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Abstract
Weyl points, as linearly double degenerated point of band structures, have been extensively researched in electronic and classical wave systems. However, Weyl points’ realization is always accompanied with delicate “lattice structures”. In this work, frequency-tunable terahertz (THz) generalized Weyl points inside the parameter space have been investigated and displayed by a specially designed photonic crystal with polydimethylsiloxane (PDMS) immersed in 4-cyano’-pentylbipenyl (5CB) liquid crystals (LCs). The reflective phase vortices as a signature of the generalized Weyl points are observed through our numerically simulations. Besides, interface states between photonic crystals and any reflective substrates are fulfilled too. Meanwhile, we could also change the orientation of LC molecule by the external magnetic field so as to tune the frequency of the first two bands’ Weyl point from 0.27698THz to 0.30013THz. This band lies in the short-range wireless communication. Thus, our proposal may be beneficial to the investigation and application of Weyl points’ properties and strongly localized states.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In 1929, one year after Dirac equation [1] born, Hermann Weyl derived the famous Weyl equation [2], which is a two-component relativistic wave equation for describing the behaviors of the massless particle Weyl Fermions [2,3]. In the past eighty years, a lot of scientists had been working on finding this kind of fundamental particle in nature. However, the goal was never achieved. It is soul-stirring that researchers found some particles in condensed-matter physics, near the linearly degenerated points of two touch bands, whose low-energy excitation could satisfy the Weyl equation [4–14]. These degenerated points are called “Weyl point”. Weyl points are frequently perceived as magnetic monopoles of quantized Berry flux in the 3D momentum space, i.e. sources or sinks. They also have chirality which can be described by the unit Chern number. So, it requires that Weyl points must be present in pairs to ensure the neutrality of Brillouin zone (BZ) [15]. Unlike 2-D Dirac point, Weyl points also possess robustness, and the perturbation could just shift their position without destroying the degeneracy [4]. Therefore, they can only be removed through pair annihilation. At the same time, the appearance of Weyl points is companied with many fantastic characteristics, such as topological protected surface states—Fermi arc [4], and Adler-Bell-Jackiw chiral anomaly [16–18]. Hence, it has attracted tremendous attentions for investigating various intriguing phenomenon, and a lot of remarkable achievements have been obtained, such as observing the Weyl points in photonic [19–29], acoustic [30–34], and plasmonic [35] systems. As we know, if we want to obtain Weyl points in momentum space, we must break either inversion or time-reversal symmetry (or both) belonging to systems [19]. This makes the systems for realizing the Weyl points have to possess well-designed three-dimensional geometries to break inversion symmetry or magnetic materials to break time-reversal symmetry under external magnetic field. Therefore, it raises the difficulty level of exploring Weyl points physics. However, to our excitement, recently, some works about the topological singular points in the synthetic dimensions rather than in the momentum space have been finished [36–38]. The synthetic dimensions own the abilities to realize physics in higher dimensions and simplify experimental designs [36,39–44]. On the other hand, the possible control over the synthetic dimensions make it easier for the experimental verification on the topological characteristics of the topological singular points [36,37]. At present, by utilizing the concept of synthetic dimensions, researchers have experimentally realized the generalized Weyl points with one-dimensional photonic crystals (PCs) [44] and theoretically discussed in two-dimensional resonator lattice [43]. In the former, researcher used two independent geometric parameters (which form a parameter space) to replace two wave vector components in the three-component Weyl Hamiltonian. This way preserves the properties of Weyl points, such as bulk-edge correspondence relation between the edge states and Weyl points, and also greatly reduce the difficulty of investigating those properties in the optical frequency regime.
