Chiral anomalies and topological phases of matter form the basis of the research presented in this dissertation. The chiral anomaly is considered both in the context of magnetic Weyl semimetals and in the context of non-Hermitian Dirac actions. Topological phases of matter play a role in this work through the research on Weyl semimetals and in the formulation of local topological markers.

The simplest example of magnetic Weyl semimetals consist of two Weyl cones separated in momentum space by a magnetisation vector which acts as an axial gauge field. We describe the emergence of axial electromagnetic fields by considering a magnetic field driven domain wall in this magnetisation. The parallel axial magnetic and axial electric fields give rise to the axial anomaly, and in turn to the chiral magnetic effect; a nonequilibrium current located at the domain wall. The chiral magnetic effect is a source of electromagnetic radiation, and a measurement of this radiation would provide evidence of the existence of the axial anomaly.

Electronic manipulation of domain walls is a central objective in spintronics. We describe how the axial anomaly, in terms of external electromagnetic fields, acts as a torque on the domain wall, and allows for electric control of the equilibrium configuration of the domain wall. We show how the axial anomaly is used to flip the chirality of the domain wall by tuning the electric field. Measuring the change in domain wall chirality constitutes a signal of the axial anomaly. We also describe how the Fermi arc boundary states of the Weyl semimetal at the domain wall result in an effective hard axis anisotropy which allows for large domain wall velocities irrespective of the intrinsic anisotropy of the material.

Our interest in non-Hermitian chiral anomalies stems from the existence of topological phases of matter in non-Hermitian models. We evaluate the chiral anomaly for a non-Hermitian Dirac theory with massless fermions with complex Fermi velocities coupled to non-Hermitian axial and vector gauge fields. The anomaly is compared with the corresponding anomaly of a Hermitianised and an anti-Hermitianised action derived from the non-Hermitian action. We find that the non-Hermitian anomaly does not correspond to the combined anomalous terms derived from the Hermitianised and anti-Hermitianised theory, as would be expected classically, resulting in new anomalous terms in the conservation laws for the chiral current.

Local topological markers are real space expressions of topological invariants evaluated by local expectation values and are important for characterising topology in noncrystalline structures. We derive analytic expressions for local topological markers for strong topological phases of matter in odd dimensions, by generalising the formulation of the even dimensional local Chern marker. This is not a straightforward task since the topological invariants in odd dimensions are basis dependent. Our solution is to express the invariants in terms of a family of parameter dependent projectors interpolating between a trivial state and the topological state of interest. The odd dimensional invariant is therefore expressed as a Chern character integrated over the combined space of the odd dimensional Brillouin zone and the one dimensional parameter space. As a result, we provide an easy-to-use chiral marker for symmetry classes with a chiral constraint, and a Chern-Simons marker for symmetry classes with either time reversal symmetry (in three dimensions) or particle hole symmetry (in one dimension). These markers are readily extended to interacting systems by considering the topological equivalence between a gapped one-particle density matrix of the interacting state and a projector corresponding to a free fermion state.

Kirala anomalier och topologiska faser av materia utgör grunden för den forskning som presenteras i denna avhandling. Den kirala anomalin studeras både i samband med magnetiska Weyl-semimetaller och i termer av en icke-Hermitisk verkansintegral. Vi undersöker topologiska faser av materia både genom Weyl-semimetaller och i formuleringen av lokala topologiska markörer. 

Den enklaste beskrivningen av magnetiska Weyl-semimetaller består av två så kallade Weyl-koner. Dessa kan separeras genom att addera en term som kan ses som ett axiellt gaugefält, och som beter sig som en magnetiseringsvektor.  Vi beskriver uppkomsten av axiella elektromagnetiska fält genom att betrakta en magnetfältsdriven domänvägg i denna magnetisering. Kombinationen av axiella magnetiska och elektriska fält resulterar i den kirala anomalin, och i sin tur till den kirala magnetiska effekten. Detta manifesteras i en icke-jämviktsström som följer med domänväggen. Den viktiga slutsatsen här är att den kirala magnetiska effekten är en källa till elektromagnetisk strålning, och en mätning av denna skulle därför fungera som en indirekt observation av den kirala anomalin.

