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The basic question which drove our whole work was to find a meaningful noncommutative gauge theory even for the time-like case ($\theta^{0 i} \neq 0$). In order to be able to tackle questions regarding unitarity, it is not sufficient to consider theories which include the noncommutative parameter only up to a finite order. The reason is that in order to investigate tree-level unitarity or the optical theorem in loops one has to know the behavior of the noncommutative theory for center-of-mass energies much greater than the noncommutative scale. Therefore an effective theory, that is by construction only valid up to the noncommutative scale, isn't sufficient for our purpose. Our model is based on two fundamental assumptions. The first assumption is given by the commutation relations \eqref{eq:ncalg}. This led to the Moyal-Weyl star-product \eqref{eq:astproduct2} which replaces all point-like products between two fields. The second assumption is to assume that the model built this way is not only invariant under the noncommutative gauge transformation but also under the commutative one. In order to obtain an action of such a model one has to replace the fields by their appropriate \swms. We chose the gauge fixed action \eqref{eq:actioncgf} as the fundamental action of our model. After having constructed the action of the NCQED including the {\swms} we were confronted with the problem of calculating the {\swms} to all orders in $\tMN$. By means of \cite{bbg} we could calculate the {\swms} order by order in the gauge field, where each order in the gauge field contains all orders in the noncommutative parameter (\cf chapter \ref{chapter:swms}). By comparing the maps with the result we obtained from an alternative ansatz \cite{bcpvz}, we realized that already the simplest {\swm} for the gauge field is not unique. In chapter \ref{chapter:ambiguities} we examined this ambiguity, which we could parametrised by an arbitrary function $\astf$. The next step was to derive the Feynman rules for our NCQED. One finds that the propagators remain unchanged so that the free theory is equal to the commutative QED. The fermion-fermion-photon vertex contains not only a phase factor coming from the Moyal-Weyl star-product but also two additional terms which have their origin in the \swms. Beside the 3-photon vertex which is already present in NCQED without {\swms} and which has also additional terms coming from the \swms, too, one has a contact vertex which couples two fermions with two photons. After having derived all the vertices we calculated the pair annihilation scattering process $e^+ e^- \rightarrow \gamma \gamma$ at Born level. By choosing the parameter $\kggg = 1$ (\cf section \ref{sec:represent}), we found that the amplitude of the pair annihilation process becomes equal to the amplitude of the NCQED without \swms. This means that, at least for this process, the NCQED excluding {\swms} is only a special case of NCQED including \swms. On the basis of the pair annihilation process, we afterwards investigated tree-level unitarity. In order to satisfy the tree-level unitarity we had to constrain the arbitrary function $\astf$. We found that the series expansion of $\astf$ has to start with unity. In addition, the even part of the function must not increase faster than $s^{-1/2} \log(s)$ for $s \rightarrow \infty$, whereas the odd part of the $\astf$-function can't be constrained, at least by the process we considered. By assuming these constrains for the $\astf$-function, we could show that tree-level unitarity is satisfied if one incorporates the uncertainties present in the energy and the momenta of the scattered particles, \ie the uncertainties of the center-of-mass energy and the scattering angles. This uncertainties are not exclusively present due to the finite experimental resolution. A delta-like center-of-mass energy as well as delta-like momenta are in general not possible because the scattered particles are never exact plane waves.
