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. 

Understanding the extreme sensitivity of the eigenvalues of non-Hermitian Hamiltonians to the boundary conditions is of great importance when analyzing non-Hermitian systems, as it appears generically and is intimately connected to the skin effect and the breakdown of the conventional bulk boundary correspondence. Here we describe a method to find the eigenvalues of one-dimensional one-band models with arbitrary boundary conditions. We use this method on several systems to find analytical expressions for the eigenvalues, which give us conditions on the parameter values in the system for when we can expect the spectrum to be insensitive to a change in boundary conditions. By stacking one-dimensional chains, we use the derived results to find corresponding conditions for insensitivity for some two-dimensional systems with periodic boundary conditions in one direction. This would be hard by using other methods to detect skin effect, such as the winding of the determinant of the Bloch Hamiltonian. Finally, we use these results to make predictions about the (dis)appearance of the skin effect in purely two-dimensional systems with open boundary conditions in both directions.

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.

Non-Hermitian (NH) Hamiltonians can be used to describe dissipative systems, notably including systems with gain and loss, and are currently intensively studied in the context of topology. A salient difference between Hermitian and NH models is the breakdown of the conventional bulk-boundary correspondence, invalidating the use of topological invariants computed from the Bloch bands to characterize boundary modes in generic NH systems. One way to overcome this difficulty is to use the framework of biorthogonal quantum mechanics to define a biorthogonal polarization, which functions as a real-space invariant signaling the presence of boundary states. Here, we generalize the concept of the biorthogonal polarization beyond the previous results to systems with any number of boundary modes and show that it is invariant under basis transformations as well as local unitary transformations. Additionally, we focus on the anisotropic Su-Schrieffer-Heeger chain and study gap closings analytically. We also propose a generalization of a previously developed method with which to find all the bulk states of the system with open boundaries to NH models. Using the exact solutions for the bulk and boundary states, we elucidate genuinely NH aspects of the interplay between the bulk and boundary at the phase transitions.

Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees of freedom we establish that the non-Hermitian extensions of higher-order topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the non-Hermitian skin effect, and also naturally encompasses genuinely non-Hermitian phenomena in the absence thereof.

In a toothpick-type cellular automaton, a shape is drawn, and then at each time-step copies of the same shape are attached at certain predetermined places. The resulting pattern exhibits unexpected growth properties. We investigate the fractal-like large-scale behavior of the Q-toothpick cellular automaton, which is built from quarter circles, with starting configurations consisting of an arbitrary number of quarter circles. In this paper, we prove that infinitely long barriers of quarter circles arise in the pattern, and divide it into non-interacting triangular parts. Furthermore, we show that the behavior of these triangular parts is described by the one-dimensional elementary cellular automaton rule 18 and is related to the Sierpinski triangle.

Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis) appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.