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This work focuses on a fundamental problem in modern numerical rela- tivity: Extracting gravitational waves in a coordinate and gauge independent way to nourish a unique and physically meaningful expression. We adopt a new procedure to extract the physically relevant quantities from the numerically evolved space-time. We introduce a general canonical form for the Weyl scalars in terms of fundamental space-time invariants, and demonstrate how this ap- proach supersedes the explicit definition of a particular null tetrad. As a second objective, we further characterize a particular sub-class of tetrads in the Newman-Penrose formalism: the transverse frames. We establish a new connection between the two major frames for wave extraction: namely the Gram-Schmidt frame, and the quasi-Kinnersley frame. Finally, we study how the expressions for the Weyl scalars depend on the tetrad we choose, in a space-time containing distorted black holes. We apply our newly developed method and demonstrate the advantage of our approach, compared with methods commonly used in numerical relativity.
One of the most popular extensions of the SM is Supersymmetry (SUSY). It is a symmetry relating fermions and bosons and also the only feasible extension to the symmetries of spacetime. With SUSY it is then possible to explain some of the open questions left by the SM while at the same time opening the possibility of gauge unification at a high scale. SUSY theories require the addition of new particles, in particular an extra Higgs doublet and at least as many new scalars as fermions in the SM. Much in the same way that the Higgs boson breaks SU (2)L symmetry, these new scalars can break any symmetry for which they carry a charge through spontaneous symmetry breaking.
Let us assume there is a local minimum of the potential that reproduces the correct phenomenol- ogy for a parameter point of a given model. By exploring whether there are other deeper minima with VEVs that break symmetries we want to conserve, like SU (3)C or U (1)EM , it is possible to exclude regions of parameter space where that happens. The local minimum with the correct phenomenology might still be metastable, so it is also necessary to calculate the probability of tunneling between minima.
In this work we propose and apply a framework to constrain the parameter space of models with many scalars through the minimization of the one-loop eff e potential and the calculation of tunneling times at zero and non zero temperature.After a brief discussion about the shortcomings of the SM and an introduction of the basics of SUSY, we introduce the theory and numerical methods needed for a successful vacuum stability analysis. We then present Vevacious, a public code where we have implemented our proposed framework. Afterwards we go on to analyze three interesting examples.
For the constrained MSSM (CMSSM) we explore the existence of charge- and color- breaking (CCB) minima and see how it constraints the phenomenological relevant region of its parameter space at T = 0. We show that the regions reproducing the correct Higgs mass and the correct relic density for dark matter all overlap with regions suffering from deeper CCB minima.
Inspired by the results for the CMSSM, we then consider the natural MSSM and check the region of parameter space consistent with the correct Higgs mass against CCB minima at T /= 0. We find that regions of parameter space with CCB minima overlap significantly with that reproducing the correct Higgs mass. When thermal eff are considered the majority of such points are then found to have a desired symmetry breaking minimum with very low survival probability. In both these studies we find that analytical conditions presented in the literature fail in dis- criminating regions of parameter space with CCB minima. We also present a way of adapting our framework so that it runs quickly enough for use with parameter fit studies.
Lastly we show a different example of using vacuum stability in a phenomenological study. For the BLSSM we investigate the violation of R-parity through sneutrino VEVs and where in parameter space does this happen. We find that previous analyses in literature fail to identify regions with R-parity conservation by comparing their results to our full numerical analysis.
A general theory for all classes of unconventional superconductors is still one of the unsolved key issues in condensed-matter physics. Actually, it is not yet fully settled if there is a common underlying pairing mechanism. Instead, it might be possible that several distinct sources for unconventional (not phonon-mediated) superconductivity have to be considered, or an electron-phonon interaction is not negligible. The focus of this thesis is on the most probable mechanism for the formation of Cooper pairs in unconventional superconductors, namely a strictly electronic one where spin fluctuations are the mediators. Studying different superconductors in this thesis, the emphasis is put on material-independent features of the pairing mechanism. In addition, the investigation of the phase diagrams enables a view on the vicinity of superconductivity. Thus, it is possible to clarify which competing quantum fluctuations enhance or weaken the propensity for a superconducting state. The broad range of superconducting materials requires the use of more than one numerical technique to study an appropriate microscopic description. This is not a problem but a big advantage because this facilitates the approach-independent description of common underlying physics. For this evaluation, the strongly correlated cuprates are simulated with the variational cluster approach. Especially the question of a pairing glue is taken into consideration. Furthermore, it is possible to distinguish between retarded and non-retarded contributions to the gap function. The cuprates are confronted with the cobaltate NaCoO and graphene. These weakly correlated materials are investigated with the functional renormalization group (fRG) and reveal a comprehensive phase diagram, including a d+id-wave superconductivity, which breaks time-reversal symmetry. The corresponding gap function is nodeless, but for NaCoO, it features a doping-dependent anisotropy. In addition, some general considerations on the kagome lattice are completing the discussion, where a sublattice interference dramatically affects the Fermi-surface instabilities, suppressing the usual spin-density wave and d+id-wave superconductivity. Thereby, some different fascinating charge and bond orders as well as a nematic are observable. In short, this thesis provides an insight to distinct classes of unconventional superconductors with appropriate simulation techniques. This facilitates to separate the material specific properties from the universal ones.
Despite its history of more than one hundred years, the phenomenon of
superconductivity has not lost any of its allure. During that time the concept
and perception of the superconducting state - both from an experimental and
theoretical point of view - has evolved in way that has
triggered increasing interest. What was initially believed to simply be the
disappearance of electrical resistivity, turned out to be a universal and
inevitable result of quantum statistics, characterized by many more
aspects apart from its zero resistivity. The insights of
BCS-theory eventually helped to uncover its deep connection to particle physics
and consequently led to the formulation of the Anderson-Higgs-mechanism. The
very core of this theory is the concept of gauge symmetry (breaking). Within the
framework of condensed-matter theory, gauge invariance is only one of several
symmetry groups which are crucial for the description and classification of
superconducting states. \\
In this thesis, we employ time-reversal, inversion, point group and spin
symmetries to investigate and derive possible Hamiltonians featuring spin-orbit
interaction in two and three spatial dimensions.
In particular, this thesis aims at a generalization of existing numerical
concepts to open up the path to spin-orbit coupled (non)centrosymmetric
superconductors in multi-orbital models.
This is done in a two-fold way: On the one hand, we formulate - based on the
Kohn-Luttinger effect - the perturbative renormalization group in the
weak-coupling limit. On the other hand, we define the spinful flow equations of
the effective action in the framework of functional renormalization, which is
valid for finite interaction strength as well. Both perturbative and functional
renormalization groups produce a low-energy effective (spinful) theory that
eventually gives rise to a particular superconducting state, which is investigated
on the level of the irreducible two-particle vertex. The symbiotic relationship
between both perturbative and functional renormalization can be traced back to
the fact that, while the perturbative renormalization at infinitesimal coupling
is only capable of dealing with the Cooper instability, the functional
renormalization can investigate a plethora of instabilities both in the
particle-particle and particle-hole channels. \\
Time-reversal and inversion are the two key symmetries, which are being used to
discriminate between two scenarios. If both time-reversal and inversion symmetry
are present, the Fermi surface will be two-fold degenerate and characterized by a
pseudospin degree of freedom. In contrast, if inversion symmetry is broken, the
Fermi surface will be spin-split and labeled by helicity. In both cases, we
construct the symmetry allowed states in the particle-particle as well as the
particle-hole channel. The methods presented are formally unified and implemented
in a modern object-oriented reusable and extendable C++ code.
This methodological implementation is employed to one member of both families of
pseudospin and helicity characterized systems. For the pseudospin case, we choose
the intriguing matter of strontium ruthenate, which has been heavily
investigated for already twenty-four years, but still keeps puzzling researchers.
Finally, as the helicity based application, we consider the oxide heterostructure
LaAlO$_{3}$/SrTiO$_{3}$, which became famous for its highly mobile two-
dimensional electron gas and is suspected to host topological superconductivity.
It is natural to consider the possibility that the most energetic particles detected (> 10^18 eV), ultra-high-energy cosmic rays (UHECRs), are originated at the most luminous transient events observed (> 10^52 erg s^-1), gamma-ray bursts (GRBs). As a result of the interaction of highly-accelerated, magnetically-confined protons and ions with the photon field inside the burst, both neutrons and UHE neutrinos are expected to be created: the former escape the source and beta-decay into protons which propagate to Earth, where they are detected as UHECRs, while the latter, if detected, would constitute the smoking gun of hadronic acceleration in the sources.
Recently, km-scale neutrino telescopes such as IceCube have finally reached the sensitivities required to probe the neutrino predictions of some of the existing GRB models. On that account, we present here a revised, self-consistent model of joint UHE proton and neutrino production at GRBs that includes a state-of-the-art, improved numerical calculation of the neutrino flux (NeuCosmA); that uses a generalised UHECR emission model where some of the protons in the sources are able to "leak out" of their magnetic confinement before having interacted; and that takes into account the energy losses of the protons during their propagation to Earth. We use our predictions to take a close look at the cosmic ray-neutrino connection and find that the current UHECR observations by giant air shower detectors, together with the upper bounds on the flux of neutrinos from GRBs, are already sufficient to put tension on several possibilities of particle emission and propagation, and to point us towards some requirements that should be fulfilled by GRBs if they are to be the sources of the UHECRs. We further refine our analysis by studying a dynamical burst model, where we find that the different particle species originate at distinct stages of the expanding GRB, each under particular conditions. Finally, we consider a possibility of new physics: the effect of neutrino decay in the flux of UHE neutrinos from GRBs. On the whole, our results demonstrate that self-consistent models of particle production are now integral to the advancement of the field, given that the full picture of the UHE Universe will only emerge as a result of looking at the multi-messenger sky, i.e., at gamma-rays, cosmic rays, and neutrinos simultaneously.
Two-particle excitations, such as spin and charge excitations, play a key role in high-Tc cuprate superconductors (HTSC). Due to the antiferromagnetism of the parent compound the magnetic excitations are supposed to be directly related to the mechanism of superconductivity. In particular, the so-called resonance mode is a promising candidate for the pairing glue, a bosonic excitation mediating the electronic pairing. In addition, its interactions with itinerant electrons may be responsible for some of the observed properties of HTSC. Hence, getting to the bottom of the resonance mode is crucial for a deeper understanding of the cuprate materials . To analyze the corresponding two-particle correlation functions we develop in the present thesis a new, non-perturbative and parameter-free technique for T=0 which is based on the Variational Cluster Approach (VCA, an embedded cluster method for one-particle Green's functions). Guided by the spirit of the VCA we extract an effective electron-hole vertex from an isolated cluster and use a fully renormalized bubble susceptibility chi0 including the VCA one-particle propagators.Within our new approach, the magnetic excitations of HTSC are shown to be reproduced for the Hubbard model within the relevant strong-coupling regime. Exceptionally, the famous resonance mode occurring in the underdoped regime within the superconductivity-induced gap of spin-flip electron-hole excitations is obtained. Its intensity and hourglass dispersion are in good overall agreement with experiments. Furthermore, characteristic features such as the position in energy of the resonance mode and the difference of the imaginary part of the susceptibility in the superconducting and the normal states are in accord with Inelastic Neutron Scattering (INS) experiments. For the first time, a strongly-correlated parameter-free calculation revealed these salient magnetic properties supporting the S=1 magnetic exciton scenario for the resonance mode. Besides the INS data on magnetic properties further important new insights were gained recently via ARPES (Angle-Resolved Photoemission-Spectroscopy) and Raman experiments which disclosed a quite different doping dependence of the antinodal compared to the near-nodal gap. This thesis provides an approach to the Raman response similar to the magnetic case for inspecting this gap dichotomy. In agreement with experiments and one-particle data obtained in the VCA, we recover the antinodal gap decreasing and the near-nodal gap increasing as a function of doping. Hence, our results prove the Hubbard model to account for these salient gap features. In summary, we develop a two-particle cluster approach which is appropriate for the strongly-correlated regime and contains no free parameter. Our results obtained with this new approach combined with the phase diagram and the one-particle excitations obtained in the VCA strongly constitute a Hubbard model description of HTSC cuprate materials.