Here, we propose a one-dimensional PCs composed of polydimethylsiloxane (PDMS) and 4-cyano’-pentylbipenyl (5CB) liquid crystals (LCs) for generating the tunable Weyl points in terahertz (THz) range. By designing the appropriate thickness of each layer in the designed unit cell, frequency of the first two bands’ Weyl point falls in the interval from 0.2THz to 0.3THz, which is useful for short range wireless communications. The numerically results show that the reflective phase of a PC exhibiting vortex structures in the parameter space around the generalized Weyl point. And the reflective phase centered on generalized Weyl points covers completely, so as to guarantee the realization of the interface states. Besides, our designed 1-D PCs have the characteristic of fine frequency tunability of synthetic Weyl points due to the introduction of 5CB LCs. Here, by changing the direction of external magnetic from 0° to 90°, we numerically realize the tunable frequency of the first generalized Weyl point from 0.27698THz to 0.30013THz.
2. Generalized Weyl points in synthetic space
As we know, the Hamiltonian in the vicinity of Weyl points can be generally written as [14]
where k is the wave vector in reciprocal space, A is a 3x4 matrix of coefficients, are the Pauli matrices describing self-spin and σ0 is the 2x2 identity matrix. The combination of σj and σ0 form a complete basis for 2x2 Hermitian matrices [25]. In general, Weyl points are defined in the 3-D momentum space. However, here we realize the Weyl points in a synthetic space. These generalized Weyl points still preserve the standard characteristics of the Weyl point, such as associated topological charges and robustness against variations in the parameters.As shown in the inset in Fig. 1(a), the unit cell of the proposed structure for forming the Weyl point, is composed of two layers of PDMS and two layers of 5CB LCs. In our simulation, the first and third layers (cyan) are made of 5CB LCs with changeable refractive index of na, and the second and forth layers (gray) are made of PDMS with refractive index of in the designing wavelength (0.1~1THz) range. The thickness of each layer has been given in Fig. 1(a). At first, we use finite element method software COMSOL Multiphysics to calculate the band structures [24] of unit cell with two layers (a layer of 5CB LCs with thickness and a layer of PDMS with thickness ), in which the eigenfrequency solver is used. For calculating the band structure, Floquet periodic boundary condition is imposed on both sides along the periodic direction of the unit cell, and Continuity periodic boundary condition is imposed on the periphery to represent infinity. By the means of scanning the Bloch wavevector in first BZ, we acquire the band dispersions in momentum space. The band dispersion is plotted in Fig. 1(b) with red dash line. Then, we set the parameters as and for four-layer PC, which doubles the length of each unit cell, so as to fold the 1-D BZ. The dispersion of this four-layer PC is shown in Fig. 1(b) with blue solid line. It is obvious that the folding manipulation of the artificial band gives a linear crossing along the Bloch vector direction, which is the precondition of the existing Weyl points in synthetic space. After that, we perform the numerical simulation of band structures in p-q space. During calculating, the Floquet wavevector is set as where Through scanning the geometrical parameters p and q, we obtain the structure of first two bands, as shown in Fig. 1(c). It can be seen that two bands form a conical intersection near the point of At the crossing point, the frequency is 0.29179THz. Combined with the intersection at the band dispersion is linear in all directions. Therefore, we call this node the generalized Weyl point. In order to confirm our results, according to [44], we calculate the effective Hamiltonian around this degenerate point as follow
Fig. 1 Realization of Weyl points in a synthetic space. (a) Photonic crystals (PCs) with different p, q values. p and q form a parameter space, which determines the geometric structure of PCs. The inset shows one unit cell of the PC, where the first and the third layers are made of 5CB LCs (cyan), and the second and the forth layers are made of PDMS (gray). The thickness of each layer is related to its position in the p-q parameter space. (b) The band dispersion of PCs with two layers in one unit cell (red dash line) and four layers in one unit cell (blue solid line). Crossing points appear inside four layers’ dispersion. Here, and (c) The dispersion of PCs in the p-q space with and Here, two bands form a conical intersection. Panels (b) and (c) together show that the band dispersions are linear in all directions around the degenerated point in synthetic space, so we call it generalized Weyl point. (d) The equal frequency contours around generalized Weyl point in p-q space and its charge “-1”.