In this thesis we study various aspects of chaos synchronization of time-delayed coupled chaotic maps. A network of identical nonlinear units interacting by time-delayed couplings can synchronize to a common chaotic trajectory. Even for large delay times the system can completely synchronize without any time shift. In the first part we study chaotic systems with multiple time delays that range over several orders of magnitude. We show that these time scales emerge in the Lyapunov spectrum: Different parts of the spectrum scale with the different delays. We define various types of chaos depending on the scaling of the maximum exponent. The type of chaos determines the synchronization ability of coupled networks. This is, in particular, relevant for the synchronization properties of networks of networks where time delays within a subnetwork are shorter than the corresponding time delays between the different subnetworks. If the maximum Lyapunov exponent scales with the short intra-network delay, only the elements within a subnetwork can synchronize. If, however, the maximum Lyapunov exponent scales with the long inter-network connection, complete synchronization of all elements is possible. The results are illustrated analytically for Bernoulli maps and numerically for tent maps. In the second part the attractor dimension at the transition to complete chaos synchronization is investigated. In particular, we determine the Kaplan-Yorke dimension from the spectrum of Lyapunov exponents for iterated maps. We argue that the Kaplan-Yorke dimension must be discontinuous at the transition and compare it to the correlation dimension. For a system of Bernoulli maps we indeed find a jump in the correlation dimension. The magnitude of the discontinuity in the Kaplan-Yorke dimension is calculated for networks of Bernoulli units as a function of the network size. Furthermore the scaling of the Kaplan-Yorke dimension as well as of the Kolmogorov entropy with system size and time delay is investigated. Finally, we study the change in the attractor dimension for systems with parameter mismatch. In the third and last part the linear response of synchronized chaotic systems to small external perturbations is studied. The distribution of the distances from the synchronization manifold, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transmitted signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments. The linear response is also quantified by means of the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is studied numerically for Bernoulli, tent and logistic maps. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devil's staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bi-directionally coupled chain of three units can completely filter out the perturbation. Thus the second moment and the bit error rate become zero.
We investigated emission properties of photonic structures with InAs/InGaAlAs/InP quantum dashes grown by molecular beam epitaxy on a distributed Bragg reflector. In high-spatial-resolution photoluminescence experiment, well-resolved sharp spectral lines are observed and single-photon emission is detected in the third telecommunication window characterized by very low multiphoton events probabilities. The photoluminescence spectra measured on simple photonic structures in the form of cylindrical mesas reveal significant intensity enhancement by a factor of 4 when compared to a planar sample. These results are supported by simulations of the electromagnetic field distribution, which show emission extraction efficiencies even above 18% for optimized designs. When combined with relatively simple and undemanding fabrication approach, it makes this kind of structures competitive with the existing solutions in that spectral range and prospective in the context of efficient and practical single-photon sources for fiber-based quantum networks applications.
Hochenergetische solare Teilchen werden bei ihrem Transport durch die Heliosphäre an turbulenten Magnetfeldern gestreut. Für das Verständnis dieses Streuprozesses ergeben sich aus heutiger Sicht zwei wesentliche Hindernisse: - Bei der Streuung hochenergetischer Teilchen an turbulenten Magnetfeldern handelt es sich um einen nichtlinearen Prozess, der durch analytische Theorien kaum zu beschreiben ist. - Der Streuprozess hängt stark von den tatsächlichen Magnetfeldern und somit auch von der Magnetfeldturbulenz ab. Unser bisheriges Verständnis der heliosphärischen Turbulenz ist leider aufgrund spärlicher experimenteller Daten deutlich eingeschränkt, was eine qualifizierte Umsetzung in analytischen und numerischen Ansätzen deutlich erschwert. Dies machte in der Vergangenheit künstliche Annahmen für die Modellerstellung notwendig. In dieser Arbeit wird der Teilchentransport mit Hilfe der Simulation von Testteilchen in einem turbulenten, magnetohydrodynamischen Plasma untersucht. Durch die Testteilchen