Explaining the baryon asymmetry of the Universe has been a long-standing problem of particle physics, with the consensus being that new physics is required as the Standard Model (SM) cannot resolve this issue. Beyond the Standard Model (BSM) scenarios would need to incorporate new sources of \(CP\) violation and either introduce new departures from thermal equilibrium or modify the existing electroweak phase transition. In this thesis, we explore two approaches to baryogenesis, i.e. the generation of this asymmetry.
In the first approach, we study the two-particle irreducible (2PI) formalism as a means to investigate non-equilibrium phenomena. After arriving at the renormalised equations of motions (EOMs) to describe the dynamics of a phase transition, we discuss the techniques required to obtain the various counterterms in an on-shell scheme. To this end, we consider three truncations up to two-loop order of the 2PI effective action: the Hartree approximation, the scalar sunset approximation and the fermionic sunset approximation. We then reconsider the renormalisation procedure in an \(\overline{\text{MS}}\) scheme to evaluate the 2PI effective potential for the aforementioned truncations. In the Hartree and the scalar sunset approximations, we obtain analytic expressions for the various counterterms and subsequently calculate the effective potential by piecing together the finite contributions. For the fermionic sunset approximation, we obtain similar equations for the counterterms in terms of divergent parts of loop integrals. However, these integrals cannot be expressed in an analytic form, making it impossible to evaluate the 2PI effective potential with the fermionic contribution. Our main results are thus related to the renormalisation programme in the 2PI formalism: \( (i) \)the procedure to obtain the renormalised EOMs, now including fermions, which serve as the starting point for the transport equations for electroweak baryogenesis and \( (ii) \) the method to obtain the 2PI effective potential in a transparent manner.
In the second approach, we study baryogenesis via leptogenesis. Here, an asymmetry in the lepton sector is generated, which is then converted into the baryon asymmetry via the sphaleron process in the SM. We proceed to consider an extension of the SM along the lines of a scotogenic framework. The newly introduced particles are charged odd under a \(\mathbb{Z}_2\) symmetry, and masses for the SM neutrinos are generated radiatively. The \(\mathbb{Z}_2\) symmetry results in the lightest BSM particle being stable, allowing for a suitable dark matter (DM) candidate. Furthermore, the newly introduced heavy Majorana fermionic singlets provide the necessary sources of \(CP\) violation through their Yukawa interactions and their out-of-equilibrium decays produce a lepton asymmetry. This model is constrained from a wide range of observables, such as consistency with neutrino oscillation data, limits on branching ratios of charged lepton flavour violating decays, electroweak observables and obtaining the observed DM relic density. We study leptogenesis in this model in light of the results of a Markov chain Monte Carlo scan, implemented in consideration of the aforementioned constraints. Successful leptogenesis in this model, to account for the baryon asymmetry, then severely constrains the available parameter space.
The prediction and the experimental discovery of topological insulators has set the stage for a novel type of electronic devices. In contrast to conventional metals or semiconductors, this new class of materials exhibits peculiar transport properties at the sample surface, as conduction channels emerge at the topological boundaries of the system.
In specific materials with strong spin-orbit coupling, a particular form of a two-dimensional topological insulator, the quantum spin Hall state, can be observed.
Here, the respective one-dimensional edge channels are helical in nature, meaning that there is a locking of the spin orientation of an electron and its direction of motion.
Due to the symmetry of time-reversal, elastic backscattering off interspersed impurities is suppressed in such a helical system, and transport is approximately ballistic.
This allows in principle for the realization of novel energy-efficient devices, ``spintronic`` applications, or the formation of exotic bound states with non-Abelian statistics, which could be used for quantum computing.
The present work is concerned with the general transport properties of one-dimensional helical states. Beyond the topological protection mentioned above, inelastic backscattering can arise from various microscopic sources, of which the most prominent ones will be discussed in this Thesis. As it is characteristic for one-dimensional systems, the role of electron-electron interactions can be of major importance in this context.
First, we review well-established techniques of many-body physics in one dimension such as perturbative renormalization group analysis, (Abelian) bosonization, and Luttinger liquid theory. The latter allow us to treat electron interactions in an exact way.
Those methods then are employed to derive the corrections to the conductance in a helical transport channel, that arise from various types of perturbations.
Particularly, we focus on the interplay of Rashba spin-orbit coupling and electron interactions as a source of inelastic single-particle and two-particle backscattering. It is demonstrated, that microscopic details of the system, such as the existence of a momentum cutoff, that restricts the energy spectrum, or the presence of non-interacting leads attached to the system, can fundamentally alter the transport signature.
By comparison of the predicted corrections to the conductance to a transport experiment, one can gain insight about the microscopic processes and the structure of a quantum spin Hall sample.
Another important mechanism we analyze is backscattering induced by magnetic moments. Those findings provide an alternative interpretation of recent transport measurements in InAs/GaSb quantum wells.
Over the last two decades, accompanied by their prediction and ensuing realization, topological non-trivial materials like topological insulators, Dirac semimetals, and Weyl semimetals have been in the focus of mesoscopic condensed matter research. While hosting a plethora of intriguing physical phenomena all on their own, even more fascinating features emerge when superconducting order is included. Their intrinsically pronounced spin-orbit coupling leads to peculiar, time-reversal symmetry protected surface states, unconventional superconductivity, and even to the emergence of exotic bound states in appropriate setups.
This Thesis explores various junctions built from - or incorporating - topological materials in contact with superconducting order, placing particular emphasis on the transport properties and the proximity effect.
We begin with the analysis of Josephson junctions where planar samples of mercury telluride are sandwiched between conventional superconducting contacts. The surprising observation of pronounced excess currents in experiments, which can be well described by the Blonder-Tinkham-Klapwijk theory, has long been an ambiguous issue in this field, since the necessary presumptions are seemingly not met. We propose a resolution to this predicament by demonstrating that the interface properties in hybrid nanostructures of distinctly different materials yet corroborate these assumptions and explain the outcome. An experimental realization is feasible by gating the contacts. We then proceed with NSN junctions based on time-reversal symmetry broken Weyl semimetals and including superconducting order. Due to the anisotropy of the electron band structure, both the transport properties as well as the proximity effect depend substantially on the orientation of the interfaces between the materials. Moreover, an imbalance can be induced in the electron population between Weyl nodes of opposite chirality, resulting in a non-vanishing spin polarization of the Cooper pairs leaking into the normal contacts. We show that such a system features a tunable dipole character with possible applications in spintronics. Finally, we consider partially superconducting surface states of three-dimensional topological insulators. Tuning such a system into the so-called bipolar setup, this results in the formation of equal-spin Cooper pairs inside the superconductor, while simultaneously acting as a filter for non-local singlet pairing. The creation and manipulation of these spin-polarized Cooper pairs can be achieved by mere electronic switching processes and in the absence of any magnetic order, rendering such a nanostructure an interesting system for superconducting spintronics. The inherent spin-orbit coupling of the surface state is crucial for this observation, as is the bipolar setup which strongly promotes non-local Andreev processes.
In this work, we studied in great detail how the unknown parameters of the SUSY seesaw model can be determined from measurements of observables at or below collider energies, namely rare flavor violating decays of leptons, slepton pair production processes at linear colliders and slepton mass differences. This is a challenging task as there is an intricate dependence of the observables on the unknown seesaw, light neutrino and mSUGRA parameters. In order to separate these different influences, we first considered two classes of seesaw models, namely quasi-degenerate and strongly hierarchical right-handed neutrinos. As a generalisation, we presented a method that can be used to reconstruct the high energy seesaw parameters, among them the heavy right-handed neutrino masses, from low energy observables alone.
The hunt for topological materials is one of the main topics of recent research in condensed matter physics. We analyze the 4-band Luttinger model, which considers the total angular momentum \(j = 3/2\) hole states of many semiconductors. Our analysis shows that this model hosts a wide array of topological phases and allows analytical calculations of the related topological surface states. The existence of these surface states is highly desired due to their strong protection against perturbations.
In the first part of the thesis, we predict the existence of either one or two two-dimensional (2D) surface states of topological origin in the three-dimensional (3D) quadratic-node semimetal phase of the Luttinger model, called the Luttinger semimetal phase. We associate the origin of these states with the inverted order of s and p-orbital states in the band structure and approximate chiral symmetry around the node. Hence, our findings are essential for many materials, including HgTe, α-Sn, and iridate compounds. Such materials are often modified with strain engineering by growing the crystal on a substrate with a different lattice constant, which adds a deformation potential to the electrons. While tensile strain is often used to drive such materials into a gapped topological insulator regime, we apply compressive strain to induce a topological semimetal regime. Here, we differentiate between Dirac and Weyl semimetals based on inversion and time-reversal symmetry being simultaneously present or not. One major part of this thesis is the theoretical study of the evolution of the Luttinger semimetal surface states in these topological semimetal phases.
The relative strength of the compressive strain and typical bulk inversion asymmetry (BIA) terms allow the definition of a symmetry hierarchy in the system. The cubic symmetric \(O_h\) Luttinger model is the highest symmetry low-energy parent model. Since the BIA terms in the Weyl semimetal phase are small in most materials, we find a narrow energy and momentum range around the Weyl points where the surface states form Fermi arcs between two Weyl nodes with opposite chirality. Consequently, we see 2D momentum planes between the Weyl points, which can be considered as effective 2D Chern insulators with chiral edge states connecting the valence and conduction band in the bulk gap. Exceeding the range of the BIA terms, the compressive strain becomes dominating, and the system behaves like a Dirac semimetal with two doubly degenerate linear Dirac nodes in the band structure. For energies larger than the compressive strain strength, the quadratic terms in the Luttinger model dominate and surface band structure is indistinguishable from an unperturbed Luttinger semimetal. To conclude this symmetry hierarchy, we analyze the limit of the Luttinger model when the remote \(j = 1/2\)
electron states show a considerable hybridization with the \(j = 3/2\) hole states around the Fermi level. Here, the Luttinger model is not valid anymore and one needs to consider more complicated models, like the 6-band Kane Hamiltonian.
