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We derive a framework to apply topological quantum chemistry in systems subject to magnetic flux. We start by deriving the action of spatial symmetry operators in a uniform magnetic field, which extends Zak's magnetic translation groups to…
Our understanding of topological insulators is based on an underlying crystalline lattice where the local electronic degrees of freedom at different sites hybridize with each other in ways that produce nontrivial band topology, and the…
Recently, novel crystalline constructions known as Cayley-Schreier lattices have been suggested as a platform for realizing arbitrary gauge fields in synthetic crystals with real hopping amplitudes. We show that Cayley-Schreier lattices can…
Systems as diverse as mechanical structures assembled from elastic components, and photonic metamaterials enjoy a common geometrical feature: a sublattice symmetry. This property realizes a chiral symmetry first introduced to characterize a…
We introduce a novel class of interaction-enabled topological crystalline insulators in two- and three-dimensional electronic systems, which we call "topological crystalline magnet." It is protected by the product of the time-reversal…
Topological insulators are a novel class of quantum materials in which time-reversal symmetry, relativistic (spin-orbit) effects and an inverted band structure result in electronic metallic states on the surfaces of bulk crystals. These…
Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported…
Topological acoustics is an emerging field that lies at the intersection of condensed matter physics, mechanical structural design and acoustics engineering. It explores the design and construction of novel artificial structures, such as…
Discovery of novel topological orders of condensed matters is of a significant interest in both fundamental and applied physics due to the associated quantum conductance behaviors and unique symmetry-protected backscattering-immune…
In the field of topological insulators, the topological properties of quantum states in samples with simple geometries, such as a cylinder or a ribbon, have been classified and understood during the last decade. Here, we extend these…
We give an introduction to topological crystalline insulators, that is, gapped ground states of quantum matter that are not adiabatically connected to an atomic limit without breaking symmetries that include spatial transformations, like…
Spatial symmetries appearing in both real and momentum space are of fundamental significance to crystals. However, in the conventional framework, every space group in real space, either symmorphic or nonsymmorphic, corresponds to a…
Symmetries crucially underlie the classification of topological phases of matter. Most materials, both natural as well as architectured, possess crystalline symmetries. Recent theoretical works unveiled that these crystalline symmetries can…
Topological phases of matter are often understood and predicted with the help of crystal symmetries, although they don't rely on them to exist. In this chapter we review how topological phases have been recently shown to emerge in amorphous…
Topological disclinations, crystallographic defects that break rotation lattice symmetry, have attracted great interest and exhibited wide applications in cavities, waveguides, and lasers. However, topological disclinations have thus far…
We develop a lattice model which exhibits topological transitions from $Z_2$ topological insulators to mirror symmetry-protected topological crystalline insulators by introducing additional spin-orbit coupling terms. The topological phase…
$f$-electron systems exhibit a subtle interplay between strong spin--orbit coupling and crystal-field effects, producing complex energy landscapes that are computationally demanding. We introduce auxiliary functions, constructed by…
Recently, the topological physics in artificial crystals for classical waves has become an emerging research area. In this Letter, we propose a unique bilayer design of sonic crystals that are constructed by two layers of coupled hexagonal…
The bulk-boundary correspondence, which links a bulk topological property of a material to the existence of robust boundary states, is a hallmark of topological insulators. However, in crystalline topological materials the presence of…
The mathematical field of topology has become a framework to describe the low-energy electronic structure of crystalline solids. A typical feature of a bulk insulating three-dimensional topological crystal are conducting two-dimensional…