The hallmark of topological phases of matter is the presence of robust boundary states. In this dissertation, a formalism is developed with which analytical solutions for these states can be straightforwardly obtained by making use of destructive interference, which is naturally present in a large family of lattice models. The validity of the solutions is independent of tight-binding parameters, and as such these lattices can be seen as a subset of solvable systems in the landscape of tight-binding models. The approach allows for a full control of the topological phase of the system as well as the dispersion and localization of the boundary states, which makes it possible to design lattice models possessing the desired topological phase from the bottom up. Further applications of this formalism can be found in the fields of higher-order topological phases—where boundary states localize to boundaries with a codimension larger than one—and of non-Hermitian Hamiltonians—which is a fruitful approach to describe dissipation, and feature many exotic features, such as the possible breakdown of bulk-boundary correspondence—where the access to exact solutions has led to new insights.

The bulk-boundary correspondence, which in topological insulators describes the relationship between the bulk invariant computed for a system with periodic boundary conditions and the number of boundary states in the corresponding system with open boundary conditions, is well-known and important for predicting the behavior of these systems. In recent years, however, the modeling of dissipative and non-equilibrium systems using non-Hermitian Hamiltonians has become increasingly popular. These systems feature many novel phenomena, but in particular the bulk-boundary correspondence breaks down since the spectrum of the system with periodic boundary conditions now differs fundamentally from the spectrum of the system with open boundary conditions. It is thus no longer possible to use the Bloch Hamiltonian to predict the appearance of boundary states.

Integral to understanding the behavior of these systems, is to understand how the boundary states behave. This is what is studied in the accompanying papers, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence, and Phase transitions and generalized biorthogonal trace polarization in non-Hermitian systems, of this thesis, where also a new kind of biorthogonal bulk-boundary correspondence is developed.

The aim of this licentiate thesis is to give the background necessary to understand the accompanying papers. It is divided into two parts. The first part describes the well-established theory of boundary states in a certain class of Hermitian systems for which there exist exact solutions that are straightforward to analyze, which then are generalized to the non-Hermitian case in the accompanying papers. The second part gives some background to non-Hermitian systems, the unusual phenomena that occur in them, and an introduction to biorthogonal quantum mechanics and why it is necessary to redefine the inner product one uses when calculating quantum mechanical probabilities.

In topological insulators, the bulk-boundary correspondence describes the relationship between the bulk invariant -- computed for a system with periodic boundary conditions -- and the number of topological boundary states in the corresponding system with open boundary conditions. This is a well-known property of these systems and is important for predicting how they will behave. In recent years, however, the modeling of dissipative and non-equilibrium systems using non-Hermitian Hamiltonians has become increasingly popular. These systems feature many novel phenomena; in particular the bulk-boundary correspondence breaks down since the spectrum of the system with periodic boundary conditions typically differs fundamentally from the spectrum of the system with open boundary conditions.        

In this thesis, the behavior of the boundary states in non-Hermitian lattice models is studied. The framework of biorthogonal quantum mechanics is used to develop the biorthogonal bulk-boundary correspondence, which predicts the (dis)appearance of the boundary states in these systems. Closely related to the drastic change in spectra between boundary conditions is the non-Hermitian skin effect. This refers to the exponential localization of almost all eigenstates to the boundaries and is typically seen in non-Hermitian lattice models. How to predict this, and how to quantify the sensitivity of the spectrum to the boundary conditions are therefore questions that are also studied in this thesis.