In the second part of this thesis, we analyze theoretically two different setups for s-wave superconductivity proximitized \(j = 3/2\) particles in Luttinger materials under a magnetic field. First, we explore a one-dimensional wire setup, where the intrinsic BIA of inversion asymmetric crystals opens a topological gap in the bulk states. In contrast to wires, modeled by a quadratic dispersion with Rashba or Dresselhaus spin-orbit coupling, we find two topological phase transitions due to the different effects of magnetic fields to \(|j_z| = 3/2\) heavy-hole (HH) and \(|j_z| = 1/2\) light-hole (LH) states. Second, we discuss a two-dimensional Josephson junction setup, where we find Andreev-bound states inside the superconducting gap. Here, the intrinsic spin-orbit coupling of the Luttinger model is sufficient to open a topological gap even in the presence of inversion symmetry. This originates from the hybridization of the light and heavy-hole bands in combination with the superconducting pairing.
Consequently, both setups can form Majorana-bound states at the boundaries of the system.
The existence of these states are highly relevant in the scientific community due to their nonabelian braiding statistics and stability against decoherence, making them a prime candidate for the realization of topological quantum computation. Majorana-bound states form at zero energy and are protected by the topological gap. We predict that our findings of the topological superconductor phase of the Luttinger model are valid for both semimetal and metal phases. Hence, our study is additionally relevant for metallic systems, like p-doped GaAs. This opens a new avenue for the search for topological superconductivity.
Next to the emergence of nearly isolated quantum systems such as ultracold atoms with unprecedented experimental tunability, the conceptualization of the eigenstate thermalization hypothesis (ETH) by Deutsch and Srednicki in the late 20th century has sparked exceptional interest in the mechanism of quantum thermalization. The ETH conjectures that the expectation value of a local observable within the quantum state of an isolated, interacting quantum system converges to the thermal equilibrium value at large times caused by a loss of phase coherence, referred to as dephasing. The thermal behavior within the quantum expectation value is traced back to the level of individual eigenstates, who locally act as a thermal bath to subsystems of the full quantum system and are hence locally indistinguishable to thermal states. The ETH has important implications for the understanding of the foundations of statistical mechanics, the quantum-to-classical transition, and the nature of quantum entanglement. Irrespective of its theoretical success, a rigorous proof has remained elusive so far. $$ \ $$
An alternative approach to explain thermalization of quantum states is given by the concept of typicality. Typicality deals with typical states \(\Psi\) chosen from a subspace of Hilbert space with energy \(E\) and small fluctuations \(\delta\) around it. It assumes that the possible microstates of this subspace of Hilbert space are uniformly distributed random vectors. This is inspired by the microcanonical ensemble in classical statistical mechanics, which assumes equal weights for all accessible microstates with energy \(E\) within an energy allowance \(\delta\). It follows from the ergodic hypothesis, which states that the time spent in each part of phase space is proportional to its volume leading to large time averages being equated to ensemble averages. In typicality, the Hilbert space of quantum mechanics is hence treated as an analogue of classical phase space where statistical and thermodynamic properties can be defined. Since typicality merely shifts assumptions of statistical mechanics to the quantum realm, it does not provide a complete understanding of the emergence of thermalization on a fundamental microscopic level. $$ \ $$
To gain insights on quantum thermalization and derive it from a microscopic approach, we exclusively consider the fundamental laws of quantum mechanics. In the joint work with T. Hofmann, R. Thomale and M. Greiter, on which this thesis reports, we explore the ETH in generic local Hamiltonians in a two-dimensional spin-\(1/2\) lattice with random nearest neighbor spin-spin interactions and random on-site magnetic fields. This isolated quantum system is divided into a small subsystem weakly coupled to the remaining part, which is assumed to be large and which we refer to as bath. Eigenstates of the full quantum system as well as the action of local operators on those can then be decomposed in terms of a product basis of eigenstates of the small subsystem and the bath. Central to our analysis is the fact that the coupling between the subsystem and the bath, represented in terms of the uncoupled product eigenbasis, is given by an energy dependent random band matrix, which is obtained from both analytical and numerical considerations. $$ \ $$
Utilizing the methods of Dyson-Brownian random matrix theory for random band matrices, we analytically show that the overlaps of eigenstates of the full quantum system with the uncoupled product eigenbasis are described by Cauchy-Lorentz distributions close to their respective peaks. The result is supported by an extensive numerical study using exact diagonalization, where the numerical parameters for the overlap curve agree with the theoretical calculation. The information on the decomposition of the eigenstates of the full quantum system enables us to derive the reduced density matrix within the small subsystem given the pure density matrix of a single eigenstate. We show that in the large bath limit the reduced density matrix converges to a thermal density matrix with canonical Boltzmann probabilities determined by renormalized energies of the small subsystem which are shifted from their bare values due the influence of the coupling to the bath. The behavior of the reduced density matrix is confirmed through a finite size scaling analysis of the numerical data. Within our calculation, we make use of the pivotal result, that the density of states of a local random Hamiltonian is given by a Gaussian distribution under very general circumstances. As a consequence of our analysis, the quantum expectation value of any local observable in the subsystem agrees with its thermal expectation value, which proves the validity of the ETH in the equilibrium phase for the considered class of random local Hamiltonians and elevates it from hypothesis to theory. $$ \ $$
Our analysis of quantum thermalization solely relies on the application of quantum mechanics to large systems, locality and the absence of integrability. With the self-averaging property of large random matrices, random matrix theory does not entail a statistical assumption, but is rather applied as a mathematical tool to extract information about the behavior of large quantum systems. The canonical distribution of statistical mechanics is derived without resorting to statistical assumptions such as the concepts of ergodicity or maximal entropy, nor assuming any characteristics of quantum states such as in typicality. In future research, with this microscopic approach it may become possible to exactly pinpoint the origin of failure of quantum thermalization, e.g. in systems that exhibit many body localization or many body quantum scars. The theory further enables the systematic investigation of equilibration, i.e. to study the time scales on which thermalization takes place.
The four-dimensional Minkowski space is known to be a good description for space-time down to the length scales probed by the latest high-energy experiments. Nevertheless, there is the viable and exciting possibility that additional space-time structure will be observable in the next generation of collider experiments. Hence, we discuss different extensions of the standard model of particle physics with an extra dimension at the TeV-scale. We assume that some of the gauge and Higgs bosons propagate in one additional spatial dimension, while matter fields are confined to a four-dimensional subspace, the usual Minkowski space. After compactification on an S^1/Z_2 orbifold, an effective four-dimensional theory is obtained where towers of Kaluza-Klein (KK) modes, in addition to the standard model fields, reflect the higher-dimensional structure of space-time. The models are elaborated from the 5D Lagrangian to the Feynman rules of the KK modes. Special attention is paid to an appropriate generalization of the Rxi-gauge and the interplay between spontaneous symmetry breaking and compactification. Confronting the observables in 5D standard model extensions with combined precision measurements at the Z-boson pole and the latest data from LEP2, we constrain the possible size R of the extra dimension experimentally. A multi-parameter fit of all relevant input parameters leads to bounds for the compactification scale M=1/R in the range 4-6 TeV at the 2 sigma confidence level and shows how the mass of the Higgs boson is correlated with the size of an extra dimension. Considering a future linear e+e- collider, we outline the discovery potential for an extra dimension using the proposed TESLA specifications as an example. As a consistency check for the various models, we analyze Ward identities and the gauge boson equivalence theorem in W-pair production and find that gauge symmetry is preserved by a complex interplay of the Kaluza-Klein modes. In this context, we point out the close analogy between the traditional Higgs mechanism and mass generation for gauge bosons via compactification. Beyond the tree-level, the higher-dimensional models studied extensively in the literature and in the first part of this thesis have to be extended. We modify the models by the inclusion of brane kinetic terms which are required as counter terms. Again, we derive the corresponding 4D theory for the KK towers paying special attention to gauge fixing and spontaneous symmetry breaking. Finally, the phenomenological implications of the new brane kinetic terms are investigated in detail.
Despite its precise agreement with the experiment, the validity of the standard model (SM) of elementary particle physics is ensured only up to a scale of several hundred GeV so far. Even more, the inclusion of gravity into an unifying theory poses a problem which cannot be solved by ordinary quantum field theory (QFT). String theory, which is the most popular ansatz for a unified theory, predicts QFT on noncommutative space-time as a low energy limit. Nevertheless, independently of the motivation given by string theory, the nonlocality inherent to noncommutative QFT opens up the possibility for the inclusion of gravity. There are no theoretical predictions for the energy scale Lambda_NC at which noncommutative effects arise and it can be assumed to lie in the TeV range, which is the energy range probed by the next generation of colliders. Within this work we study the phenomenological consequences of a possible realization of QFT on noncommutative space-time relying on this assumption. The motivation for this thesis was given by the gap in the range of phenomenological studies of noncommutative effects in collider experiments, due to the absence in the literature of Large Hadron Collider (LHC) studies regarding noncommutative QFTs. In the first part we thus performed a phenomenological analysis of the hadronic process pp -> Z gamma -> l^+l^- gamma at the LHC and of electron-positron pair annihilation into a Z boson and a photon at the International Linear Collider (ILC). The noncommutative extension of the SM considered within this work relies on two building blocks: the Moyal-Weyl star-product of functions on ordinary space-time and the Seiberg-Witten maps. The latter relate the ordinary fields and parameters to their noncommutative counterparts such that ordinary gauge transformations induce noncommutative gauge transformations. This requirement is expressed by a set of inhomogeneous differential equations (the gauge equivalence equations) which are solved by the Seiberg-Witten maps order by order in the noncommutative parameter Theta. Thus, by means of the Moyal-Weyl star-product and the Seiberg-Witten maps a noncommutative extension of the SM as an effective theory as expansion in powers of Theta can be achieved, providing the framework of our phenomenological studies. A consequence of the noncommutativity of space-time is the violation of rotational invariance with respect to the beam axis. This effect shows up in the azimuthal dependence of cross sections, which is absent in the SM as well as in other models beyond the SM. Thus, the azimuthal dependence of the cross section is a typical signature of noncommutativity and can be used in order to discriminate it against other new physics effects. We have found this dependence to be best suited for deriving the sensitivity bounds on the noncommutative scale Lambda_NC. By studying pp -> Z gamma -> l^+l^- gamma to first order in the noncommutative parameter Theta, we show in the first part of this work that measurements at the LHC are sensitive to noncommutative effects only in certain cases, giving bounds on the noncommutative scale of Lambda_NC > 1.2 TeV. Our result improved the bounds present in the literature coming from past and present collider experiments by one order of magnitude. In order to explore the whole parameter range of the noncommutativity, ILC studies are required. By means of e^+e^- -> Z gamma -> l^+l^- gamma to first order in Theta we have shown that ILC measurements are complementary to LHC measurements of the noncommutative parameters. In addition, the bounds on Lambda_NC derived from the ILC are significantly higher and reach Lambda_NC > 6 TeV. The second part of this work arose from the necessity to enlarge the range of validity of our model towards higher energies. Thus, we expand the neutral current sector of the noncommutative SM to second order in $\theta$. We found that, against the general expectation, the theory must be enlarged by additional parameters. The new parameters enter the theory as ambiguities of the Seiberg-Witten maps. The latter are not uniquely determined and differ by homogeneous solutions of the gauge equivalence equations. The expectation was that the ambiguities correspond to field redefinitions and therefore should vanish in scattering matrix elements. However, we proved that this is not the case, and the ambiguities do affect physical observables. Our conjecture is, that every order in Theta will introduce new parameters to the theory. However, only the experiment can decide to what extent efforts with still higher orders in Theta are reasonable and will also give directions for the development of theoretical models of noncommutative QFTs.
