Crystalline topological phases: From corner charge to hinge state

This thesis is concerned with the properties of insulating crystals. At first glance, it might not seem very interesting to study the electronic properties of such materials, as there is no conduction possible. Why then would one devote an entire thesis to this topic? The answer is that there exists a class of peculiar insulators, which are insulating in the interior of the material, but conducting at the edges. We mostly consider crystalline (topological) insulators, and more specifically insulators possessing a rotational symmetry. The atoms of such a material form a lattice that is two-, three-, four- or six-fold rotational symmetric. If this symmetry is broken, any conductive edge states present disappear. Apart from conduction edge states, rotational symmetry can also protect a quantized corner charge. These protected edge states and corner charges are not just an edge effect: they can be predicted if we know the configuration of the interior of the crystal. This follows from the so-called bulk-boundary correspondence, which states that the existence of protected states on the edge of a material, and quantized corner charges, can be related to the configuration of electrons in the interior of the material. An important question, which we answer for some specific cases in this thesis, is how we can exactly calculate the value of the corner charge given the interior of a crystal. We also provide a recipe to construct, by stacking the two-dimensional material graphene, a higher-order topological insulator, and we show the existence of a hybrid-order topological insulator, which features surface states protected by different symmetries. Finally, we investigate a sheet of graphene in a strong magnetic field and irradiated by circularly polarized light