Att finna verktyg att manipulera domänväggar med är ett centralt mål inom spinntronik. Vi beskriver explicit hur den axiella anomalin fungerar som ett vridmoment på domänväggen, och därmed möjliggör för en elektrisk kontroll av domänväggen. Vi visar speciellt hur det är möjligt att vända domänväggens kiralitet genom att ändra elektriska fältets styrka. En mätning av förändringen i domänväggens kiralitet utgör en signal för den axiella anomalin. Vi beskriver vidare hur utnyttjande av Fermiytans egenskaper vid domänväggen möjliggör stora domänväggshastigheter oberoende av materialets inneboende anisotropi.

Vårt intresse för icke-Hermitiska kirala anomalier bottnar i förekomsten av icke-Hermitiska topologiska faser. Vi utvärderar den kirala anomalin för en icke-Hermitisk modell med masslösa fermioner med komplexa Fermi-hastigheter kopplade till icke-Hermitiska axiella och vektoriella gaugefält. Klassiskt förväntar vi oss att den icke-Hermitska anomalin består av två bidrag, ett härrörande från den hermitska delen och det andra från den anti-hermitska. Detta är dock inte fallet, vilket resulterar i nya anomala termer i konserveringslagarna för den kirala strömmen.

Normalt karakteriseras topologiska faser av icke-lokala invarianter som beräknas i momentrummet. Detta är praktiskt då systemen är translationsinvarianta, men för icke-kristallina material är det önskvärt med lokala storheter. Vi härleder analytiska uttryck för en uppsättning lokala topologiska markörer giltiga i udda dimensioner. Detta är möjligt genom att generalisera formuleringen av den lokala Chern-markören för jämna dimensioner. Eftersom de topologiska invarianterna i udda dimensioner är basberoende är detta en icke-trivial uppgift.  Vi uttrycker invarianterna i termer av projektorer som interpolerar mellan ett trivialt tillstånd och det topologiska tillståndet av intresse. Mer specifikt uttrycks vår invariant som en integral av en Chern-karaktär över ett rum som kombinerar Brillouin-zonen och parameterrymden. Resultatet är de lättanvända chirala, och Chern-Simons markörerna som kan appliceras på ett flertal olika symmetriklasser. Vi beskriver även hur dessa markörer lätt kan utvidgas till växelverkande system.

While topology is a property of a quantum state itself, most existing methods for characterizing the topology of interacting phases of matter require direct knowledge of the underlying Hamiltonian. We offer an alternative by utilizing the one-particle density matrix formalism to extend the concept of the Chern, chiral, and Chern-Simons markers to include interactions. The one-particle density matrix of a free-fermion state is a projector onto the occupied bands, defining a Brillouin zone bundle of the given topological class. This is no longer the case in the interacting limit, but as long as the one-particle density matrix is gapped, its spectrum can be adiabatically flattened, connecting it to a topologically equivalent projector. The corresponding topological markers thus characterize the topology of the interacting phase. Importantly, the one-particle density matrix is defined in terms of a given state alone, making the local markers numerically favorable, and providing a valuable tool for characterizing topology of interacting systems when only the state itself is available. To demonstrate the practical use of the markers we use the chiral marker to identify the topology of midspectrum eigenstates of the Ising-Majorana chain across the transition between the ergodic and many-body localized phases. We also apply the chiral marker to random states with a known topology, and compare it with the entanglement spectrum degeneracy.

Topological phases of matter are ubiquitous in crystals, but less is known about their existence in amorphous systems, that lack long-range order. We review the recent progress made on defining amorphous topological phases, their new phenomenology. We discuss the open questions in the field which promise to significantly enlarge the set of materials and synthetic systems benefiting from the robustness of topological matter.

Local topological markers, topological invariants evaluated by local expectation values, are valuable forcharacterizing topological phases in materials lacking translation invariance. The Chern marker-the Chernnumber expressed in terms of the Fourier transformed Chern character-is an easily applicable local markerin even dimensions, but there are no analogous expressions for odd dimensions. We provide general analyticexpressions for local markers for free-fermion topological states in odd dimensions protected by localsymmetries: aChiral marker, a localZmarker which in case of translation invariance is equivalent to thechiral winding number, and aChern-Simons marker, a localZ2marker characterizing all nonchiral phases inodd dimensions. We achieve this by introducing a one-parameter familyP theta of single-particle densitymatrices interpolating between a trivial state and the state of interest. By interpreting the parameter theta as anadditional dimension, we calculate the Chern marker for the familyP theta. We demonstrate the practical use ofthese markers by characterizing the topological phases of two amorphous Hamiltonians in three dimensions:a topological superconductor (Zclassification) and a topological insulator (Z2classification).

