Higher-order topological phases with crystalline symmetries

Topological insulators (TIs) that are insulating in the bulk but conducting at surfaces have been thoroughly studied for decades. Recently, a new class of TIs, dubbed the higher-order topological insulator (HOTI), has been discovered. Unlike conventional TIs, it is gapped at the boundaries but host robust features at the boundaries of boundaries. For example, the quadrupole insulator, which is the first discovered HOTI, displays gapless corner states and quantized 1/2 electronic charge localized at the corners of the lattice. The topologies in HOTIs are usually protected by crystalline symmetries. In this dissertation, we systematically study the charge fractionalization in 2D HOTIs under rotation symmetries. We find that, under the Cn rotation symmetries, the electronic charge fractionalizes in units of 1/n at corners and bulk disclination defects in symmetric HOTIs. Using spatially localized Wannier representations, we provide an intuitive microscopic theory to account for the mechanism of charge fractionalization. Via the K-theory classification framework, we constructed topological indices defined in the crystal momentum space, which relate the fractional charge to the symmetry representations at high symmetry points in the Brillouin zone. In addition, we propose a new observable indicator for detecting HOTIs in metamaterial systems, which relies on spatial distributions of states in a given band and hence requires only crystalline symmetries. This indicator can identify non-trivial HOTIs that would be neglected by the conventional method of searching for gapless boundary states. We then study the gapless topological semimetals having crystalline symmetries. We identify a new type of Weyl semimetals, the higher-order Weyl semimetal (HOWSM), which hosts arc-like gapless states at both the surfaces and the hinges in a bounded 3D lattice. A 2nd-order Weyl node can be viewed as the critical point between a Chern insulator phase and a HOTI phase. We provide tight-binding models for the 2nd-order Weyl semimetals having C4z rotation symmetry and inversion symmetries. To understand the physics implications of 2nd-order Weyl nodes, we first study the hybridization between Weyl nodes of different orders, from which an exotic insulating phase emerges, displaying independent surface Dirac cones and hinge arcs. We then discuss the unique electromagnetic response in the 2nd-order Weyl semimetals. Our work serves as the first step to the discovery as well as the understanding of HOWSMs.U of I OnlyAuthor requested U of Illinois access only (OA after 2yrs) in Vireo ETD syste