Atomic Design of Three-Dimensional Photonic $Z_2$ Dirac and Weyl Points
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We present a study on the existence of synthetic Weyl points with various planar symmetries in phononic crystals. We design a set of phononic crystals to display Weyl points at high-symmetry points with different symmetries in both square and honeycomb lattices. We demonstrate that, in a square lattice, double Weyl points are protected by C4 rotation symmetry, while in a honeycomb lattice, they are protected by C3 rotation symmetry. Additionally, we investigated the effects of symmetry on double Weyl points. The results indicate that double Weyl points would split into two single Weyl points along high-symmetry lines if we break the corresponding symmetries. The distributions of Weyl points in various symmetries are presented systematically. Finally, gapless surface states and the robust one-way acoustic transport in a square lattice are demonstrated in a double Weyl system.
Square lattice
Gapless playback
Lattice (music)
Weyl transformation
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Three-dimensional (3D) Dirac and Weyl semimetals are novel states of quantum matter. We classify stable 3D Dirac and Weyl semimetals with reflection and rotational symmetry in the presence of time reversal symmetry and spin-orbit coupling, which belong to seventeen different point groups. They have two classes of reflection symmetry, with the mirror plane parallel and perpendicular to rotation axis. In both cases two types of Dirac points, existing through accidental band crossing (ABC) or at a time reversal invariant momentum (TBC), are determined by four different reflection symmetries. We classify those two types of Dirac points with a combination of different reflection and rotational symmetries. We further classify Dirac and Weyl line nodes to show in which types of mirror plane they can exist. Finally we discuss that Weyl line nodes and Dirac points can exist at the same time taking ${\mathrm{C}}_{4\mathrm{v}}$ symmetry as an example.
Point reflection
Weyl semimetal
Reflection symmetry
Brillouin zone
Mirror symmetry
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Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by 2\pi/3 and inversion, rotation by 2\pi/3 and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by 2\pi/3 and vertical reflection. All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by 2\pi/3 . Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.
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We show that acoustic double Weyl points are observed in a square lattice by breaking inversion symmetry. We also show that the double Weyl points are protected by the ${C}_{4}$ rotation symmetry but are unaffected by translation symmetry along the $z$ direction. When ${C}_{4}$ rotation symmetry is broken, the double Weyl point will split into two single Weyl points in the x-y plane. Gapless surface states and backscattering immune properties are demonstrated in double and single Weyl systems. The topologically protected one-way propagation of sound waves is demonstrated experimentally. The acoustic Weyl points obtained in the easily fabricated square lattice structure will provide a platform to study the topological properties and lead to potential applications in acoustic devices.
Square lattice
Gapless playback
Point reflection
Lattice (music)
Weyl transformation
Translational symmetry
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An infinite set of conformally invariant solutions of the two-dimensional quantum field theory, possessing a global symmetry Zn is constructed. These solutions can describe the critical behavior of Zn symmetric statistical systems.
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In time-reversal-breaking centrosymmetric systems, the appearance of Weyl points can be guaranteed by an odd number of even/odd-parity occupied bands at eight inversion-symmetry-invariant momenta. Here, based on symmetry analysis and first-principles calculations, we demonstrate that for time-reversal-invariant systems with ${S}_{4}$ symmetry, the Weyl semimetal phase can be characterized by the inequality between a well-defined invariant $\ensuremath{\eta}$ and an ${S}_{4}$ indicator ${z}_{2}$. By applying this criterion, we find that some candidates, previously predicted to be topological insulators, are actually Weyl semimetals in the noncentrosymmetric space group with ${S}_{4}$ symmetry. Our first-principles calculations show that four pairs of Weyl points are located in the ${k}_{x,y}=0$ planes, with each plane containing four same-chirality Weyl points. An effective model has been built and captures the nontrivial topology in these materials. Our strategy to find the Weyl points by using symmetry indicators and invariants opens a new route to search for Weyl semimetals in time-reversal-invariant systems.
Weyl semimetal
Point reflection
Parity (physics)
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Weyl semimetal has the massless and chiral low-energy electronic excitation charateristic, and its quasi-particle behavior can be described by Weyl equation, and may lead to appealing transport properties, such as Fermi arc surface state, negative magnetic resistance, chiral Landau level, etc. By analogous with Weyl semimetal, one has realized Weyl point degeneracy of electromagnetic wave in an ideal Weyl metamaterial. In this article, by breaking the mirror symmetry of the saddle-shaped meta-atom structure, we theoretically investigate chirality-dependent split and shift effect of Weyl point frequencies which would otherwise be identical. The frequency shift can be tuned by the symmetry-broken intensity. Finally, we study the Fermi arc surface state connecting two Weyl points on <inline-formula><tex-math id="Z-20200717194227">\begin{document}$\left\langle {001} \right\rangle $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200195_Z-20200717194227.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200195_Z-20200717194227.png"/></alternatives></inline-formula> crystal surface.
Weyl semimetal
Saddle point
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We present a spectral rigidity result for the Dirac operator on lens spaces. More specifically, we show that each homogeneous lens space and each three dimensional lens space $L(q;p)$ with $q$ prime is completely characterized by its Dirac spectrum in the class of all lens spaces.
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