Wave functions on periodic lattices are commonly described by Bloch band theory. Besides Abelian Bloch states labeled by a momentum vector, hyperbolic lattices support non-Abelian Bloch states that have so far eluded analytical treatments. By …

We extend the notion of topologically protected semi-metallic band crossings to hyperbolic lattices in negatively curved space. Due to their distinct translation group structure, such lattices support non-Abelian Bloch states which, unlike …

We introduce and experimentally realize hyperbolic matter as a novel paradigm for topological states, made of particles moving in the hyperbolic plane with negative curvature. Curvature of space is emulated through a hyperbolic lattice using …

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. For spinless particles, the stability of a triple nodal point requires two ingredients: rotation symmetry …

Electron band structures, which describe the energy-momentum relation for electrons in solids, can exhibit robust crossings called "nodes". Such nodes famously occur in graphene or in Weyl semimetals, and often facilitate special transport phenomena, such as the decrease of resistivity of Weyl semimetals in applied parallel magnetic field.

We study a class of topological materials which in their momentum-space band structure exhibit threefold degeneracies known as triple points. Focusing specifically on $\mathcal{PT}$-symmetric crystalline solids with negligible spin-orbit coupling, we …

Weyl semimetals in three-dimensional crystals provide the paradigm example of topologically protected band nodes. It is usually taken for granted that a pair of colliding Weyl points annihilate whenever they carry opposite chiral charge. In stark …

An Alice string is a topological defect with a very peculiar feature. When a defect with a monopole charge encircles an Alice string, the monopole charge changes sign. In this paper, we generalize this notion to the momentum space of periodic media …

Nodal lines inside the momentum space of three-dimensional crystalline solids are topologically stabilized by a $\pi$-flux of Berry phase. Nodal-line rings in $\mathcal{PT}$-symmetric systems with negligible spin-orbit coupling (here described as …

Topological semimetals are characterized by topologically protected band-structure nodes. One prominent example is the Weyl semimetal, characterized by Weyl points carrying topological Chern numbers. In this dissertation, we explore the topology of …