We apply an antiferromagnetic symmetry breaking implementation of the dynamical cluster approximation (DCA) to investigate the two-dimensional hole-doped Kondo lattice model (KLM) with hopping $t$ and coupling $J$. The DCA is an approximation at the level of the self-energy. Short range correlations on a small cluster, which is self-consistently embedded in the remaining bath electrons of the system, are handled exactly whereas longer ranged spacial correlations are incorporated on a mean-field level. The dynamics of the system, however, are retained in full. The strong temporal nature of correlations in the KLM make the model particularly suitable to investigation with the DCA. Our precise DCA calculations of single particle spectral functions compare well with exact lattice QMC results at the particle-hole symmetric point. However, our DCA version, combined with a QMC cluster solver, also allows simulations away from particle-hole symmetry and has enabled us to map out the magnetic phase diagram of the model as a function of doping and coupling $J/t$. At half-filling, our results show that the linear behaviour of the quasi-particle gap at small values of $J/t$ is a direct consequence of particle-hole symmetry, which leads to nesting of the Fermi surface. Breaking the symmetry, by inclusion of a diagonal hopping term, results in a greatly reduced gap which appears to follow a Kondo scale. Upon doping, the magnetic phase observed at half-filling survives and ultimately gives way to a paramagnetic phase. Across this magnetic order-disorder transition, we track the topology of the Fermi surface. The phase diagram is composed of three distinct regions: Paramagnetic with {\it large} Fermi surface, in which the magnetic moments are included in the Luttinger sum rule, lightly antiferromagnetic with large Fermi surface topology, and strongly antiferromagnetic with {\it small} Fermi surface, where the magnetic moments drop out of the Luttinger volume. We draw on a mean-field Hamiltonian with order parameters for both magnetisation and Kondo screening as a tool for interpretation of our DCA results. Initial results for fixed coupling and doping but varying temperature are also presented, where the aim is look for signals of the energy scales in the system: the Kondo temperature $T_{K}$ for initial Kondo screening of the magnetic moments, the Neel temperature $T_{N}$ for antiferromagnetic ordering, a possible $T^{*}$ at which a reordering of the Fermi surface is observed, and finally, the formation of the coherent heavy fermion state at $T_{coh}$.
The Galactic Starburst Region NGC 3603 : exciting new insights on the formation of high mass stars
(2004)
One of the most fundamental, yet still unsolved problems in star formation research is addressed by the question "How do high mass stars form?". While most details related to the formation and early evolution of low mass stars are quite well understood today, the basic processes leading to the formation of high mass stars still remain a mystery. There is no doubt that low mass stars like our Sun form via accretion of gas and dust from their natal environment. With respect to the formation of high mass stars theorists currently discuss two possible scenarios controversely: First, similar to stars of lower masses, high mass stars form by continuous (time variable) accretion of large amounts of gas and dust through their circumstellar envelopes and/or disks. Second, high mass stars form by repeated collisions (coalescence) of protostars of lower masses. Both scenarios bear difficulties which impose strong constrains on the final mass of the young star. To find evidences for or against one of these two theoretical models is a challenging task for observers. First, sites of high mass star formation are much more distant than the nearby sites of low mass star formation. Second, high mass stars form and evolve much faster than low mass star. In particular, they contract to main sequence, hydrogen burning temperatures and densities on time scales which are much shorter than typical accretion time scales. Third, as a consequence of the previous point, young high mass stars are usually deeply embedded in their natal environment throughout their (short) pre-main sequence phase. Therefore, high mass protostars are rare, difficult to find and difficult to study. In my thesis I undertake a novel approach to search for and to characterize high mass protostars, by looking into a region where young high mass stars form in the violent neighbourhood of a cluster of early type main sequence stars. The presence of already evolved O type stars provides a wealth of energetic photons and powerful stellar winds which evaporate and disperse the surrounding interstellar medium, thus "lifting the courtains" around nearby young stars at a relatively early evolutionary stage. Such premises are given in the Galactic starburst region NGC 3603. Nevertheless, a large observational effort with different telescopes and instruments -- in particular, taking advantage of the high angular resolution and high sensitivity of near and mid IR instruments available at ESO -- was necessary to achieve the goals of my study. After a basic introduction on the topic of (high mass) star formation in Chapter 1, a short overview of the investigated region NGC 3603 and its importance for both galactic and extragalactic star formation studies is given in Chapter 2. Then, in Chapter 3, I report on a comprehensive investigation of the distribution and kinematics of the molecular gas and dust associated with the NGC 3603 region. In Chapter 4 I thoroughly address the radial extent of the NGC 3603 OB cluster and the spatial distribution of the cluster members. Together with deep Ks band imaging data, a detailed survey of NGC 3603 at mid IR wavelengths allows to search the neighbourhood of the cold molecular gas and dust for sources with intrinsic mid IR excess (Chapter 5). In Chapter 6 I characterize the most prominent sources of NGC 3603 IRS 9 and show that these sources are bona-fide candidates for high mass protostars. Finally, a concise summary as well as an outlook on future prospects in high mass star formation research is given in Chapter 7.
In this thesis, the electronic transport properties of mesoscopic condensed matter systems based on graphene are investigated by means of numerical as well as analytical methods. In particular, it is analyzed how the concepts of quantum interference and disorder, which are essential to mesoscopic devices in general, are affected by the unique electronic and transport properties of the graphene material system. We consider the famous Aharonov–Bohm effect in ring-shaped transport geometries, and, besides providing an overview over the recent developments on the subject, we study the signatures of fundamental phenomena such as Klein tunneling and specular Andreev reflection, which are specific to graphene, in the magnetoconductance oscillations. To this end, we introduce and utilize a variant of the well-known recursive Green’s function technique, which is an efficient numerical method for the calculation of transport observables in effectively non-interacting open quantum systems in the framework of a tight binding model. This technique is also applied to study the effects of a specific kind of disorder, namely short-range resonant scatterers, such as strongly bound adatoms or molecules, that can be modeled as vacancies in the graphene lattice. This numerical analysis of the conductance in the presence of resonant scatterers in graphene leads to a non-trivial classification of impurity sites in the graphene lattice and is further substantiated by an independent analytical treatment in the framework of the Dirac equation. The present thesis further contains a formal introduction to the topic of non-equilibrium quantum transport as appropriate for the development of the numerical technique mentioned above, a general introduction to the physics of graphene with a focus on the particular phenomena investigated in this work, and a conclusion where the obtained results are summarized and open questions as well as potential future developments are highlighted.
Testing Models with Higher Dimensional Effective Interactions at the LHC and Dark Matter Experiments
(2013)
Dark matter and non-zero neutrino masses are possible hints for new physics beyond the Standard Model of particle physics. Such potential consequences of new physics can be described by effective field theories in a model independent way. It is possible that the dominant contribution to low-energy effects of new physics is generated by operators of dimension d>5, e.g., due to an additional symmetry. Since these are more suppressed than the usually discussed lower dimensional operators, they can lead to extremly weak interactions even if new physics appears at comparatively low scales. Thus neutrino mass models can be connected to TeV scale physics, for instance. The possible existence of TeV scale particles is interesting, since they can be potentially observed at collider experiments, such as the Large Hadron Collider. Hence, we first recapitulate the generation of neutrino masses by higher dimensional effective operators in a supersymmetric framework. In addition, we discuss processes that can be used to test these models at the Large Hadron Collider. The introduction of new particles can affect the running of gauge couplings. Hence, we study the compatibilty of these models with Grand Unified Theories. The required extension of these models can imply the existence of new heavy quarks, which requires the consideration of cosmological constraints. Finally, higher dimensional effective operators can not only generate small neutrino masses. They also can be used to discuss the interactions relevant for dark matter detection experiments. Thus we apply the methods established for the study of neutrino mass models to the systematic discussion of higher dimensional effective operators generating dark matter interactions.