werden auch die nichtlinearen Streuprozesse korrekt wiedergegeben, wodurch das erste hier genannte Hindernis überwunden wird. Dies wurde auch bereits in früheren numerischen Untersuchungen erfolgreich angewendet. Die Modellierung der Turbulenz für den Fall des Teilchentransports erfolgt in dieser Arbeit erstmalig auf Grundlage der magnetohydrodynamischen Gleichungen. Dabei handelt es sich um die mathematisch korrekte Wiedergabe der Magnetfeldturbulenz unterhalb der Ionen-Gyrofrequenz mit nur geringen numerischen Einschränkungen. Darüber hinaus erlaubt ein auf das physikalische Szenario anpassbarer Turbulenztreiber eine noch realistischere Simulation der Turbulenz. Durch diesen universell gültigen, numerischen Ansatz können für das zweite hier angegebene Hindernis jegliche künstlichen Annahmen vermieden werden. Die drei im Rahmen dieser Arbeit erstmals zusammengeführten Methoden (Testteilchen, magnetohydrodynamische Turbulenz, Turbulenztreiber) ermöglichen somit eine Untersuchung und Analyse von Transport- und Turbulenzphänomenen mit herausragender Qualität, die insbesondere für den Fall des Teilchentransports einen direkten Anschluss an experimentelle Ergebnisse ermöglichen. Wichtige Ergebnisse im Rahmen dieser Arbeit sind: - der Nachweis der Drei-Wellen-Wechselwirkung für schwache und einsetzende starke Turbulenz. - eine Analyse der Anisotropie der Turbulenz im Bezug auf das Hintergrundmagnetfeld in Abhängigkeit vom Treibmodell. Insbesondere die Anisotropie ist experimentell bislang kaum erfassbar. - eine Untersuchung der Auswirkung der Gyroresonanzen auf die Diffusionskoeffizienten hochenergetischer solarer Teilchen in allgemeiner Form. - die Simulation des Teilchentransports in der Heliosphäre auf Grundlage experimenteller Messdaten. Die genauere Analyse der Simulationsergebnisse ermöglicht insgesamt einen Zugang zum Verständnis des Transports, der durch experimentelle Untersuchungen nicht erfassbar ist. Bei der Simulation wurden lediglich die Magnetfeldstärke sowie die untersuchte Teilchenenergie vorgegeben. Aus der Analyse der Simulationsergebnisse ergibt sich dieselbe mittlere freie Weglänge, wie sie auch durch andere Verfahren direkt aus den Messergebnissen gewonnen werden konnte. Auch die vorwiegende Ausrichtung der hochenergetischen Teilchen parallel und antiparallel zum Hintergrundmagnetfeld in der Simulation entspricht experimentellen Untersuchungen. Es zeigt sich, dass diese allein aus den resonanten Streuprozessen der Teilchen mit den Magnetfeldern resultiert. Des Weiteren werden die Art der Diffusion, der Energieverlust der Teilchen während des Transportprozesses sowie die Gültigkeit der quasilinearen Theorie untersucht.
We review the particle physics ingredients affecting the normalization, shape, and flavor composition of astrophysical neutrinos fluxes, such as different production modes, magnetic field effects on the secondaries muons, pions, and kaons, and flavor mixing, where we focus on p? interactions. We also discuss the interplay with neutrino propagation and detection, including the possibility to detect flavor and its application in particle physics, and the use of the Glashow resonance to discriminate p? from pp interactions in the source. We illustrate the implications on fluxes and flavor composition with two different models: 1 the target photon spectrum is dominated by synchrotron emission of coaccelerated electrons and 2 the target photon spectrum follows the observed photon spectrum of gamma-ray bursts. In the latter case, the multimessenger extrapolation from the gamma-ray fluence to the expected neutrino flux is highlighted.
In this discussion session, the sensitivity and optimization of future long-baseline experiments is addressed, with a special emphasis on feasible projects and the description in terms of the error on the parameters. In addition, a statement on the precision interesting for \(ν_e → ν_τ\) and \(ν_μ → ν_τ\)oscillation measurements is obtained. A special topic is the impact of the recent T2K hint for non-zero \(θ_{13}\).
In the course of the growth of the Internet and due to increasing availability of data, over the last two decades, the field of network science has established itself as an own area of research. With quantitative scientists from computer science, mathematics, and physics working on datasets from biology, economics, sociology, political sciences, and many others, network science serves as a paradigm for interdisciplinary research.
One of the major goals in network science is to unravel the relationship between topological graph structure and a network’s function. As evidence suggests, systems from the same fields, i.e. with similar function, tend to exhibit similar structure. However, it is still vague whether a similar graph structure automatically implies likewise function. This dissertation aims at helping to bridge this gap, while particularly focusing on the role of triadic structures.
