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.

Metamaterial, optiska och fotoniska system samt elektriska och akustiska system uppför sig alla, som bekant, på ett helt annat sätt än isolerade kvantsystem; detsamma gäller öppna kvantsystem som står i kontakt med omgivningen. Den främsta anledningen till detta är växelverkan med omgivningen, som kan tillföra energi och partiklar till och från systemet. Insikten att dessa system ändå kan beskrivas med en formalism som strukturellt liknar kvantmekaniken för isolerade system har väckt ett stort intresse för det som har kommit att kallas icke-hermitisk fysik.

Sedan dess har resultat från kvantteorin kring topologiska faser tillämpats på icke-hermitiska system.  Denna teori beskriver uppkomsten av materiafaser som framstår som identiska inom det tidigare paradigmet av symmetrier och symmetribrytning, men som skiljer sig åt av topologiska skäl. Ett exempel på detta är olika topologiska isolatorer som kännetecknas av topologiska invarianter: diskreta tal som kan beräknas utifrån deras bandstruktur.  Fasövergångar mellan olika topologiska isolatorer sker via topologiskt robusta nodpunkter i bandstrukturen, vilka bär på förändringarna i invarianterna.

Här använder vi homotopiteori för att härleda sådana invarianter för icke-hermitiska system.  Icke-hermitiska nodpunkter kallas vanligtvis exceptionella punkter och skyddas av den icke-abelska flätgruppen, vilket innebär att de topologiskt skiljer sig avsevärt från sina hermitiska motsvarigheter. Vi finner att exceptionella punkter av högre ordning skyddas av vindningstal, att system som är underkastade PT-symmetri (paritet och tidsomvändning) uppvisar en kombinerad egenvärdes- och egenvektortopologi,  och  att tvådimensionella rumsliga symmetrier (kristallografiska grupper) kan tvinga fram exceptionella punkter.

Vi härleder vidare fermionfördubblingssatser för icke-hermitiska topologiska system, vilka begränsar det tillåtna antalet och sammansättningen av exceptionella punkter i ett material. Dessa satser skiljer sig från sina hermitiska motsvarigheter på grund av den icke-abelska strukturen i exceptionella punkter. Vi visar, teoretiskt och i fotonikexperiment, att detta kan leda till topologiska monopoler, vilka är förbjudna i hermitiska system.

Metamaterials, optical and photonic setups, as well as electric and acoustic systems all (clearly) behave very differently from isolated quantum systems; as do open quantum systems that are in contact with an environment. The primary reason is precisely this coupling to an environment, which can deliver energy and particles into and out of the system. The insight that these systems can nevertheless be described in a formalism that structurally resembles quantum mechanics of isolated systems, has fuelled a wave of interest in what has come to be known as non-hermitian physics.

More recently, results from the quantum theory of topological phases started being applied to the non-hermitian realm. This theory describes the emergence of phases of matter that seem identical in its preceding paradigm of symmetries and symmetry breaking, but that differ for topological reasons. Exemplarily, there are different topological insulators that are distinguished by topological invariants, discrete numbers that can be calculated from their band structure. Phase transitions between them happen via topologically robust nodal points in the band structure, which carry the changes in invariants.

Here we use homotopy theory to derive such invariants for non-hermitian systems. Non-hermitian nodal points are commonly called exceptional points, and are protected by the non-abelian braid group, thus topologically extremely different from their hermitian counterparts. We find that exceptional points of higher order are protected by winding numbers, that systems subject to PT-symmetry (parity and time reversal) show combined eigenvalue and eigenvectortopology, and that two-dimensional spatial symmetries (called wallpaper symmetries) can enforce exceptional points.

We further derive fermion doubling theorems for non-hermitian topological systems, which constrain the allowed number and composition of exceptional points in a material. These theorems differ from their hermitian analogues due tothe non-abelian structure inherent to exceptional points. We show, theoretically and in photonics experiments, that this can lead to topological monopoles, which are forbidden in hermitian systems.