Electronic order Topology in crystals and symmetry-breaking from interactions

Electrons in matter are entirely different from their lonely counterparts in particle physics. This thesis explores two of the most fundamental types of emergent electronic order, both made possible by the environment provided to electrons by a lattice of atomic nuclei.
The first is topological order in insulating crystals, which results in robust electronic states along the edges or at corners, protected by symmetries of the atomic lattice. We demonstrate a new method for identifying and distinguishing between different topological phases of crystals with a rotational symmetry.
Next, we turn to charge density waves, a type of electronic order that breaks one or more symmetries of the crystal lattice. This charge-ordered phase is the result of coupling between electrons and vibrational modes of the lattice known as phonons. In response to this coupling, electrons form standing waves of charge density and the lattice undergoes a periodic distortion, changing the crystal structure. We analyse in detail the charge density waves that form in the material VSe2. By incorporating the momentum-dependence of the electron-phonon coupling strength, our theoretical model reproduces and elucidates confounding experimental data for the bulk crystal. Thinning VSe2 down to a single layer, we find a dimensional crossover to a unique state with two charge density waves which are driven by distinct mechanisms.