After a general introduction to the main concepts of network science, existing work devoted to the relevance of triadic substructures is reviewed. A major challenge in modeling triadic structure is the fact that not all three-node subgraphs can be specified independently
of each other, as pairs of nodes may participate in multiple of those triadic subgraphs.
In order to overcome this obstacle, we suggest a novel class of generative network models based on so called Steiner triple systems. The latter are partitions of a graph’s vertices into pair-disjoint triples (Steiner triples). Thus, the configurations on Steiner triples can be specified independently of each other without overdetermining the network’s link
structure.
Subsequently, we investigate the most basic realization of this new class of models. We call it the triadic random graph model (TRGM). The TRGM is parametrized by a probability distribution over all possible triadic subgraph patterns. In order to generate a network instantiation of the model, for all Steiner triples in the system, a pattern is drawn from the distribution and adjusted randomly on the Steiner triple. We calculate the degree distribution of the TRGM analytically and find it to be similar to a Poissonian distribution. Furthermore, it is shown that TRGMs possess non-trivial triadic structure. We discover inevitable correlations in the abundance of certain triadic subgraph
patterns which should be taken into account when attributing functional relevance to particular motifs – patterns which occur significantly more frequently than expected at random. Beyond, the strong impact of the probability distributions on the Steiner triples on the occurrence of triadic subgraphs over the whole network is demonstrated. This interdependence allows us to design ensembles of networks with predefined triadic substructure. Hence, TRGMs help to overcome the lack of generative models needed for assessing the relevance of triadic structure.
We further investigate whether motifs occur homogeneously or heterogeneously distributed over a graph. Therefore, we study triadic subgraph structures in each node’s neighborhood individually. In order to quantitatively measure structure from an individual node’s perspective, we introduce an algorithm for node-specific pattern mining for both directed unsigned, and undirected signed networks. Analyzing real-world datasets, we find that there are networks in which motifs are distributed highly heterogeneously, bound to the proximity of only very few nodes. Moreover, we observe indication for the potential sensitivity of biological systems to a targeted removal of these critical vertices. In addition, we study whole graphs with respect to the homogeneity and homophily of their node-specific triadic structure. The former describes the similarity of subgraph distributions in the neighborhoods of individual vertices. The latter quantifies whether connected vertices
are structurally more similar than non-connected ones. We discover these features to be characteristic for the networks’ origins. Moreover, clustering the vertices of graphs regarding their triadic structure, we investigate structural groups in the neural network of C. elegans, the international airport-connection network, and the global network of diplomatic sentiments between countries. For the latter we find evidence for the instability of triangles considered socially unbalanced according to sociological theories.
Finally, we utilize our TRGM to explore ensembles of networks with similar triadic substructure in terms of the evolution of dynamical processes acting on their nodes. Focusing on oscillators, coupled along the graphs’ edges, we observe that certain triad motifs impose a clear signature on the systems’ dynamics, even when embedded in a larger
network structure.
Numerical Simulations of Heavy Fermion Systems: From He-3 Bilayers to Topological Kondo Insulators
(2014)
Even though heavy fermion systems have been studied for a long time, a strong interest in heavy fermions persists to this day. While the basic principles of local moment formation, Kondo effect and formation of composite quasiparticles leading to a Fermi liquid, are under- stood, there remain many interesting open questions. A number of issues arise due to the interplay of heavy fermion physics with other phenomena like magnetism and superconduc- tivity.
In this regard, experimental and theoretical investigations of He-3 can provide valuable insights. He-3 represents a unique realization of a quantum liquid. The fermionic nature of He-3 atoms, in conjunction with the absence of long-range Coulomb repulsion, makes this material an ideal model system to study Fermi liquid behavior.
Bulk He-3 has been investigated for quite some time. More recently, it became possible to prepare and study layered He-3 systems, in particular single layers and bilayers. The pos- sibility of tuning various physical properties of the system by changing the density of He-3 and using different substrate materials makes layers of He-3 an ideal quantum simulator for investigating two-dimensional Fermi liquid phenomenology.