Active Galactic Nuclei (AGNs) are among the most powerful and most intensively studied objects in the Universe. AGNs harbor a mass accreting supermassive black hole (SMBH) in their center and emit radiation throughout the entire electromagnetic spectrum. About 10% show relativistic particle outflows, perpendicular to the so-called accretion disk, which are known as jets. Blazars, a subclass of AGN with jet orientations close to the line-of-sight of the observer, are highly variable sources from radio to TeV energies and dominate the γ- ray sky. The overall observed broadband emission of blazars is characterized by two distinct emission humps. While the low-energy hump is well described by synchrotron radiation of relativistic electrons, both leptonic processes such as inverse Compton scattering and hadronic processes such as pion-photoproduction can explain the radiation measured in the high-energy hump. Neutrinos, neutral, nearly massless particles, which only couple to the weak force 1 are exclusively produced in hadronic interactions of protons accelerated to relativistic energies. The detection of a high-energy neutrino from an AGN would provide an irrefutable proof of hadronic processes happening in jets. Recently, the IceCube neutrino observatory, located at the South Pole with a total instrumented volume of about one km 3 , provided evidence for a diffuse high-energy neutrino flux. Since the atmospheric neutrino spectrum falls steeply with energy, individual events with the clearest signature of coming from an extraterrestrial origin are those at the highest energies. These events are uniformly distributed over the entire sky and are therefore most likely of extragalactic nature. While the neutrino event (known as “BigBird”) with a reconstructed energy of ∼ 2 PeV has already been detected in temporal and spatial agreement with a single blazar in an active phase, still, the chance coincidence for such an association is only on the order of ∼ 5%. The neutrino flux at these high energies is low, so that even the brightest blazars only yield a Poisson probability clearly below unity. Such a small probability is in agreement with the observed all-sky neutrino flux otherwise, the sky would already be populated with numerous confirmed neutrino point sources. In neutrino detectors, events are typically detected in two different signatures 2 . So-called shower-like electron neutrino events produce a large particle cascade, which leads to a pre- cise energy measurement, but causes a large angular uncertainty. Track-like muon neutrino events, however, only produce a single trace in the detector, leading to a precise localization but poor energy reconstruction. The “BigBird” event was a shower-like neutrino event, tem- porally coincident with an activity phase of the blazar PKS 1424−418, lasting several months. Shower-like neutrino events typically lead to an angular resolution of ∼ 10 ◦ , while track-like events show a localization uncertainty of only ∼ 1 ◦ . Considering the potential detection of a track-like neutrino event in agreement with an activity phase of a single blazar lasting only days would significantly decrease the chance coincidence of such an association. In this thesis, a sample of bright blazars, continuously monitored by Fermi/LAT in the MeV to GeV regime, is considered as potential neutrino candidates. I studied the maximum possible neutrino ex- pectation of short-term blazar flares with durations of days to weeks, based on a calorimetric argumentation. I found that the calorimetric neutrino output of most short-term blazar flares is too small to lead to a substantial neutrino detection. However, for the most extreme flares, Poisson probabilities of up to ∼ 2% are reached, so that the possibility of associated neutrino detections in future data unblindings of IceCube and KM3NeT seems reasonable. On 22 September 2017, IceCube detected the first track-like neutrino event (named IceCube- 170922A) coincident with a single blazar in an active phase. From that time on, the BL Lac object TXS 0506+056 was subject of an enormous multiwavelength campaign, revealing an en- hanced flux state at the time of the neutrino arrival throughout several different wavelengths. In this thesis, I first studied the long-term flaring behavior of TXS 0506+056, using more than nine years of Fermi/LAT data. I found that the activity phase in the MeV to GeV regime already started in early 2017, months before the arrival of IceCube-170922A. I performed a calorimetric analysis on a 3-day period around the neutrino arrival time and found no sub- stantial neutrino expectation from such a short time range. By computing the calorimetric neutrino prediction for the entire activity phase of TXS 0506+056 since early 2017, a possible association seems much more likely. However, the post-trial corrected chance coincidence for a long-term association between IceCube-170922A and the blazar TXS 0506+056 is on the level of ∼ 3.5 σ, establishing TXS 0506+056 as the most promising neutrino point source candidate in the scientific community. Another way to explain a high-energy neutrino signal without an observed astronomical counterpart, would be the consideration of blazars at large cosmological distances. These high-redshift blazars are capable of generating the observed high-energy neutrino flux, while their γ-ray emission would be efficiently downscattered by Extragalactic Background Light (EBL), making them almost undetectable to Fermi/LAT. High-redshift blazars are impor- tant targets, as they serve as cosmological probes and represent one of the most powerful classes of γ-ray sources in the Universe. Unfortunately, only a small number of such objects could be detected with Fermi/LAT so far. In this thesis, I perform a systematic search for flaring events in high-redshift γ-ray blazars, which long-term flux is just below the sensitiv- ity limit of Fermi/LAT. By considering a sample of 176 radio detected high-redshift blazars, undetected at γ-ray energies, I was able to increase the number of previously unknown γ-ray blazars by a total of seven sources. Especially the blazar 5BZQ J2219−2719, at a distance of z = 3.63 was found to be the most distant new γ-ray source identified within this thesis. In the final part of this thesis, I studied the flaring behavior of bright blazars, previously considered as potential neutrino candidates. While the occurrence of flaring intervals in blazars is of purely statistical nature, I found potential differences in the observed flaring behavior of different blazar types. Blazars can be subdivided into BL Lac (BLL) objects, Flat-Spectrum Radio Quasar (FSRQ) and Blazars Candidates of Uncertain type (BCU). FSRQs are typ- ically brighter than BL Lac or BCU type blazars, thus longer flares and more complicated substructures can be resolved. Although BL Lacs and BCUs are capable of generating signifi- cant flaring episodes, they are often identified close to the detection threshold of Fermi/LAT. Long-term outburst periods are exclusively observed in FSRQs, while BCUs can still con- tribute with flare durations of up to ten days. BL Lacs, however, are only detected in flaring states of less than four days. FSRQs are bright enough to be detected multiple times with time gaps between two subsequent flaring intervals ranging between days and months. While BL Lacs can show time gaps of more than 100 days, BCUs are only observed with gaps up to 20 days, indicating that these objects are detected only once in the considered time range of six years. The newly introduced parameter “Boxyness” describes the averaged flux in an identified flaring state and does highly depend on the shape of the considered flare. While perfectly box-like flares (flares which show a constant flux level over the entire time range) correspond to an averaged flux which is equal the maximum flare amplitude, irregular shaped flares generate a smaller averaged flux. While all blazar types show perfectly box-shaped daily flares, BL Lacs and BCUs are typically not bright enough to be resolved for multiple days. The work presented in this thesis illustrates the challenging state of multimessenger neu- trino astronomy and the demanding hunt for the first extragalactic neutrino point sources. In this context, this work discusses the multiwavelength emission behavior of blazars as a promising class of neutrino point sources and allows for predictions of current and future source associations
The observation of neutrino masses and lepton mixing has highlighted the incompleteness of the Standard Model of particle physics. In conjunction with this discovery, new questions arise: why are the neutrino masses so small, which form has their mass hierarchy, why is the mixing in the quark and lepton sectors so different or what is the structure of the Higgs sector. In order to address these issues and to predict future experimental results, different approaches are considered. One particularly interesting possibility, are Grand Unified Theories such as SU(5) or SO(10). GUTs are vertical symmetries since they unify the SM particles into multiplets and usually predict new particles which can naturally explain the smallness of the neutrino masses via the seesaw mechanism. On the other hand, also horizontal symmetries, i.e., flavor symmetries, acting on the generation space of the SM particles, are promising. They can serve as an explanation for the quark and lepton mass hierarchies as well as for the different mixings in the quark and lepton sectors. In addition, flavor symmetries are significantly involved in the Higgs sector and predict certain forms of mass matrices. This high predictivity makes GUTs and flavor symmetries interesting for both, theorists and experimentalists. These extensions of the SM can be also combined with theories such as supersymmetry or extra dimensions. In addition, they usually have implications on the observed matter-antimatter asymmetry of the universe or can provide a dark matter candidate. In general, they also predict the lepton flavor violating rare decays mu -> e gamma, tau -> mu gamma and tau -> e gamma which are strongly bounded by experiments but might be observed in the future. In this thesis, we combine all of these approaches, i.e., GUTs, the seesaw mechanism and flavor symmetries. Moreover, our request is to develop and perform a systematic model building approach with flavor symmetries and to search for phenomenological implications. This provides a new perspective in model building since it allows us to screen models by its predictions on the theoretical and phenomenological side, i.e., we can apply further model constraints to single out a desired model. The results of our approach are, e.g., diverse lepton flavor and GUT models, a systematic scan of lepton flavor violation, new mass matrices, a new understanding of lepton mixing angles, a general extension of the idea of quark-lepton complementarity theta_12=pi/4-epsilon/sqrt{2} and for the first time the QLC relation in an SU(5) GUT.
In this work the supersymmetric seesaw model and its effects on low-energy leptonic observables and thermal leptogenesis have been systematically investigated. Precision measurements will increase the sensitivity on lepton-flavor violating decays, particularly on Br(l_j->l_i gamma) and also on electric and magnetic dipole moments in the near future. In order to improve also the accuracy of theoretical predictions for these processes, we have performed a full one-loop calculation of the underlying supersymmetric processes taking into account the lepton masses. Since the mechanism of soft supersymmetry breaking (SSB) is completely unknown, a novel analysis beyond the often studied minimal Supergravity scenarios has been performed. This way it has been demonstrated that in the considered mSUGRA, AMSB, GMSB and gaugino mediated scenarios, the ongoing search for Br(mu->e gamma) can constrain fundamental SSB parameters and/or the seesaw parameters. On the other hand, the basic parameters of thermal leptogenesis, such as the CP asymmetry in the decays of the lightest right-handed Majorana neutrino, provide probes of the unknown complex orthogonal R-matrix of the seesaw model.
Since its popularization due to Randall and Sundrum (RS) one decade ago, and in connection with the $AdS/CFT$ correspondence in particular, 5D warped background spacetime has been one of the most fruitful new ideas in physics beyond the standard model (SM), leading to new insights into symmetry breaking and the properties of strongly interacting theories inaccessible to direct perturbative calculations, while at the same time relating gravity to phenomenological model building. This has, among others, led to a renewed interest in models of electroweak symmetry breaking without physical scalar fields in the guise of so-called 'warped higgsless' models, which could provide an alternative to the famed Higgs mechanism of electroweak symmetry breaking which is part of the Standard Model of particle physics. However, little emphasis was put on reconciling these models with the strong evidence from astrophysical observations that one or several new, as yet unknown, stable particle species exist which form the cold dark matter content of the universe. The nature of dark matter and electroweak symmetry breaking are among the most prominent puzzles subject to experimental scrutiny at the Tevatron, direct search experiments, and in the near future at the LHC, which compels us the believe that both issues should be addressed together in any alternative scenario beyond the Standard Model. In this thesis we have investigated phenomenological implications which arise for cosmology and collider physics when the electroweak symmetry breaking sector of warped higgsless models is extended to include warped supersymmetry with conserved $R$ parity. The goal was to find the simplest supersymmetric extension of these models which still has a realistic light spectrum including a viable dark matter candidate. To accomplish this, we have used the same mechanism which is already at work for symmetry breaking in the electroweak sector to break supersymmetry as well, namely symmetry breaking by boundary conditions. While supersymmetry in five dimensions contains four supercharges and is therefore directly related to 4D $\mathcal{N}=2$ supersymmetry, half of them are broken by the background leaving us with ordinary $\mathcal{N}=1$ theory in the massless sector after Kaluza-Klein expansion. We thus use boundary conditions to model the effects of a breaking mechanism for the remaining two supercharges. The simplest viable scenario to investigate is a supersymmetric bulk and IR brane without supersymmetry on the UV brane. Even though parts of the light spectrum are effectively projected out by this mechanism, we retain the rich phenomenology of complete $\mathcal{N}=2$ supermultiplets in the Kaluza-Klein sector. While the light supersymmetric spectrum consists of electroweak gauginos which get their $\mathcal{O}(100\mbox{ GeV})$ masses from IR brane electroweak symmetry breaking, the light gluinos and squarks are projected out on the UV brane. The neutralinos, as mass eigenstates of the neutral bino-wino sector, are automatically the lightest gauginos, making them LSP dark matter candidates with a relic density that can be brought to agreement with WMAP measurements without extensive tuning of parameters. For chargino masses close to the experimental lower bounds at around $m_{\chi^+}\approx 100\dots 110$ GeV, the dark matter relic density points to LSP masses of around $m_\chi\approx 90$ GeV. At the LHC, the standard particle content of our model shares most of the key features of known warped higgsless models. We have performed Monte Carlo simulations of warped higgsless LSP and NLSP production at a benchmark point using \nameomega/\namewhizard, concentrating on $\ptmiss$ in association with third generation quarks. After background reduction cuts on the quark momenta and angles, we get hadronic cross sections of $\sigma>100\mbox{ fb}$ at $14\mbox{ TeV}$ with characteristic $\ptmiss$ distributions for $\chi\chi t\overline{t}$ final states, while the final states with $b\overline{b}$ pairs have much lower event rates and shapes which are hard to discern in experiments. Our results suggest that the discovery of warped higgsless LSP dark matter at the LHC via missing energy is within reach for the first few $\mbox{ fb}^{-1}$ at $14$ TeV if $b$ and in particular $t$ identification is reliable.