The chiral anomaly underlies a broad number of phenomena, from enhanced electronic transport in topological metals to anomalous currents in the quark-gluon plasma. The discovery of topological states of matter in non-Hermitian systems raises the question of whether there are anomalous conservation laws that remain unaccounted for. To answer this question, we consider both two and four space-time dimensions, presenting a unified formulation to calculate anomalous responses in Hermitianized, anti-Hermitianized, and non-Hermitian systems of massless electrons with complex Fermi velocities coupled to non-Hermitian gauge fields. Our results indicate that the quantum conservation laws of chiral currents of non-Hermitian systems are not related to those in Hermitianized and anti-Hermitianized systems, as would be expected classically, due to different anomalous terms that we derive. We further present some physical consequences of our non-Hermitian anomaly that may have implications for a broad class of emerging experimental systems that realize non-Hermitian Hamiltonians.

We show how the axial (chiral) anomaly induces a spin torque on the magnetization in magnetic Weyl semimetals. The anomaly produces an imbalance in left- and right-anded chirality carriers when non-orthogonal electric and magnetic fields are applied. Such imbalance generates a spin density which exerts a torque on the magnetization, the strength of which can be controlled by the intensity of the applied electric field. We show how this results in an electric control of the chirality of domain walls, as well as in an improvement of the domain wall dynamics, by delaying the onset of the Walker breakdown. The measurement of the electric field mediated changes in the domain wall chirality would constitute a direct proof of the axial anomaly. Additionally, we show how quantum fluctuations of electronic Fermi arc states bound to the domain wall naturally induce an effective magnetic anisotropy, allowing for high domain wall velocities even if the intrinsic anisotropy of the magnetic Weyl semimetal is small. 

A space-time dependent node separation in Weyl semimetals acts as an axial vector field. Coupled with domain wall motion in magnetic Weyl semimetals, this induces axial electric and magnetic fields localized at the domain wall. We show how these fields can activate the axial (chiral) anomaly and provide a direct experimental signature of it. Specifically, a domain wall provides a spatially dependent Weyl node separation and an axial magnetic field B-5, and domain wall movement, driven by an external magnetic field, gives the Weyl node separation a time dependence, inducing an axial electric field E-5. At magnetic fields beyond the Walker breakdown, E-5. B-5 becomes nonzero and activates the axial anomaly that induces a finite axial charge density-imbalance in the number of left- and right-handed fermions-moving with the domain wall. This axial density in turn produces, via the chiral magnetic effect, an oscillating current flowing along the domain wall plane, resulting in a characteristic radiation of electromagnetic waves emanating from the domain wall. A detection of this radiation would constitute a direct measurement of the axial anomaly induced by axial electromagnetic fields.

We present a framework for characterizing higher-order topological phases directly from the one-particle density matrix, without any reference to an underlying Hamiltonian. Our approach extends the mode-shell correspondence, originally formulated for single-particle Hamiltonians, to Gaussian states subject to chiral constraints. In this correspondence, the mode index counts topological boundary modes, while the shell index quantifies the bulk topology in a region surrounding the modes, providing a bulk-boundary diagnostic. In one-dimensional topological insulators, the shell index reduces to the local chiral marker, recovering the winding number in the translation-invariant limit. We apply the mode-shell correspondence to a C4-symmetric higher-order topological insulator with a chiral constraint and show that a fractional shell index implies that the higher-order phase is intrinsic. The one-particle density matrix is formulated in real space, so the mode-shell correspondence also applies to models without translation invariance. By introducing structural disorder into the C4-symmetric higher-order insulator, we show that the mode-shell correspondence remains a meaningful diagnostic in amorphous structures. The mode-shell correspondence generalizes to interacting states with a gapped bulk spectrum in the one-particle density matrix, providing a practical and diverse route to characterize higher-order topology from the quantum state itself.