In particular, bilayers of He-3 have recently been found to exhibit heavy fermion behavior. As a function of temperature, a crossover from an incoherent state with decoupled layers to a coherent Fermi liquid of composite quasiparticles was observed. This behavior has its roots in the hybridization of the two layers. The first is almost completely filled and subject to strong correlation effects, while the second layer is only partially filled and weakly correlated. The quasiparticles are formed due to the Kondo screening of localized moments in the first layer by the second-layer delocalized fermions, which takes place at a characteristic temperature scale, the coherence scale Tcoh.
Tcoh can be tuned by changing the He-3 density. In particular, at a certain critical filling,
the coherence scale is expected to vanish, corresponding to a divergence of the quasiparticle effective mass, and a breakdown of the Kondo effect at a quantum critical point. Beyond the critical point, the layers are decoupled. The first layer is a local moment magnet, while the second layer is an itinerant overlayer.
However, already at a filling smaller than the critical value, preempting the critical point, the onset of a finite sample magnetization was observed. The character of this intervening phase remained unclear.
Motivated by these experimental observations, in this thesis the results of model calcula- tions based on an extended Periodic Anderson Model are presented. The three particle ring exchange, which is the dominant magnetic exchange process in layered He-3, is included in the model. It leads to an effective ferromagnetic interaction between spins on neighboring sites. In addition, the model incorporates the constraint of no double occupancy by taking the limit of large local Coulomb repulsion.
By means of Cellular DMFT, the model is investigated for a range of values of the chemical potential µ and inverse temperature β = 1/T . The method is a cluster extension to the Dy- namical Mean-Field Theory (DMFT), and allows to systematically include non-local correla- tions beyond the DMFT. The auxiliary cluster model is solved by a hybridization expansion CTQMC cluster solver, which provides unbiased, numerically exact results for the Green’s function and other observables of interest.
As a first step, the onset of Fermi liquid coherence is studied. At low enough temperature, the self-energy is found to exhibit a linear dependence on Matsubara frequency. Meanwhile, the spin susceptibility crossed over from a Curie-Weiss law to a Pauli law. Both observations serve as fingerprints of the Fermi liquid state.
The heavy fermion state appears at a characteristic coherence scale Tcoh. This scale depends strongly on the density. While it is rather high for small filling, for larger filling Tcoh is increas- ingly suppressed. This involves a decreasing quasiparticle residue Z ∼ Tcoh and an enhanced mass renormalization m∗/m ∼ Tcoh−1. Extrapolation leads to a critical filling, where the co-
herence scale is expected to vanish at a quantum critical point. At the same time, the effective mass diverges. This corresponds to a breakdown of the Kondo effect, which is responsible for the formation of quasiparticles, due to a vanishing of the effective hybridization between the layers.
Taking only single-site DMFT results into account, the above scenario seems plausible. However, paramagnetic DMFT neglects the ring exchange interaction completely. In or- der to improve on this, Cellular DMFT simulations are conducted for small clusters of size Nc = 2 and 3. The results paint a different physical picture. The ring exchange, by favor- ing a ferromagnetic alignment of spins, competes with the Kondo screening. As a result, strong short-range ferromagnetic fluctuations appear at larger values of µ. By lowering the temperature, these fluctuations are enhanced at first. However, for T < Tcoh they are increas- ingly suppressed, which is consistent with Fermi liquid coherence. However, beyond a certain threshold value of µ, fluctuations persist to the lowest temperatures. At the same time, while not apparent in the DMFT results, the total occupation n increases quite strongly in a very narrow range around the same value of µ. The evolution of n with µ is always continuous, but hints at a discontinuity in the limit Nc → ∞. This first-order transition breaks the Kondo effect. Beyond the transition, a ferromagnetic state in the first layer is established, and the second layer becomes a decoupled overlayer.