This Thesis explores hybrid structures on the basis of quantum spin Hall insulators, and in particular the interplay of their edge states and superconducting and magnetic order. Quantum spin Hall insulators are one example of topological condensed matter systems, where the topology of the bulk bands is the key for the understanding of their physical properties. A remarkable consequence is the appearance of states at the boundary of the system, a phenomenon coined bulk-boundary correspondence. In the case of the two-dimensional quantum spin Hall insulator, this is manifested by so-called helical edge states of counter-propagating electrons with opposite spins. They hold great promise, \emph{e.g.}, for applications in spintronics -- a paradigm for the transmission and manipulation of information based on spin instead of charge -- and as a basis for quantum computers. The beginning of the Thesis consists of an introduction to one-dimensional topological superconductors, which illustrates basic concepts and ideas. In particular, this includes the topological distinction of phases and the accompanying appearance of Majorana modes at their ends. Owing to their topological origin, Majorana modes potentially are essential building-blocks for topological quantum computation, since they can be exploited for protected operations on quantum bits. The helical edge states of quantum spin Hall insulators in conjunction with $s$-wave superconductivity and magnetism are a suitable candidate for the realization of a one-dimensional topological superconductor. Consequently, this Thesis investigates the conditions in which Majorana modes can appear. Typically, this happens between regions subjected to either only superconductivity, or to both superconductivity and magnetism. If more than one superconductor is present, the phase difference is of paramount importance, and can even be used to manipulate and move Majorana modes. Furthermore, the Thesis addresses the effects of the helical edge states on the anomalous correlation functions characterizing proximity-induced superconductivity. It is found that helicity and magnetism profoundly enrich their physical structure and lead to unconventional, exotic pairing amplitudes. Strikingly, the nonlocal correlation functions can be connected to the Majorana bound states within the system. Finally, a possible thermoelectric device on the basis of hybrid systems at the quantum spin Hall edge is discussed. It utilizes the peculiar properties of the proximity-induced superconductivity in order to create spin-polarized Cooper pairs from a temperature bias. Cooper pairs with finite net spin are the cornerstone of superconducting spintronics and offer tremendous potential for efficient information technologies.
In this thesis I present results concerning realistic calculations of correlated fermionic many-body systems. One of the main objectives of this work was the implementation of a hybridization expansion continuous-time quantum Monte Carlo (CT-HYB) algorithm and of a flexible self-consistency loop based on the dynamical mean-field theory (DMFT). DMFT enables us to treat strongly correlated electron systems numerically. After the implementation and extensive testing of the program we investigated different problems to answer open questions concerning correlated systems and their numerical treatment.
This thesis deals with the chaotic dynamics of nonlinear networks consisting of semiconductor lasers which have time-delayed self-feedbacks or mutual couplings. These semiconductor lasers are simulated numerically by the Lang-Kobayashi equations. The central issue is how the chaoticity of the lasers, measured by the maximal Lyapunov exponent, changes when the delay time is changed. It is analysed how this change of chaoticity with increasing delay time depends on the reflectivity of the mirror for the self-feedback or the strength of the mutal coupling, respectively. The consequences of the different types of chaos for the effect of chaos synchronization of mutually coupled semiconductor lasers are deduced and discussed. At the beginning of this thesis, the master stability formalism for the stability analysis of nonlinear networks with delay is explained. After the description of the Lang-Kobayashi equations and their linearizations as a model for the numerical simulation of semiconductor lasers with time-delayed couplings, the artificial sub-Lyapunov exponent $\lambda_{0}$ is introduced. It is explained how the sign of the sub-Lyapunov exponent can be determined by experiments. The notions of "strong chaos" and "weak chaos" are introduced and distinguished by their different scaling properties of the maximal Lyapunov exponent with the delay time. The sign of the sub-Lyapunov exponent $\lambda_{0}$ is shown to determine the occurence of strong or weak chaos. The transition sequence "weak to strong chaos and back to weak chaos" upon monotonically increasing the coupling strength $\sigma$ of a single laser's self-feedback is shown for numerical calculations of the Lang-Kobayashi equations. At the transition between strong and weak chaos, the sub-Lyapunov exponent vanishes, $\lambda_{0}=0$, resulting in a special scaling behaviour of the maximal Lyapunov exponent with the delay time. Transitions between strong and weak chaos by changing $\sigma$ can also be found for the Rössler and Lorenz dynamics. The connection between the sub-Lyapunov exponent and the time-dependent eigenvalues of the Jacobian for the internal laser dynamics is analysed. Counterintuitively, the difference between strong and weak chaos is not directly visible from the trajectory although the difference of the trajectories induces the transitions between the two types of chaos. In addition, it is shown that a linear measure like the auto-correlation function cannot unambiguously reveal the difference between strong and weak chaos either. Although the auto-correlations after one delay time are significantly higher for weak chaos than for strong chaos, it is not possible to detect a qualitative difference. If two time-scale separated self-feedbacks are present, the shorter feedback has to be taken into account for the definition of a new sub-Lyapunov exponent $\lambda_{0,s}$, which in this case determines the occurence of strong or weak chaos. If the two self-feedbacks have comparable delay times, the sub-Lyapunov exponent $\lambda_{0}$ remains the criterion for strong or weak chaos. It is shown that the sub-Lyapunov exponent scales with the square root of the effective pump current $\sqrt{p-1}$, both in its magnitude and in the position of the critical coupling strengths. For networks with several distinct sub-Lyapunov exponents, it is shown that the maximal sub-Lyapunov exponent of the network determines whether the network's maximal Lyapunov exponent scales strongly or weakly with increasing delay time. As a consequence, complete synchronization of a network is excluded for arbitrary networks which contain at least one strongly chaotic laser. Furthermore, it is demonstrated that the sub-Lyapunov exponent of a driven laser depends on the number of the incoherently superimposed inputs from unsynchronized input lasers. For networks of delay-coupled lasers operating in weak chaos, the condition $|\gamma_{2}|<\mathrm{e}^{-\lambda_{\mathrm{m}}\,\tau}$ for stable chaos synchronization is deduced using the master stability formalism. Hence, synchronization of any network depends only on the properties of a single laser with self-feedback and the eigenvalue gap of the coupling matrix. The characteristics of the master stability function for the Lang-Kobayashi dynamics is described, and consequently, the master stability function is refined to allow for precise practical prediction of synchronization. The prediction of synchronization with the master stability function is demonstrated for bidirectional and unidirectional networks. Furthermore, the master stability function is extended for two distinct delay times. Finally, symmetries and resonances for certain values of the ratio of the delay times are shown for the master stability function of the Lang-Kobyashi equations.
One key scientific program of the MAGIC telescope project is the discovery and detection of blazars. They constitute the most prominent extragalactic source class in the very high energy (VHE) Gamma-ray regime with 29 out of 34 known objects (as of April 2010). Therefore a major part of the available observation time was spent in the last years on high-frequency peaked blazars. The selection criteria were chosen to increase the detection probability. As the X-ray flux is believed to be correlated to the VHE Gamma-ray flux, only X-ray selected sources with a flux F(X) > 2 μJy at 1 keV were considered. To avoid strong attenuation of the Gamma-rays in the extragalactic infrared background, the redshift was restricted to values between z < 0.15 and z < 0.4, depending on the declination of the objects. The latter determines the zenith distance during culmination which should not exceed 30° (for z < 0.4) and 45° (for z < 0.15), respectively. Between August 2005 and April 2009, a sample of 24 X-ray selected high-frequency peaked blazars has been observed with the MAGIC telescope. Three of them were detected including 1ES 1218+304 being the first high-frequency peaked BL Lacertae object (HBL) to be discovered with MAGIC in VHE Gamma-rays. One previously detected object was not confirmed as VHE emitter in this campaign by MAGIC. A set of 20 blazars previously not detected will be treated more closely in this work. In this campaign, during almost four years ~ 450 hrs or ~ 22% of the available observation time for extragalactic objects were dedicated to investigate the baseline emission of blazars and their broadband spectral properties in this emission state. For the sample of 20 objects in a redshift range of 0.018 < z < 0.361 integral flux upper limits in the VHE range on the 99.7% confidence level (corresponding to 3 standard deviations) were calculated resulting in values between 2.9% and 14.7% of the integral flux of the Crab Nebula. As the distribution of significances of the individual objects shows a clear shift to positive values, a stacking method was applied to the sample. For the whole set of 20 objects, an excess of Gamma-rays was found with a significance of 4.5 standard deviations in 349.5 hours of effective exposure time. For the first time a signal stacking in the VHE regime turned out to be successful. The measured integral flux from the cumulative signal corresponds to 1.4% of the Crab Nebula flux above 150 GeV with a spectral index α = −3.15±0.57. None of the objects showed any significant variability during the observation time and therefore the detected signal can be interpreted as the baseline emission of these objects. For the individual objects lower limits on the broad-band spectral indices αX−Gamma between the X-ray range at 1 keV and the VHE Gamma-ray regime at 200 GeV were calculated. The majority of objects show a spectral behaviour as expected from the source class of HBLs: The energy output in the VHE regime is in general lower than in X-rays. For the stacked blazar sample the broad-band spectral index was calculated to αX−Gamma = 1.09, confirming the result found for the individual objects. Another evidence for the revelation of the baseline emission is the broad-band spectral energy distribution (SED) comprising archival as well as contemporaneous multi-wavelength data from the radio to the VHE band. The SEDs of known VHE Gamma-ray sources in low flux states matches well the SED of the stacked blazar sample.
In the field of spintronics, spin manipulation and spin transport are the main principles that need to be implemented. The main focus of this thesis is to analyse semiconductor systems where high fidelity in these principles can be achieved. To this end, we use numerical methods for precise results, supplemented by simpler analytical models for interpretation.
The material system of 2D topological insulators, HgTe/CdTe quantum wells, is interesting not only because it provides a topologically distinct phase of matter, physically manifested in its protected transport properties, but also since within this system, ballistic transport of high quality can be realized, with Rashba spin-orbit coupling and electron densities that are tunable by electrical gating. Extending the Bernvevig-Hughes-Zhang model for 2D topological insulators, we derive an effective four-band model including Rashba spin-orbit terms due to an applied potential that breaks the spatial inversion symmetry of the quantum well. Spin transport in this system shows interesting physics because the effects of Rashba spin-orbit terms and the intrinsic Dirac-like spin-orbit terms compete. We show that the resulting spin Hall signal can be dominated by the effect of Rashba spin-orbit coupling. Based on spin splitting due to the latter, we propose a beam splitter setup for all-electrical generation and detection of spin currents. Its working principle is similar to optical birefringence. In this setup, we analyse spin current and spin polarization signals of different spin vector components and show that large in-plane spin polarization of the current can be obtained. Since spin is not a conserved quantity of the model,
we first analyse the transport of helicity, a conserved quantity even in presence of Rashba spin-orbit terms. The polarization defined in terms of helicity is related to in-plane polarization of the physical spin.
Further, we analyse thermoelectric transport in a setup showing the spin Hall effect. Due to spin-orbit coupling, an applied temperature gradient generates a transverse spin current, i.e. a spin Nernst effect, which is related to the spin Hall effect by a Mott-like relation. In the metallic energy regimes, the signals are qualitatively explained by simple analytic models. In the insulating regime, we observe a spin Nernst signal that originates from the finite-size induced overlap of edge states.
