside the first layer, a k represents the wave vector in the first layer, and x represents the distance from the truncated plane to the starting plane of the first layer. The corresponding magnetic field is: To show the validity of this method, we compare the reflection phase obtained from Eq. (C14) to that from the full wave simulation (where we use the transfer matrix method). As the working frequency is inside the bandgap and if the number of unit cell is large enough, the reflection phase will converge. In the full wave simulation, we have ensured that the number of unit cells is large enough. In Fig. 8, red circles and blue solid line represent the reflection phases obtained with Eq.  Also changing the truncation position or the working frequency just shift the reflection phases and do not change the conclusion above. As an example, we change the truncating position and now the PCs are truncated at the starting plane of the first layer. The corresponding reflection phases in the p q  space are shown in Fig. 9(a), where the parameters used are the same as those in Fig. 3(a)and the working frequency is also fixed at the frequency of the Weyl point. Compared with Fig. 3(a), the reflection phase in Fig. 9(a) changes locally but the charge of the vortex is still preserved. If the truncating position of the PC is inside other layers, we can obtain the same conclusion following a similar proof as the one given above.
We now analyze Eq. (C14) in more detail. When f r    , we have c   , which is independent of the value of r , so the reflection phase maintains a constant for a fixed  ; while for f r   , we can also get the same result, but with c  0 instead. Now we consider the dispersion of the interface states between PCs and a mirror with reflection phase of 0 in the parameter space. As shown in Fig. 9(b), the interface states form a surface which rotates 180°around the Weyl point as frequency increasing. Such feature is discussed before [55], however as effective Hamiltonians are used in Fig. 9(b), the intersection between the interface states and the bulk states is a straight line here.
In the above derivation, we use the transfer matrix method which is specialized to our systems.
However, the existence of the reflection phase vortex pinning at the Weyl point is not specialized to our systems. To see this point, let us focus on the band gap region. In general, the surface property of a truncated bulk is characterized by the impedance matrix [56]. However, if the surface property is dominated by one plane wave component and then the surface property can be approximately described by a scalar surface impedance or equivalently the reflection phase [57]. Let us then assume that inside the boundary region between a system with Weyl points and a reflecting substrate, dashed lines). Here the working frequency is at around 303THz. We neglect the dispersion of the silver film as the frequency range of interest is narrow. The thickness of the two silver films in this measurement are estimated to be around 20nm. Fig. S1(a) shows the measured reflection spectrum. The distance between these two silver films is obtained through fitting with Eq. (2). Substitute into Eq. (1), we now obtained the reflection phase of the silver films as shown in Fig. S1(b), which is about ( 0.95 0.0471)   around 303THz. Figure S1 (a), the reflection spectrum of the Fabry-Perot (FP) cavity composed of two silver films, and the corresponding experimental setup is shown in the inset. The reflection spectrum is measured above the top glass.
(b), the deduced reflection phase of silver film from the reflection spectrum in (a).
The measurement of the reflection phase of the PCs is similar to the above discussion.
The setup is shown in Fig. S2(a). Half of the PCs are deposited with silver films with thickness 200 10 nm nm  . This silver film is thick enough to reflect all the incident waves with reflection phase  . The silver film above the air gap is designed to be 7nm such that the incident wave can partially pass through it. The first measurement is performed inside the region where the FP cavity is composed of the two silver films, through which we can obtain the distance between these two silver films and the reflection phase of the silver film above the air gap. The reflection phase of the silver films above the air gap are measured to be ( 0.  . The average value of  in our experiment is 0.72% and the mean square difference is 1.3% . After subtracting the effect of this small global shift, we are now able to compare the reflection phases and the frequencies of the interface states with numerical simulation. As an example, we show reflection phases from experimental measurements (red diamonds) and from numerical simulation (blue curve) of one PC in Fig. S3(d). For this PC, the experimental measurement agrees well with the numerical simulation.

Section II: Interface State Measurement
The interface states are at the boundary between the PCs and a silver films. In these measurements, we deposit 20nm-thick silver films on the top of the PCs. And the PCs are truncated at the half plane of the first layer. A sketch of the experimental setup is shown in Fig. S3(a), where the gray, blue and red layers represent silver, HfO2 and SiO2, respectively. The reflection spectrum is measured on the top of the silver layer. An example of the reflection spectrum is shown in Fig. S3(b) with the blue line. Here the gray vertical strips represent the bulk band regions. As the interface state enhances the absorption, the interface state here corresponds to the reflection dip inside the gap region as indicated by the red triangles in Fig. S3(b). Although we have only shown the 'Fermi arc like interface states' between PCs and silver films at the frequency of the Weyl points with charge +1 in the text, the trajectory of interface states actually extend to all the frequencies in the gap region. In Fig. S4(a) and (b), we show respectively, the reflection phases in the p-q space with the frequency of the Weyl point with charge -1 and a frequency above those of all the Weyl points.
The vortex structures can be seen in both frequencies and they guarantee the existence of interface states. The interface states form a helicoid surface and such behavior has also been observed in Weyl semimetals [3].
In the main text, we have proved that the interface states always exist independent of the properties of the reflecting substrate. However, the trajectory of the interface states must connect one Weyl point to another Weyl point of an opposite charge, but which Weyl point it connects to actually depends on the property of the reflecting substrate (i.e. how the boundary is set up). As an example, we consider two reflecting substrates with reflection phases of 0 and  , respectively. Black lines in Fig. S4(a) represent the interface states when the reflection phase of the reflecting substrate is 0; while white lines in Fig. S4(b) represent the interface states when the reflection phase of the mirror is  . Indeed, the interface states connect different Weyl points as the reflection phase of the reflecting substrate changes.

Section IV: Absorption of interface states
The optical interface states are widely used in various systems [4] , as its strong confinement of light will enhance the interaction of light and matters. The synthetic Weyl points proposed here give a flexible way to construct optical interface states between PCs and arbitrary reflecting substrates. As an example, we consider a system consisting of a PC and a 15nm-thick silver film. This system exhibits a interface states between the PC and the silver film. The field intensity of this system is shown in Fig.   S5 (from Palik database [5]). We can see that the intensity is greatly enhanced at the frequency of the cavity mode (f=247.6 THz), which falls inside the band gap region whose edge frequencies are labeled by the black dashed lines in Fig. S5(a). As the field intensities are greatly enhanced at the cavity mode resonances, interface states behavior as absorption peaks in the absorption spectra. Now we consider the same PCs-Ag film configurations but with different p and q values, and the absorption in the p-q space is shown in Fig. S5(b). Here the working frequency is fixed at the frequency of the synthetic Weyl point. The trajectory of the interface states at this fixed frequency is also shown in Fig. S5(b) with the black triangles. We can see the maximum of absorption can reach 95%, and the trajectory of the absorption peaks coincide with that of the interface states. There is a small derivation near the synthetic Weyl point between those two trajectories due to the finite size of the PCs used in the simulations (The gaps vanishes at the Weyl point). The p and q values of the PC used in Fig. S5(a) are also marked by the white triangle in Fig. S5(b) for reference.