These observations provide a quite appealing interpretation of the experimental results. As a function of chemical potential, the Kondo breakdown quantum critical point is preempted by a first-order transition, where the layers decouple and the first layer turns into a ferromagnet. In the experimental situation, where the filling can be tuned directly, the discontinuous transition is mirrored by a phase separation, which interpolates between the Fermi liquid ground state at lower filling and the magnetic state at higher filling. This is precisely the range of the intervening phase found in the experiments, which is characterized by an onset of a finite sample magnetization.
Besides the interplay of heavy fermion physics and magnetic exchange, recently the spin- orbit coupling, which is present in many heavy fermion materials, attracted a lot of interest. In the presence of time-reversal symmetry, due to spin-orbit coupling, there is the possibility of a topological ground state.
It was recently conjectured that the energy scale of spin-orbit coupling can become dom- inant in heavy fermion materials, since the coherence scale and quasiparticle bandwidth are rather small. This can lead to a heavy fermion ground state with a nontrivial band topology; that is, a topological Kondo insulator (TKI). While being subject to strong correlation effects, this state must be adiabatically connected to a non-interacting, topological state.
The idea of the topological ground state realized in prototypical Kondo insulators, in par- ticular SmB6, promises to shed light on some of the peculiarities of these materials, like a residual conductivity at the lowest temperatures, which have remained unresolved so far.
In this work, a simple two-band model for two-dimensional topological Kondo insulators is devised, which is based on a single Kramer’s doublet coupled to a single conduction band. The model is investigated in the presence of a Hubbard interaction as a function of interaction strength U and inverse temperature β. The bulk properties of the model are obtained by DMFT, with a hybridization expansion CTQMC impurity solver. The DMFT approximation of a local self-energy leads to a very simple way of computing the topological invariant.
The results show that with increasing U the system can be driven through a topological phase transition. Interestingly, the transition is between distinct topological insulating states, namely the Γ-phase and M-phase. This appearance of different topological phases is possible due to the symmetry of the underlying square lattice. By adiabatically connecting both in- teracting states with the respective non-interacting state, it is shown that the transition indeed drives the system from the Γ-phase to the M-phase.
A different behavior can be observed by pushing the bare position of the Kramer’s doublet to higher binding energies. In this case, the non-interacting starting point has a trivial band topology. By switching on the interaction, the system can be tuned through a quantum phase transition, with a closing of the band gap. Upon reopening of the band gap, the system is in the Γ-phase, i. e. a topological insulator. By increasing the interaction strength further, the system moves into a strongly correlated regime. In fact, close to the expected transition to the M phase, the mass renormalization becomes quite substantial. While absent in the para- magnetic DMFT simulations conducted, it is conceivable that instead of a topological phase transition, the system undergoes a time-reversal symmetry breaking, magnetic transition.
The regime of strong correlations is studied in more detail as a function of temperature, both in the bulk and with open boundary conditions. A quantity which proved very useful is the bulk topological invariant Ns, which can be generalized to finite interaction strength and temperature. In particular, it can be used to define a temperature scale T ∗ for the onset of the topological state. Rescaling the results for Ns, a nice data collapse of the results for different values of U, from the local moment regime to strongly mixed valence, is obtained. This hints at T ∗ being a universal low energy scale in topological Kondo insulators. Indeed, by comparing T ∗ with the coherence scale extracted from the self-energy mass renormalization, it is found that both scales are equivalent up to a constant prefactor. Hence, the scale T ∗ obtained from the temperature dependence of topological properties, can be used as an independent measure for Fermi liquid coherence. This is particularly useful in the experimentally relevant mixed valence regime, where charge fluctuations cannot be neglected. Here, a separation of the energy scales related to spin and charge fluctuations is not possible.
The importance of charge fluctuations becomes evident in the extent of spectral weight transfer as the temperature is lowered. For mixed valence, while the hybridization gap emerges, a substantial amount of spectral weight is shifted from the vicinity of the Fermi level to the lower Hubbard band. In contrast, this effect is strongly suppressed in the local moment regime.