In the part on methods, we discuss two complementary methods for construction of effective semiconductor models, the envelope function theory and the method of invariants. Further, we present elements of transport theory, with some emphasis on spin-dependent signals. We show the connections of the adiabatic theorem of quantum mechanics to the semiclassical theory of electronic transport and to the characterization of topological phases. Further, as application of the adiabatic theorem to a control problem, we show that universal control of a single spin in a heavy-hole quantum dot is experimentally realizable without breaking time reversal invariance,
but using a quadrupole field which is adiabatically changed as control knob. For experimental realization, we propose a GaAs/GaAlAs quantum well system.
Due to their potential application for quantum computation, quantum dots have attracted a lot of interest in recent years. In these devices single electrons can be captured, whose spin can be used to define a quantum bit (qubit). However, the information stored in these quantum bits is fragile due to the interaction of the electron spin with its environment. While many of the resulting problems have already been solved, even on the experimental side, the hyperfine interaction between the nuclear spins of the host material and the electron spin in their center remains as one of the major obstacles. As a consequence, the reduction of the number of nuclear spins is a promising way to minimize this effect. However, most quantum dots have a fixed number of nuclear spins due to the presence of group III and V elements of the periodic table in the host material. In contrast, group IV elements such as carbon allow for a variable size of the nuclear spin environment through isotopic purification. Motivated by this possibility, we theoretically investigate the physics of the central spin model in carbon based quantum dots. In particular, we focus on the consequences of a variable number of nuclear spins on the decoherence of the electron spin in graphene quantum dots.
Since our models are, in many aspects, based upon actual experimental setups, we provide an overview of the most important achievements of spin qubits in quantum dots in the first part of this Thesis. To this end, we discuss the spin interactions in semiconductors on a rather general ground. Subsequently, we elaborate on their effect in GaAs and graphene, which can be considered as prototype materials. Moreover, we also explain how the central spin model can be described in terms of open and closed quantum systems and which theoretical tools are suited to analyze such models.
Based on these prerequisites, we then investigate the physics of the electron spin using analytical and numerical methods. We find an intriguing thermal flip of the electron spin using standard statistical physics. Subsequently, we analyze the dynamics of the electron spin under influence of a variable number of nuclear spins. The limit of a large nuclear spin environment is investigated using the Nakajima-Zwanzig quantum master equation, which reveals a decoherence of the electron spin with a power-law decay on short timescales. Interestingly, we find a dependence of the details of this decay on the orientation of an external magnetic field with respect to the graphene plane. By restricting to a small number of nuclear spins, we are able to analyze the dynamics of the electron spin by exact diagonalization, which provides us with more insight into the microscopic details of the decoherence. In particular, we find a fast initial decay of the electron spin, which asymptotically reaches a regime governed by small fluctuations around a finite long-time average value. Finally, we analytically predict upper bounds on the size of these fluctuations in the framework of quantum thermodynamics.
We investigate the single particle static and dynamic properties at zero temperature within the Hubbard an three-band-Hubbard model for the superconducting copper oxides. Based on the recently proposed self-energy functional approach (SFA) [M.Potthoff, Eur. Phys. J. B 32 429 (2003)], we present an extension of the cluster-perturbation theory (CPT) to systems with spontaneous broken symmetry. Our method accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via the exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte-Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum of the insulating cuprates. Comparison of the dispersion of the low-energy excitations with recent experimental results of angular resolved photoemission spectroscopy (ARPES) allows us to fix a consistent parameter set for the one-band Hubbard model with an additional hopping parameter t' along the lattice diagonal. The doping dependence of the single-particle excitations is studied within the t-t-U Hubbard model with special emphasis on the electron doped compounds. We show, that the ARPES results on the band structure and the Fermi surface of Nd{2-x}Ce_xCuOCl_{4-\delta} are naturally obtained within the t-t-U Hubbard model without further need for readjustment or fitting of parameters, as proposed in recent theoretical considerations. We present a theory for the photon energy and polarization dependence of ARPES intensities from the CuO2 plane in the framework of strong correlation models. The importance of surface states for the observed experimental facts is considered. We show that for electric field vector in the CuO_2 plane the ‘radiation characteristics’ of the O 2p_{\sigma} and Cu 3d_{x^2-y^2} orbitals are strongly peaked along the CuO_2 plane, i.e. most photoelectrons are emitted at grazing angles. This suggests that surface states play an important role in the observed ARPES spectra, consistent with recent data from Sr_2CuCl_2O_2. We show that a combination of surface state dispersion and Fano resonance between surface state and the continuum of LEED-states may produce a precipitous drop in the observed photoelectron current as a function of in-plane momentum, which may well mimic a Fermi-surface crossing. This effect may explain the simultaneous ‘observation’ of a hole-like and an electron-like Fermi surfaces in Bi_2Sr_2CaCu_2O_{8+\delta} at different photon energies.
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.
In this thesis, I establish new relations between quantum information measures in a two-dimensional CFT and geometric objects in a three-dimensional AdS space employing the AdS/CFT correspondence. I focus on two quantum information measures: the computational cost of quantum circuits in a CFT and Berry phases in two entangled CFTs. In particular, I show that these quantities are associated with geometric objects in the dual AdS space.
In this PhD thesis, the effect of strain on heteroepitaxial growth is investigated by means of Kinetic Monte Carlo simulations. In this context the lattice misfit, arising from the different lattice constants of the adsorbate and the substrate material, is of particular interest. As a consequence, this lattice misfit leads to long-range elastic strain effects having strong influence on the entire growing crystal and its resulting surface morphology. The main focus of this work is the investigation of different strain relaxation mechanisms and their controlling parameters, revealing interesting consequences on the subsequent growth. Since epitaxial growth is carried out under conditions far away from thermodynamic equilibrium, it is strongly determined by surface kinetics. At this point the relevant kinetic microscopic processes are described, followed by theoretical considerations of heteroepitaxial growth disclosing an overview over several independent methodological streams, used to model epitaxy in different time and length scales, as well as the characterization of misfit dislocations and the classification of epitaxial growth modes based on thermodynamic considerations. The epitaxial growth is performed by means of Kinetic Monte Carlo simulations which allows for the consideration of long range effects in systems with lateral extension of few hundred atoms. By using an off-lattice simulation model the particles are able to leave their predefined lattice sites, which is an indispensable condition for simulating strain relaxation mechanisms. The main idea of our used model is calculating the activation energy of all relevant thermally activated processes by using simple pair potentials and then realizing the dynamics by performing each event according to its probability by means of a rejection-free algorithm method. In addition, the crystal relaxation procedure, the grid-based particle access method, which accelerates the simulation enormously, and the efficient implementation of the algorithm are discussed. To study the influence of long range elastic strain effects, the main part of this work was realized on the two dimensional triangular lattice, which can be treated as a cross section of the real three dimensional case. Chapter 4 deals with the formation of misfit dislocations as a strain relaxation mechanism and the resulting consequences on the subsequent heteroepitaxial growth. We can distinguish between two principally different dislocation formation mechanisms, depending strongly on the sign as well as on the magnitude of the misfit, but also the surface kinetics need to be taken into account. Additionally, the dislocations affect the lattice spacings of the crystal whose observed progression is in qualitative good agreement with experimental results. Furthermore, the dislocations influence the subsequent growth of the adsorbate film, since the potential energy of an adatom is modulated by buried dislocations. A clear correlation between the lateral positions of buried dislocations and the positions of mounds grown on the surface can be observed. In chapter 5, an alternative strain relaxation mechanism is studied: the formation of three dimensional islands enables the particles to approach their preferred lattice spacing. We demonstrate that it is possible to adjust within our simulation model each of the three epitaxial growth modes: Volmer–Weber, Frank–van der Merve or layer-by-layer, and Stranski–Krastanov growth mode. Moreover, we can show that the emerging growth mode depends in principle on two parameters: on the one hand the interaction strength of adsorbate particles with each other, compared to the interaction of adsorbate with substrate particles, and on the other hand the lattice misfit between adsorbate and substrate particles. A sensible choice of these two parameters allows the realization of each growth mode within the simulations. In conclusion, the formation of nanostructures controlled by an underlying dislocation network can be applied in the concept of self-organized pattern formation as well as by the tendency to form ordered arrays of strain-induced three dimensional grown islands. In chapter 6, we extend our model to three dimensions and investigate the effect of strain on growth on bcc(100) surfaces. We introduce an anisotropic potential yielding a stable bcc lattice structure within the off-lattice representation. We can show that the strain built up in submonolayer islands is mainly released at the island edges and the lattice misfit has strong influence on the diffusion process on the plane surface as well as on the situation at island edges with eminent consequences on the appearance of submonolayer islands.
In this PhD thesis, we study the heteroepitaxial crystal growth by means of Monte Carlo simulations. Of particular interest in this work is the influence of the lattice mismatch of the adsorbates relative to the substrate on surface structures. In the framework of an off-lattice model, we consider one monolayer of adsorbate and investigate the emerging nanopatterns in equilibrium and their formation during growth. In chapter 1, a brief introduction is given, which describes the role of computer simulations in the field of the physics of condensed matter. Chapter 2 is devoted to some technical basics of experimental methods of molecular beam epitaxy and the theoretical description. Before a model for the simulation can be designed, it is necessary to make some considerations of the single processes which occur during epitaxial growth. For that purpose we look at an experimental setup and extract the main microscopic processes. Afterwards a brief overview of different theoretical concepts describing that physical procedures is given. In chapter 3, the model used in the simulations is presented. The aim is to investigate the growth of an fcc crystal in the [111] direction. In order to keep the simulation times within a feasible limit a simple pair potential, the Lennard-Jones potential, with continuous particle positions is used, which are necessary to describe effects resulting from the atomic mismatch in the crystal. Furthermore the detailed algorithm is introduced which is based on the idea to calculate the barrier of each diffusion event and to use the barriers in a rejection-free method. Chapter 4 is attended to the simulation of equilibrium. The influence of different parameters on the emerging structures in the first monolayer upon the surface, which is completely covered with two adsorbate materials, is studied. Especially the competition between binding energy and strain leads to very interesting pattern formations like islands or stripes. In chapter 5 the results of growth simulations are presented. At first, we introduce a model in order to realize off-lattice Kinetic Monte Carlo simulations. Since the costs in simulation time are enormous, some simplifications in the calculation of diffusion barriers are necessary and therefore the previous model is supplemented with some elements from the so-called ball and spring model. The next point is devoted to the calculation of energy barriers followed by the presentation of the growth simulations. Binary systems with only one sort of adsorbate are investigated as well as ternary systems with two different adsorbates. Finally, a comparison to the equilibrium simulations is drawn. Chapter 6 contains some concluding remarks and gives an outlook to possible further investigations.
We model Milky Way like isolated disk galaxies in high resolution three-dimensional hydrodynamical simulations with the adaptive mesh refinement code Enzo. The model galaxies include a dark matter halo and a disk of gas and stars. We use a simple implementation of sink particles to measure and follow collapsing gas, and simulate star formation as well as stellar feedback in some cases. We investigate two largely different realizations of star formation. Firstly, we follow the classical approach to transform cold, dense gas into stars with an fixed efficiency. These kind of simulations are known to suffer from an overestimation of star formation and we observe this behavior as well. Secondly, we use our newly developed FEARLESS approach to combine hydrodynamical simulations with a semi-analytic modeling of unresolved turbulence and use this technique to dynamically determine the star formation rate. The subgrid-scale turbulence regulated star formation simulations point towards largely smaller star formation efficiencies and henceforth more realistic overall star formation rates. More work is necessary to extend this method to account for the observed highly supersonic turbulence in molecular clouds and ultimately use the turbulence regulated algorithm to simulate observed star formation relations.
