C6光子晶體之雙狄拉克錐存在性與其拓樸邊界態

Abstract

在第一部份，我們證明了滿足C6對稱的二維三角光子晶體上，雙狄拉克錐可存在於布里元區中。延續前人於C6v對稱結構上的工作，雙狄拉克錐可由偶極子的E1模態和四極子的E2模態意外簡併在缺少六條對稱軸的C6光子晶體上被製造出來。利用單位晶格具有C6對稱的特性並奠基在緊束縛近似方法上，Γ點附近的頻散關係可得其解析解，產生兩組重疊的狄拉克錐以及四重根，並以多種破壞局部或者全域鏡像對稱的C6光子晶體為實例驗證之。在建構了雙狄拉克錐之後，調整幾何參數可得到零拓樸以及非零拓樸之能帶；結合這兩種類別的光子晶體，於其中的能隙可出現不易受擾動的螺旋邊界態，實現了Z2拓樸。有趣的是，這種邊界態可由單位晶格中介電常數的互換，在同一種幾何下將TM模態切換為TE模態。 在第二部份，我們探討PT二維三角光子晶體上的拓樸邊界態，以緊束縛近似方法去處理二階段的非厄米哈密頓算符。在PT未破損的情況下以自旋陳數去描述不同的拓樸性質，此可視為Bernevig-Hughes-Zhang模型之延伸。結合兩種不同拓樸的PT未破損光子晶體，螺旋邊界態可存在於其重疊的能隙之間。倘若PT破損，則非厄米擾動會使能帶拉平形成共軛虛根以及例外曲線將PT破損與PT未破損的區域分隔開來；於此情況下，頻帶的能隙會關閉使得PT破損的體態與邊界態混合在一起。

In the first part, we show that a double Dirac cone exists at the center of Brillouin zone in two-dimensional photonic crystals of triangular lattice with C6 symmetry. Following the case on the photonic lattice with C6v symmetry, the double Dirac cone in the photonic lattice with C6 symmetry, which lacks mirror symmetry about six symmetrical lines, can be constructed by the accidental degeneracy of E1 and E2 modes of two degenerate bands with dipole and quadrupole distributions, respectively. Based on the tight-binding approximation, the dispersion relation around the Γ point can be analytically solved by taking into account the C6 symmetry of unit structure on the triangular lattice, giving rise to four-fold degenerate solutions that represent an identical pair of Dirac cones. For illustration, we examine several types of photonic lattice with C6 symmetry by breaking locally and globally, respectively, the mirror symmetry. After the construction of the double Dirac cone, changing the parameters of geometry may form topological trivial and nontrivial bands, and then robust helical edge states can be produced in the frequency gap of photonic crystals composed of trivial and nontrivial regions, which realizes a Z2 topology. An interesting phenomenon is that these edge states can be switched from transverse magnetic harmonic modes to transverse electric harmonic modes in the same geometry of photonic crystals when we interchange the two dielectric constants in the unit cells. In the second part, we investigate the topological edge states in parity-time symmetric photonic crystals with C6v symmetry. A two-level non-Hermitian model that incorporates the gain and loss in the tight-binding approximation was employed to describe the dispersion of the PT symmetric system. In the unbroken PT phase, the topological property is characterized by the spin Chern numbers for the non-Hermitian Hamiltonian, which is considered an extended form of the Bernevig-Hughes-Zhang model. The helical edge states can exist in the common band gap for two topologically distinct PT unbroken lattices. In the broken PT phase, the non-Hermitian perturbation deforms the dispersion by merging the frequency bands into complex conjugate pairs and forming the exceptional contours that feature the PT phase transition. In this situation, the band gap closes and the edge states are mixed with the PT broken bulk states.