In addition to the bulk properties, the spectral function for open boundaries is studied as a function of temperature, both in the local moment and mixed valence regime. This allows an investigation of the emergence of topological edge states with temperature. The method used here is the site-dependent DMFT, which is a generalization of the conventional DMFT to inhomogeneous systems. The hybridization expansion CTQMC algorithm is used as impurity solver.
By comparison with the bulk results for the topological quantity Ns, it is found that the
temperature scale for the appearance of the topological edge states is T ∗, both in the mixed valence and local moment regime.
The extragalactic gamma-ray sky is dominated by blazars, active galactic nuclei (AGN) with a relativistic jet that is closely aligned with the line of sight. Galaxies develop an active nucleus if the central supermassive black hole (BH) accretes large amounts of ambient matter and magnetic flux. The inflowing mass accumulates around the plane perpendicular to the accretion flow's angular momentum. The flow is heated through viscous friction and part of the released energy is radiated as blackbody or non-thermal radiation, with luminosities that can dominate the accumulated stellar luminosity of the host galaxy. A fraction of the accretion flow luminosity is reprocessed in a surrounding field of ionised gas clouds. These clouds, revolving around the central BH, emit Doppler-broadened atomic emission lines. The region where these broad-line-emitting clouds are located is called broad-line region (BLR).
About one in ten AGN forms an outflow of radiation and relativistic particles, called a relativistic jet. According to the Blandford-Znajek mechanism, this is facilitated through electromagnetic processes in the magnetosphere of a spinning BH. The latter induces a magnetospheric poloidal current circuit, generating a decelerating torque on the BH and inducing a toroidal magnetic field. Consequently, rotational energy of the BH is converted to Poynting flux streaming away mainly along the rotational axis and starting the jet. One possibility for particle acceleration near the jet base is realised by magnetospheric vacuum gaps, regions temporarily devoid of plasma, such that an intermittent electric field arises parallel to the magnetic field lines, enabling particle acceleration and contributing to the mass loading of the jets.
Magnetised structures, containing bunches of relativistic electrons, propagate away from the galactic nucleus along the jets. Assuming that these electrons emit synchrotron radiation and that they inverse-Compton (IC) up-scatter abundant target photons, which can either be the synchrotron photons themselves or photons from external emitters, the emitted spectrum can be theoretically determined. Additionally taking into account that these emission regions move relativistically themselves and that the emission is Doppler-boosted and beamed in forward direction, the typical two-hump spectral energy distribution (SED) of blazars is recovered.
There are however findings that challenge this well-established model. Short-time variability, reaching down to minute scales at very high energy gamma rays, is today known to be a widespread phenomenon of blazars, calling for very compact emission regions. In most models of such optically thick emission regions, the gamma-ray flux is usually pair-absorbed exponentially, without considering the cascade evolving from the pair-produced electrons. From the observed flux, it is often concluded that emission emanates from larger distances where the region is optically thin, especially from outside of the BLR. Only in few blazars gamma-ray attenuation associated with pair absorption in the BLR was clearly reported.
With the advent of sophisticated high-energy or very high energy gamma-ray detectors, like the Fermi Large Area Telescope or the Major Atmospheric Gamma-ray Imaging Cherenkov telescopes, besides the extraordinarily fast variability spectral features have been found that cannot be explained by conventional models reproducing the two-hump SED. Two such narrow spectral features are discussed in this work. For the nearby blazar Markarian 501, hints to a sharp peak around 3 TeV have been reported from a multi-wavelength campaign carried out in July 2014, while for 3C 279 a spectral dip was found in 2018 data, that can hardly be described with conventional fitting functions. In this work it is examined whether these spectral peculiarities of blazar jet emission can be explained, if the full radiation reprocessing through an IC pair cascade is accounted for.
Such a cascade is the multiple concatenation of IC scattering events and pair production events. In the cascades generally considered in this work, relativistic electrons and high-energy photons are injected into a fixed soft target photon field. A mathematical description for linear IC pair cascades with escape terms is delivered on the basis of preliminary works. The steady-state kinetic equations for the electrons and for the photons are determined, whereby it is paid attention to an explicit formulation and to motivating the correct integration borders of all integrals from kinematic constraints. In determining the potentially observable gamma-ray flux, both the attenuated injected flux and the flux evolving as an effect of IC up-scattering, pair absorption and escape are incorporated, giving the emerging spectra very distinct imprints.