where and This effective Hamiltonian is the standard Weyl form. And the generalized Weyl point located at carries a “charge” of −1, as shown in Fig. 1(d), according to the usual definition [4]. Because the p-q parameter space is not periodic for our 1-D photonic crystals, it is not applicative that the total charges of Weyl points must vanish in periodic systems. All the above results are calculated without external magnetic field, so
3. Tunability of generalized Weyl points
Different from the previous works, in this paper, we add an extra degree freedom to realize the tunability of band dispersion of PCs. Accordingly, the frequency of generalized Weyl points can be changed. Up to now, a lot of efforts to frequency-tunability have been employed and investigated, such as tuning mechanical geometry [45] and using phase change materials [46,47]. Here, specifically, we use PDMS and 5CB LCs to replace HfO2 and SiO2 used in [44]. The PDMS has good adaptability in terahertz, and the permittivity of 5CB LCs is tunable under external magnetic field. Therefore, we put the slabs made of PDMS into the container filled with 5CB LCs, as shown in Fig. 2(a). The container is made of photosensitive resin.
Fig. 2 (a) The schematic illustration of the terahertz wave’s propagation and polarization, where the electric field is polarized along the z-direction(blue) and the magnetic field is along the y-direction(green). The plane wave propagates along x-direction (red). The LCs are randomly distributed inside the box. (b) The schematic illustration of measuring the reflective phase around the generalized Weyl points under the external magnetic field.
It is well-known that the nematic LCs exhibit strong birefringence—ordinary refractive index (refractive index along short axis) no and extraordinary refractive index (refractive index along long axis) ne, thus providing a mechanism for dynamically tuning the electromagnetic properties of the PCs. At the first stage without magnetic field, nematic LC molecules are randomly distributed inside the box (shown in the dotted box in Fig. 2(a)), leading to an average refractive index nave as follows [48]:
The effective permittivity is expected to continuously vary from the permittivity along the short axis [49] (), to the permittivity along the long axis [49] (), with the evolution of orientation of magnetic field. The average effective permittivity, calculated from Eq. (3), is between the permittivity along long and short axes of LC molecules. Accordingly, the ordinary refractive index, average refractive index and extraordinary refractive index are 1.60, 1.66 and 1.76 respectively. Figure 2(b) is the schematic illustration of our tunable system. When magnetic field is parallel to z-axis, 5CB LCs exhibit extraordinary refractive index. In the contrast, when magnetic field is parallel to y-axis, 5CB LCs exhibit ordinary refractive index.
We numerically calculate the band dispersion of 1-D PCs under three different cases. The results are shown in Fig. 3. In Figs. 3(a) and 3(b), we can see that frequency of degenerate point is blue shift when orientation of external magnetic field change from along the long axis to short axis. The optical length L of unit cell will decrease, if we reduce the refractive index of compositional material. So, it is easy to understand why the phenomenon happens. When degenerate point in k space moves, correspondingly, the degenerate point in p-q space will move too, as shown in Figs. 3(c)-3(e) respectively.
Fig. 3 The position of generalized Weyl points of band-1 and band-2 when change the direction of external magnetic field. Panel (a) and (b) show three kinds of band dispersion. The blue, red and green lines represent external magnetic along long axis, short axis and no external field respectively. (c), (d) and (e) The dispersion of PCs in p-q parameter space with In three cases, two bands all form conical intersection. (c) External field along y- axis. (d) No external field. (e) External field along z-axis.
4. Reflective vortex phase around generalized Weyl points
We now consider the reflective phase of a normal incident TE wave (as depicted in Fig. 2(a)), where the PC is semi-infinite. The working frequency of TE wave is chosen to be the frequency of the generalized Weyl point in first two bands. From the band structure in p-q space, we can get that the frequency is inside the band gap, except for Hence, the reflection coefficient can be theoretically written as with φ as a function of p and q. In Fig. 4, we exhibit the reflection coefficient and reflective phase in the whole p-q space, which includes three different cases of external magnetic field. The boundary is at the surface of the first layer.
Fig. 4 The reflection coefficient and phase around generalized Weyl points of first two bands in p-q parameter space. (a), (d) The external magnetic field along y-axis. (b), (e) No external magnetic field. (c), (f) The external magnetic field along z-axis. Here, the reflection coefficient at is minimal because of the intersection of two bands at the generalized Weyl point. Reflective phase exhibits vortex structure, and continuously change from π to -π with the increase of.
