A central topic in condensed matter research during the last decades has been the study and classification of topological phases of matter. Topological insulators in particular, a subset of symmetry protected topological phases, have been investigated for over a decade. In recent years, several extensions to this formalism have been proposed to study more unconventional systems.In this thesis we explore two of these extensions, where key assumptions in the original formalism are removed. The first case is critical systems, which have no energy gap. Conventional topological invariants are discontinuous at topological transitions, and therefore not well-defined for critical systems. We propose a method for generalizing conventional topological invariants to critical systems and show robustness to disorder that preserves criticality. The second case involves non-Hermitian systems, which appear in effective descriptions of dissipation, where we study the entanglement spectrum and its connection to topological invariants. Furthermore, by introducing non-Hermiticity to critical systems we show how the winding numbers that characterize some topological phases of the non-Hermitian system, as well as topological signatures in the entanglement spectrum, can be obtained from the related critical model.

Non-Hermitian physics has introduced phenomena like the skin effect and exceptional points, challenging traditional views of topological phases. This thesis contrasts two classification frameworks for non-Hermitian systems. The first approach employs K-theory classifications based on line-gaps and point-gaps, which explain the topological origins of the skin effect but fail to adequately describe exceptional points. The second, more nuanced approach uses homotopy theory, where the braiding of complex energies around exceptional points is integral to the classification. This method provides a detailed account of the behaviour of exceptional points, such as the splitting and merging of these points. We also study systems restricted by PT-symmetry, a symmetry that ensures that each eigenvalue is real or has a complex conjugate pair, which leads to the formation of an exceptional cone in the parameter space of two-band models. This cone features prominently in the accompanying papers, especially when introducing the concept of non-defective exceptional points and in the homotopy classification of PT-symmetric models.

Topology is a branch of mathematics that studies properties that remain unchanged under continuous deformations. In physics, topology is used to describe phenomena that are robust against small perturbations. A well-known example is topological insulators—materials that act as insulators in their interior while conducting along their surface. These conducting states are protected by topological properties and persist even when the material is slightly modified. Over the past few decades, significant effort has been devoted to understanding and classifying different types of topological phases, which describe the various ways in which such robust properties can emerge in nature.

In recent years, interest has grown in dissipative systems, where energy losses play a central role. These systems are described using non-Hermitian Hamiltonians, which extend the conventional quantum mechanical framework.

This dissertation explores how non-Hermitian physics affects the topology and classification of topological phases. In particular, we investigate a type of topological charge known as exceptional points, which arise exclusively in non-Hermitian systems. These points are characterized by a topological charge that describes how energy bands intertwine around them. We focus specifically on how certain symmetries can stabilize exceptional points and shape their properties. Finally, we examine multifold exceptional points—a more intricate class of these singularities—and their topological characteristics.