论文标题
半经典制度中的拓扑绝缘子
Topological insulators in semiclassical regime
论文作者
论文摘要
我们研究了$ 2 \ times 2 $ systems $（H d_t + \ \ \ \ \ \ \ \ \ \ \ \ \ \ mthbb {r}^2 $ on Magiclassical $ h \ rightarrow 0 $，我们的Dirac运算符$ \ Mathcal {d} $是拓扑绝缘器之间接口的标准模型：它代表无效能量的半学，而在正 /负能量下的不同拓扑阶段。它的半经典符号具有两个相反的特征值，沿圆锥形沿曲线$γ$的位置 /力量的圆锥形相交，其中材料的行为就像半学。 我们证明，wavepacket求解$（H d_t + \ Mathcal {d}）ψ_t= 0 $，最初集中在$γ$上的相空间中，分为两部分。第一部分与预定的方向和速度相干地沿着$γ$连贯。这是著名的“边缘状态”的动态表现。第二部分立即崩溃。 我们的方法包括：（i）傅立叶积分运算符减少； （ii）对规范模型进行仔细的WKB分析； （iii）重建程序。它为行进模式的速度和轮廓产生混凝土公式。作为应用，我们在分析上描述了磁性，弯曲和紧张的拓扑绝缘子模型的动态边缘状态。
We study solutions of $2 \times 2$ systems $(h D_t + \mathcal{D}) Ψ_t = 0$ on $\mathbb{R}^2$ in the semiclassical regime $h \rightarrow 0$. Our Dirac operator $\mathcal{D}$ is a standard model for interfaces between topological insulators: it represents a semimetal at null energy, and distinct topological phases at positive / negative energies. Its semiclassical symbol has two opposite eigenvalues, intersecting conically along the curve $Γ$ of positions / momenta where the material behaves like a semimetal. We prove that wavepackets solving $(h D_t + \mathcal{D}) Ψ_t = 0$, initially concentrated in phase space on $Γ$, split in two parts. The first part travels coherently along $Γ$ with predetermined direction and speed. It is the dynamical manifestation of the famous "edge state". The second part immediately collapses. Our approach consists of: (i) a Fourier integral operator reduction; (ii) a careful WKB analysis of a canonical model; (iii) a reconstruction procedure. It yields concrete formulas for the speed and profile of the traveling modes. As applications, we analytically describe dynamical edge states for models of magnetic, curved, and strained topological insulators.