When doing the numerical simulation, we set the number of unit cell to be 20 for the better reflection, as well as saving computed costing, and For different p and q values, we can find that the width of band gaps is distinct in Fig. 3. But, all band gaps are small relative to the working frequency. This is why the reflection coefficient just reaches to 1 at the edge of p-q space, as shown in Figs. 4(a)-4(c). The reflective phase distribution shows a vortex structure, with the Weyl point at the vortex center. The topological charge of this vortex is given by winding number of the phase gradient [50], which is the same as that of the Weyl point. When we change the direction of magnetic field, the vortex structure remains unchanged, as shown in Figs. 4(d)-4(f).
Besides, we also do a study about change of reflective phase with the increase of working frequency. The reflective phase inside band gaps of two-layers PCs covers π [51,52]. Here, for our designed four-layers PCs, the reflective phase can cover 2π inside band gaps, if p and q values are centered on generalized Weyl point, as shown in Fig. 4(e) (dashed black circle). To illustrate this, we chose four pairs of p and q values in numerical simulations, such as (p, q) = (0.75, 0), (0, 0.75), (−0.75, 0) and (0, −0.75), which have been shown in Fig. 5 and signed with pentagram, triangle, diamond and square respectively. We could observe that the reflective phase of these four positions can cover 2π completely. In simulations, we set the number of unit cells to 40 so as to obtain an apparent phenomenon.
Fig. 5 The reflective phases of different position centered on generalized Weyl point in parameter space. Here, (p, q) = (0.75, 0), (0, 0.75), (−0.75, 0) and (0, −0.75) are signed with pentagram, triangle, diamond and square in Fig. 4(e). The reflective phase of these positions covers 2π inside band gap. Gray strips on both sides denote bulk bands.
5. Interface states protected by generalized Weyl points
The condition of the existence of interface states [52] can be given by
where and represent the reflective phases of the PC and the reflection substrate, respectively. From the above results, we have shown that the reflective phases centered on generalized Weyl point cover 2π completely from -π to π. So, Eq. (4) can be always satisfied, no matter what the reflective phase of reflection substrate is.As an example, we use polyimide as reflection substrate in our calculations. In order to acquire an obvious interface state, we set the thickness of polyimide layer as 2d. In numerical simulation, we set and and set the number of unit cell as 40. Of course, we can change the value of p casually, due to the existence of LCs. Here, in 0.2THz~0.3THz, the dielectric constant of the polyimide is and its loss angle tangent is [53]. And the polyimide is deposited on the surface of first layer of PC. The simulation result is shown in Fig. 6. At first, we calculate the reflective phases of the PC and polyimide solely. Then, we invert the reflective phase of polyimide, as shown in Fig. 4(a). It is evident that the intersection (green arrow in Fig. 4(a)) comes up in phase curves, meaning two reflective phases opposite. Therefore, their sum is zero, which guarantees the existence of interface state. As we know, the existence of interface states makes Electromagnetic energy be localized at the interface. Hence, it will produce a reflection deep in our system, as exhibited in Fig. 6(b), and the green arrow still indicates the position of interface state. For the purpose of showing the local effect of the interface state more intuitively, we calculate the electric field distribution of the whole structure, as displayed in Fig. 6(c). The Electric field is localized near the interface coincide with our prediction. The Fig. 6(d) shows the structure of numerical simulation. According to above results, the reflective phases of PC always exist a value satisfying the Eq. (4), so as to guarantee the existence of interface states, regardless of the properties of substrate.
Fig. 6 The interface state between the designed PC with generalized Weyl points and polyimide substrate. (a) The reflective phase of PC (blue line) and polyimide substrate (red line). (b) The reflection efficiency with (red line) and without (blue line) polyimide layer on the surface of PC. (c) The distribution of Electric field along the PC. In the interface (indicated by red dashed line), Electric field is the strongest. (d) The structure to measure the interface states. Red dashed line is interface between PC (right side) and substrate (left side, blue).
