Violation of bulk-edge correspondence in two continuum models

The principle of bulk-edge correspondence for topological insulators was studied and later proven for both the integer quantum Hall effect (IQHE) as well as models of crystalline structures. In essence, the bulk shows the characteristics of an insulator, and a topological invariant can be associated with it. This invariant is in correspondence with the number of conducting edge states on the metallic surface of the material. The question then arises if there are limits to this correspondence - which we will answer affirmatively: We will see that in the case of appropriately regularized models defined on the continuum, the number of edge modes in presence of a sharp boundary depends on the boundary condition. Therefore, bulk-edge correspondence is explicitly violated for the cases we study. More explicitly, the two models we will study are the two-dimensional rotating shallow water model regularized by an odd viscous term and the Dirac Hamiltonian in dimension 2 regularized by a mass term. In each of these cases, we will show that a continuous family of boundary conditions shows a dependence of the number of edge modes on the parameter parametrizing our family resulting in a rich phase diagram. The origin of the violation of bulk-edge correspondence is then explained by scattering theory and Levinson’s theorem. We will see that the latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there. To this anomaly at infinity, we will attribute the violation of bulk-edge correspondence. After the introduction, the thesis starts with the second chapter introducing the concept of bulk-edge correspondence in the periodic setting. In there we will study the integer quantum Hall effect as well as a Dirac type model describing graphene on a honeycomb lattice. The interesting thing about this model is that it shows a phase transition when varying its parameters. We will then also introduce a relative version of Levinson’s theorem in this setting which will later be generalized and applied in the continuum case. In the third chapter, we will motivate the two-dimensional rotating shallow water model and give some background on the regulator in form of an odd viscous term we add. Furthermore, we give a characterisation of all possible local self-adjoint boundary conditions for this model and study some of the edge modes that occur in the specific family we choose to examine more closely. The fourth chapter contains the main part of this thesis and in there we will show the explicit violation of bulk-edge correspondence at the example of a 1-parameter family of boundary conditions for the shallow water model. We will introduce the bulk index and see that the number of edge modes in the finite depends on the boundary condition. This will lead us to study edge modes and the winding near infinity and to the observation of an anomaly there. In fact, the overall winding, including infinity, will still correspond with the bulk index, but the count of the number of edge modes will not. Before concluding, we will consider the massive Dirac Hamiltonian in the fifth chapter. We will do the same steps as in the shallow water model and be able to observe a similar violation