Much effort is dedicated to the numerical solution of the electrons' kinetic equation via iterative schemes. It is explained why pointwise iteration from higher to lower Lorentz factors is more efficient than iterating the whole set of sampling points. The algorithm is parallelised at two positions. First, several workers can perform pointwise iterations simultaneously. Second, the most demanding integral is cut into a number of part integrals which can be determined by multiple workers. Through these measures, the Python code can be readily applied to simulate steady-state IC pair cascades with escape.
In the case of Markarian 501 the developed framework is as follows. The AGN hosts an advection-dominated accretion flow with a normalised accretion rate of several \(10^{-4}\) and an electron temperature near \(10^{10}\) K. On the one hand, the accretion flow illuminates the few ambient gas clouds with approximate radius \(10^{11}\) m, which reprocess a fraction 0.01 of the luminosity into hydrogen and helium emission lines. On the other hand, the gamma rays from the accretion flow create electrons and positrons in a sporadically active vacuum gap in the BH magnetosphere. In the active gap, a power of roughly 0.001 of the Blandford-Znajek power is extracted from the rotating BH through a gap potential drop of several \(10^{18}\) V, generating ultra-relativistic electrons, which subsequently are multiplied by a factor of about \(10^6\) through interaction with the accretion flow photons. This electron beam propagates away from the central engine and encounters the photon field of one passing ionised cloud. The resulting IC pair cascade is simulated and the evolving gamma-ray spectrum is determined. Just above the absorption troughs due to the hydrogen lines, the spectrum exhibits a narrow bump around 3 TeV. When the cascaded emission is added to the emission generated at larger distances, the observed multi-wavelength SED including the sharp peak at 3 TeV is reproduced, underlining that radiation processes beyond conventional models are motivated by distinct spectral features.
The dip in the spectrum of 3C 279 is addressed by a similar cascade model. Three types of injection are considered, varying in the ratio of the photon density to the electron density and varying in the spectral shape. The IC pair cascade is assumed to happen either in the dense BLR photon field with a luminosity of several \(10^{37}\) W and a radial size of few \(10^{14}\) m or in the diluted photon field outside of the BLR. The latter scenario is however rejected as the spectral slope around several 100 MeV and the dip at few 10 GeV cannot be reconciled within this model. The radiation cascaded in the BLR can explain the observational data, irrespective of the assumed injected rate. It is therefore concluded that for this period of gamma-ray emission, the radiation production happens at the edge of the BLR of 3C 279.
Both investigations show that IC pair cascades can account for fine structure seen in blazar SEDs. It is insufficient to restrict the radiation transport to pure exponential absorption of an injection term. Pair production and IC up-scattering by all generations of photons and electrons in the optically thick regime critically shape the emerging spectra. As the advent of future improved detectors will provide more high-precision spectra, further observations of narrow spectral features can be expected. It seems therefore recommendable to incorporate cascading into conventional radiation production models or to extend the model developed in this work by synchrotron radiation.
Im Rahmen eines selbst-konsistenten Outer-Gap-Modells der Pulsar-Magnetosphäre wurde die elektromagnetische sehr hochenergetische Strahlung des Crab-Pulsars simuliert. Dies wurde parallel anhand zweier verschiedener Fälle getan, die sich in den angenommenen Gleichungen für die elektrische Feldstärke und für den Krümmungsradius der magnetischen Feldlinien unterscheiden. Die Kinetik der geladenen Teilchen bei ihrer Propagation durch die Outer Gap wurde unter Einbeziehung von Krümmungsstrahlung, inverser Compton-Streuung und Triple Paarbildung betrachtet. Das theoretisch simulierte Spektrum wird mit von Fermi-LAT und von den MAGIC Teleskopen gemessenen Daten verglichen.