6. Summary
In this work, through rational design and numerical simulations, we realize and identify the existence of the generalized Weyl points in 0.27698THz ~0.30018THz, which are in the atmospheric window for THz wave and have potential advantages in THz communications. The reflective phases of the semi-infinite multilayered PC show a vortex structure centered on generalized Weyl points covering from –π to π completely, guaranteeing the existence of interface states between our designed PCs and reflecting substrates owning arbitrary reflective phases. Meanwhile, by introducing 5CB LCs into the structure, we endow the generated generalized Weyl points with the tunability for the designed PC in real time, under the action of external magnetic field. Such tunability allows for the control of interface states, which may facilitate various applications, such as strong local states. On the other hand, the manner of introducing materials with variable properties also can be applied to two-dimensional and three-dimensional photonic crystals. This is beneficial for opening up a new way for the investigation and application of Weyl points in terahertz.
Funding
National Natural Science Foundation of China (61775050, 11505043, 11804073), and Fundamental Research Funds for the Central Universities (JD2017JGPY0005, JZ2018HGBZ0309 and JZ2018HGTB0240).
References
1. P. A. M. Dirac, “The quantum theory of the electron,” Proc. R. Soc. Lond. A Math. Phys. Sci. 117(778), 610–624 (1928). [CrossRef]
2. H. Weyl, “Elektron und Gravitation. I,” Z. Phys. 56(5), 330–352 (1929). [CrossRef]
3. X. G. Wan, “Topological Weyl semimetals,” Phsics 44(7), 427–439 (2015).
4. X. G. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, “Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates,” Phys. Rev. B Condens. Matter Mater. Phys. 83(20), 205101 (2011). [CrossRef]
5. A. A. Burkov and L. Balents, “Weyl semimetal in a topological insulator multilayer,” Phys. Rev. Lett. 107(12), 127205 (2011). [CrossRef] [PubMed]
6. G. B. Halász and L. Balents, “Time-reversal invariant realization of the Weyl semimetal phase,” Phys. Rev. B Condens. Matter Mater. Phys. 85(3), 035103 (2012). [CrossRef]
7. A. A. Zyuzin, S. Wu, and A. A. Burkov, “Weyl semimetal with broken time reversal and inversion symmetries,” Phys. Rev. B Condens. Matter Mater. Phys. 85(16), 165110 (2012). [CrossRef]
8. J. Liu and D. Vanderbilt, “Weyl semimetals from noncentrosymmetric topological insulators,” Phys. Rev. B Condens. Matter Mater. Phys. 90(15), 155316 (2014). [CrossRef]
9. D. Bulmash, C. X. Liu, and X. L. Qi, “Prediction of a Weyl semimetal in Hg1−x−yCdxMnyTe,” Phys. Rev. B Condens. Matter Mater. Phys. 89(8), 081106 (2014). [CrossRef]
10. M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, and T. Miyake, “Weyl node and spin texture in trigonal tellurium and selenium,” Phys. Rev. Lett. 114(20), 206401 (2015). [CrossRef] [PubMed]
11. B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental discovery of Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031013 (2015). [CrossRef]
12. S. Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C. C. Lee, S. M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349(6248), 613–617 (2015). [CrossRef] [PubMed]
13. B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V. N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nat. Phys. 11(9), 724–727 (2015). [CrossRef]
14. A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-II Weyl semimetals,” Nature 527(7579), 495–498 (2015). [CrossRef] [PubMed]
15. B. Nielsen and M. Ninomiya, “Absence of neutrinos on a lattice,” Nucl. Phys. B 193(1), 173–194 (1981). [CrossRef]
