Linear response theory is a powerful calculation tool in quantum field theory. We apply this framework to a variety of models originating from distinct areas in theoretical physics and for different reasons. In the context of black hole holography, we consider a quench model where we investigate effective thermalization, as well as the boundary signal of so-called evanescent modes which indicate the presence of a black hole-like object in the bulk. The problem of quantum thermalization plays a central role within the holographic duality between thermal states in the boundary field theory and black hole-like objects in the bulk. However, quantum thermalization is also an interesting question in itself from a fundamental point of view. Inspired by recent progress in understanding how operators in quantum field theories thermalize, which occurs even when considering integrable models, we investigate the so-called operator thermalization hypothesis. We focus on gauge theories at finite temperature with a large number of fields which present a phase transition between the low-temperature and high-temperature regimes. In a separate application of linear response theory, we investigate transport properties in a family of Weyl semimetal systems. Concretely, we develop a general analytic method to compute the magneto-optical conductivity of these systems in the presence of an external magnetic field aligned with the tilt of the spectrum. Last, we examine non-Hermitian Weyl-like systems as potential analogue black hole models and suggest a specific parity-time-symmetric dissipative Hamiltonian displaying analogue Hawking radiation.

Topology has manifestations in physics ranging from the field of condensed matter to photonics. This dissertation provides a two-fold study on the impact of topology in Hermitian and non-Hermitian band structures. Salient examples include the notion of topological invariants and knots, which are both used to describe characteristics of eigenvalue intersections. The first part focuses on Hermitian topological phases of matter, where general methods predicting transport properties in both gapped and gapless phases are presented. The second part turns to non-Hermitian phases and revolves around the topological properties of their exceptional eigenvalue degeneracies. Through a generic construction originating in knot theory, it is shown that such degeneracies take the form of knots, which furthermore bound open Fermi surfaces coinciding with the respective Seifert surfaces. This construction is then extended and applied in a similar fashion to parity-time-symmetric systems, where the exceptional points form surfaces and curves of any topology, as well as points. These theoretical descriptions constitute a fruitful platform to study dissipative systems—in particular in optics where parity-time symmetry implies a balance between gain and loss in photonic crystals—but also give rise to interesting connections to gravity in the context of analogue black holes.

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.