16. H. B. Nielsen and M. Ninomiya, “The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal,” Phys. Lett. B 130(6), 389–396 (1983). [CrossRef]
17. X. C. Huang, L. X. Zhao, Y. J. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, “Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs,” Phys. Rev. X 5(3), 031023 (2015). [CrossRef]
18. C. L. Zhang, S. Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, G. Bian, N. Alidoust, C. C. Lee, S. M. Huang, T. R. Chang, G. Chang, C. H. Hsu, H. T. Jeng, M. Neupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin, C. Zhang, H. Z. Lu, S. Q. Shen, T. Neupert, M. Zahid Hasan, and S. Jia, “Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal,” Nat. Commun. 7(1), 10735 (2016). [CrossRef] [PubMed]
19. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7(4), 294–299 (2013). [CrossRef]
20. J. Bravo-Abad, L. Lu, L. Fu, H. Buljan, and M. Soljačić, “Weyl POINTS IN PHOTONIC-CRYSTAL SUPerlattices,” 2D Mater. 2(3), 034013 (2015). [CrossRef]
21. L. Y. Wang, S. K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” Phys. Rev. A (Coll. Park) 93(6), 061801 (2016). [CrossRef] [PubMed]
22. M. Xiao, Q. Lin, and S. Fan, “Hyperbolic Weyl point in reciprocal chiral metamaterials,” Phys. Rev. Lett. 117(5), 057401 (2016). [CrossRef] [PubMed]
23. W. J. Chen, M. Xiao, and C. T. Chan, “Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states,” Nat. Commun. 7(1), 13038 (2016). [CrossRef] [PubMed]
24. M. L. Chang, M. Xiao, W. J. Chen, and C. T. Chan, “Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals,” Phys. Rev. B 95(12), 125136 (2017). [CrossRef]
25. J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M. C. Rechtsman, “Experimental observation of optical Weyl points and fermi arc-like surface states,” Nat. Phys. 13(6), 611–617 (2017). [CrossRef]
26. Z. Yang, M. Xiao, F. Gao, L. Lu, Y. Chong, and B. Zhang, “Weyl points in a magnetic tetrahedral photonic crystal,” Opt. Express 25(14), 15772–15777 (2017). [CrossRef] [PubMed]
27. M. Zhou, L. Ying, L. Lu, L. Shi, J. Zi, and Z. Yu, “Electromagnetic scattering laws in Weyl systems,” Nat. Commun. 8(1), 1388 (2017). [CrossRef] [PubMed]
28. B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y. Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation of topological surface-state arcs in photonic metamaterials,” Nat. Commun. 8(1), 97 (2017). [CrossRef] [PubMed]
29. B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “Ideal Weyl points and helicoid surface states in artificial photonic crystal structures,” Science 359(6379), 1013–1016 (2018). [CrossRef] [PubMed]
30. M. Xiao, W. J. Chen, W. Y. He, and C. T. Chan, “Synthetic gauge flux and Weyl points in acoustic systems,” Nat. Phys. 11(11), 920–924 (2015). [CrossRef]
31. Z. Yang and B. Zhang, “Acoustic Type-II Weyl nodes from stacking dimerized chains,” Phys. Rev. Lett. 117(22), 224301 (2016). [CrossRef] [PubMed]
32. F. Li, X. Q. Huang, J. Y. Lu, J. H. Ma, and Z. Y. Liu, “Weyl points and Fermi arcs in a chiral phononic crystal,” Nat. Phys. 14(1), 30–34 (2017). [CrossRef]
33. H. Ge, X. Ni, Y. Tian, S. K. Gupta, M. H. Lu, X. Lin, W. D. Huang, C. T. Chan, and Y. F. Chen, “Experimental observation of acoustic Weyl points and topological surface states,” Phys. Rev. Appl. 10(1), 014017 (2018). [CrossRef]
34. H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu, “Topological negative refraction of surface acoustic waves in a Weyl phononic crystal,” Nature 560(7716), 61–64 (2018). [CrossRef] [PubMed]
35. W. Gao, B. Yang, M. Lawrence, F. Fang, B. Béri, and S. Zhang, “Photonic Weyl degeneracies in magnetized plasma,” Nat. Commun. 7(1), 12435 (2016). [CrossRef] [PubMed]
36. P. Roushan, C. Neill, Y. Chen, M. Kolodrubetz, C. Quintana, N. Leung, M. Fang, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, J. Mutus, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. White, A. Polkovnikov, A. N. Cleland, and J. M. Martinis, “Observation of topological transitions in interacting quantum circuits,” Nature 515(7526), 241–244 (2014). [CrossRef] [PubMed]
37. M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M. Sandberg, J. Gao, M. R. Vissers, D. P. Pappas, A. Polkovnikov, and K. W. Lehnert, “Measuring a topological transition in an artificial spin-1/2 system,” Phys. Rev. Lett. 113(5), 050402 (2014). [CrossRef] [PubMed]
38. R. P. Riwar, M. Houzet, J. S. Meyer, and Y. V. Nazarov, “Multi-terminal Josephson junctions as topological matter,” Nat. Commun. 7(1), 11167 (2016). [CrossRef] [PubMed]
39. O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, “Quantum simulation of an extra dimension,” Phys. Rev. Lett. 108(13), 133001 (2012). [CrossRef] [PubMed]
40. L. Yuan, Y. Shi, and S. Fan, “Photonic gauge potential in a system with a synthetic frequency dimension,” Opt. Lett. 41(4), 741–744 (2016). [CrossRef] [PubMed]
41. B. Lian and S. C. Zhang, “Five-dimensional generalization of the topological Weyl semimetal,” Phys. Rev. B 94(4), 041105 (2016). [CrossRef]
42. F. Mei, Z. Y. Xue, D. W. Zhang, L. Tian, C. Lee, and S. L. Zhu, “Witnessing topological Weyl semimetal phase in a minimal circuit-qed lattice,” Quantum Sci. Technol. 1(1), 015006 (2016). [CrossRef]
43. Q. Lin, M. Xiao, L. Yuan, and S. Fan, “Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension,” Nat. Commun. 7(1), 13731 (2016). [CrossRef] [PubMed]
44. Q. Wang, M. Xiao, H. Liu, S. N. Zhu, and C. T. Chan, “Optical interface states protected by synthetic weyl points,” Phys. Rev. X 7(3), 031032 (2017). [CrossRef]
45. P. Zhu, W. Yang, R. Wang, S. Gao, B. Li, and Q. Li, “Direct writing of flexible barium titanate/polydimethylsiloxane 3D photonic crystals with mechanically tunable terahertz properties,” Adv. Opt. Mater. 5(7), 1600977 (2017). [CrossRef]
46. Y. Ke, I. Balin, N. Wang, Q. Lu, A. I. Tok, T. J. White, S. Magdassi, I. Abdulhalim, and Y. Long, “Two-dimensional SiO2/VO2 photonic crystals with statically visible and dynamically infrared modulated for smart window deployment,” ACS Appl. Mater. Interfaces 8(48), 33112–33120 (2016). [CrossRef] [PubMed]
47. Z. Guo, X. Yang, F. Shen, Q. Zhou, J. Gao, and K. Guo, “Active-tuning and polarization-independent absorber and sensor in the infrared region based on the phase change material of Ge2Sb2Te5 (GST),” Sci. Rep. 8(1), 12433 (2018). [CrossRef] [PubMed]
48. K. C. Lim, J. D. Margerum, and A. M. Lackner, “Liquid crystal millimeter wave electronic phase shifter,” Appl. Phys. Lett. 62(10), 1065–1067 (1993). [CrossRef]
49. T. R. Tsai, C. Y. Chen, C. L. Pan, R. P. Pan, and X. C. Zhang, “Terahertz time-domain spectroscopy studies of the optical constants of the nematic liquid crystal 5CB,” Appl. Opt. 42(13), 2372–2376 (2003). [CrossRef] [PubMed]
50. N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51(3), 591–648 (1979). [CrossRef]
51. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef] [PubMed]
52. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014). [CrossRef]
53. T. B. Zhou, F. R. Hu, J. Xiao, L. H. Zhang, F. Liu, T. Chen, J. H. Niu, and X. M. Xiong, “Design of a polarization-insensitive and broadband terahertz absorber using metamaterials,” Wuli Xuebao 63(17), 178103 (2014).