In this thesis, the low-temperature regime of replica symmetry breaking in the SK-model has been thoroughly investigated. In order to access this regime and to perform self-consistence calculations with high accuracy at high orders of replica symmetry breaking, a formalism has been developed which reduces the numerical effort to the absolute minimum. The central idea of its derivation is the identification of asymptotic regions in which the recursion relations can be solved analytically. The new object in the numerical treatment is then the correction to this asymptotic regime, represented by a sequence of so-called kernel correction functions. This method increased the effciency of the numerics considerably so that up to 200 orders of RSB could be calculated at zero temperature and zero external field, and up to 60 (65) orders of RSB for finite temperature (external field). The remarkable high precision of these calculations allowed the extraction of several quantities with accuracy exceeding the literature values by several orders of magnitude. The results of the numerical calculations have been analyzed in great detail. Especially the convergence behavior of various observables and of the order function with respect to the RSB order has been investigated since the high but finite RSB regime has been addressed in the present work for the first time. Several unexpected features of finite order replica symmetry breaking have been observed.
Relativistic effects crucially influence the fundamental properties of many quantum materials. In the accelerated reference frame of an electron, the electric field of the nuclei is transformed into a magnetic field that couples to the electron spin. The resulting interaction between an electron spin and its orbital angular momentum, known as spin-orbit coupling (SOC), is hence fundamental to the physics of many condensed matter phenomena. It is particularly important quantitatively in low-dimensional quantum systems, where its coexistence with inversion symmetry breaking can lead to a splitting of spin degeneracy and spin momentum locking. Using the paradigm of Landau Fermi liquid theory, the physics of SOC can be adequately incorporated in an effective single particle picture. In a weak coupling approach, electronic correlation effects beyond single particle propagator renormalization can trigger Fermi surface instabilities such as itinerant magnetism, electron nematic phases, superconductivity, or other symmetry broken states of matter.
In this thesis, we use a weak coupling-based approach to study the effect of SOC on Fermi surface instabilities and, in particular, superconductivity. This encompasses a weak coupling renormalization group formulation of unconventional superconductivity as well as the random phase approximation. We propose a unified formulation for both of these two-particle Green’s function approaches based on the notion of a generalized susceptibility.
In the half-Heusler semimetal and superconductor LuPtBi, both SOC and electronic correlation
effects are prominent, and thus indispensable for any concise theoretical description. The metallic and weakly dispersive surface states of this material feature spin momentum locked Fermi surfaces, which we propose as a possible domain for the onset of unconventional surface superconductivity. Using our framework for the analysis of Fermi surface instability and combining it with ab-initio density functional theory calculations, we analyse the surface band structure of LuPtBi, and particularly its propensity towards Cooper pair formation. We study how the presence of strong SOC modifies the classification of two-electron wave functions as well as the screening of electron-electron interactions. Assuming an electronic mechanism, we identify a chiral superconducting condensate featuring Majorana edge modes to be energetically favoured over a wide range of model parameters.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
Magnetism is a phenomenon ubiquitously found in everyday life. Yet, together with superconductivity and superfluidity, it is among the few macroscopically realized quantum states. Although well-understood on a quasi-classical level, its microscopic description is still far from being solved. The interplay of strong interactions present in magnetic condensed-matter systems and the non-trivial commutator structure governing the underlying spin algebra prevents most conventional approaches in solid-state theory to be applied.
On the other hand, the quantum limit of magnetic systems is fertile land for the development of exotic phases of matter called spin-liquids. In these states, quantum fluctuations inhibit the formation of magnetic long-range order down to the lowest temperatures. From a theoretical point of view, spin-liquids open up the possibility to study their exotic properties, such as fractionalized excitations and emergent gauge fields. However, despite huge theoretical and experimental efforts, no material realizing spin-liquid properties has been unambiguously identified with a three-dimensional crystal structure. The search for such a realization is hindered by the inherent difficulty even for model calculations. As most numerical techniques are not applicable due to the interaction structure and dimensionality of these systems, a methodological gap has to be filled.
In this thesis, to fill this void, we employ the pseudo-fermion functional renormalization group (PFFRG), which provides a scheme to investigate ground state properties of quantum magnetic systems even in three spatial dimensions.
We report the status quo of this established method and extend it by alleviating some of its inherent approximations. To this end, we develop a multi-loop formulation of PFFRG, including hitherto neglected terms in the underlying flow equations consistently, rendering the outcome equivalent to a parquet approximation. As a necessary prerequisite, we also significantly improve the numerical accuracy of our implementation of the method by switching to a formulation respecting the asymptotic behavior of the vertex functions as well as employing state-of-the-art numerical algorithms tailored towards PFFRG. The resulting codebase was made publicly accessible in the open-source code PFFRGSolver.jl.
We subsequently apply the technique to both model systems and real materials. Augmented by a classical analysis of the respective models, we scan the phase diagram of the three-dimensional body-centered cubic lattice up to third-nearest neighbor coupling and the Pyrochlore lattice up to second-nearest neighbor. In both systems, we uncover in addition to the classically ordered phases, an extended parameter regime, where a quantum paramagnetic phase appears, giving rise to the possibility of a quantum spin liquid.
Additionally, we also use the nearest-neighbor antiferromagnet on the Pyrochlore lattice as well as the simple cubic lattice with first- and third-nearest neighbor couplings as a testbed for multi-loop PFFRG, demonstrating, that the inclusion of higher loop orders has quantitative effects in paramagnetic regimes and that the onset of order can be signaled by a lack of loop convergence.
Turning towards material realizations, we investigate the diamond lattice compound MnSc\(_2\)S\(_4\), explaining on grounds of ab initio couplings the emergence of a spiral spin liquid at low temperatures, but above the ordering transition.
In the Pyrochlore compound Lu\(_2\)Mo\(_2\)O\(_5\)N\(_2\), which is known to not magnetically order down to lowest temperatures, we predict a spin liquid state displaying a characteristic gearwheel pattern in the spin structure factor.
This thesis studies connections between quantum information measures and geometric features of spacetimes within the AdS/CFT correspondence. These studies are motivated by the idea that spacetime can be thought of as an effect emerging from an underlying entanglement structure in the AdS/CFT correspondence. In particular, I study generalized entanglement measures in two-dimensional conformal field theories and their holographic duals. Unlike the ordinary entanglement entropy of a spatial subregion typically used in the AdS/CFT context, the generalization considered here measures correlations between different fields as well as between spatial degrees of freedom. I present a new gauge invariant definition of the generalized entanglement entropy applicable to both mixed and pure states as well as explicit results for thermal states of the S_N-orbifold theory of the D1/D5 system. Along the way, I develop computation techniques for conformal blocks on the torus and apply them to the calculation of the ordinary entanglement entropy for large central charge CFTs at finite size and finite temperature. The generalized Ryu-Takayanagi formula arising from these studies provides further support for the idea that entanglement and geometry are intrinsically linked in AdS/CFT. The results show that the holographic dual to the generalized entanglement entropy given by the length of a geodesic winding around black hole horizons or naked singularities probes subregions of spacetime that are inaccessible to Ryu-Takayanagi surfaces, thereby solving the puzzle of how these features of the spacetime are encoded in the boundary theory. Furthermore, I investigate quantum circuits embedded in two-dimensional conformal field theories as well as computational complexity measures therein. These investigations are motivated by conjectures relating computational complexity in conformal field theories to geometric features of black hole geometries. In this thesis, I study quantum circuits built up from conformal transformations. I investigate examples of computational complexity measures in these circuits related to geometric actions on coadjoint orbits of the Virasoro group and to the Fubini-Study metric. I then work out relations between these computational complexity measures and the dual gravitational theory. Moreover, I construct a bulk dual to the circuits in consideration and use this construction to study geometric realizations of computational complexity measures from first principles. The results of this part on the one hand rule out some possibilities for dual realizations of computational complexity in two-dimensional CFTs put forward in previous work while on the other hand providing a new robust dual realization of a computational complexity measure based on the Fubini-Study distance.
At a hadron collider as the LHC or the Tevatron the production of a photon in association with a leptonically decaying vector boson represents an important class of processes. These processes stand out due to a very clean signal of a photon and two leptons. Furthermore they
provide direct access to the photon–vector-boson couplings and thus an easy opportunity to test the
gauge sector of the Standard Model. Within the scope of this work we present a full calculation of the next-to-leading-order corrections which include the O (αs) corrections of the strong interaction as well as the electroweak corrections of O (α) including all photon-induced contributions. For the creation of matrix elements we use methods based on Feynman diagrams. The IR singularities are treated with the dipole subtraction technique. In order to separate photons from jets, a quark-to-photon fragmentation function ´a la Glover / Morgan or Frixione’s cone isolation is employed. Moreover, two different scenarios for charged leptons in the fi state were considered. The fi scenario for dressed leptons assumes that a charged lepton and a photon will be recombined if they are collinear. In the second scenario for bare muons it is assumed that leptons and photon can be separated in a detector also if they are collinear.
For our calculation we implemented all corrections into a fl Monte Carlo program. Be- sides the computation of the total cross section this program is also able to generate diff tial distributions of several experimentally motivated observables. Apart from the expected large electroweak corrections in the high transverse-momentum regions and sizeable corrections in the resonance regions of the transverse or the invariant masses we found photon-induced corrections up to several 10% for high transverse momenta. Within run I at the LHC for 7/8 TeV the experimental accuracy for Vγ production was roughly 10%. Due to the higher luminosity at run II this accuracy
will be reduced to the level of a few percent so that corrections of the same order within the theoretical predictions might become relevant. In this work we present results for the total cross section at the LHC for 7, 8 and 14 TeV and the corresponding distributions
for 14 TeV.
Supersymmetry is currently the best motivated extension of the Standard Model and will be subject to extensive studies in the upcoming generation of colliders. The e-e- mode would be a straight forward extension to the currently planed International Linear Collider, planned to operate in e+e- mode. The low background in this mode may prove advantageous in the study of CP- and Lepton Flavour Violtation. In this work a CP sensitive observable based on transverse beam polarisation is introduced and the impact of neutralino mixing on the total cross section in cas of non-vanishing CP-violtating phases is studied in representative scenarios including non-GUT scenarios. Additionally, the mixing of sleptons is studied in the context of LFV, an analytical approximation is developed, and possible background free measurements of these effects are investigated.
The mechanism of spontaneous symmetry breaking is essential to provide masses to the W and Z gauge bosons and fermions of the SM. We hope to elucidate this mechanism at the next generation of colliders. While the SM has been tested with astonishing precision it is believed to be an effective theory of a more fundamental Great Unified Theory. SUSY is one of the most attractive extensions of the SM of particle physics. Therefore, the search for SUSY is a top priority at the next generation of colliders. Once Higgs bosons are discovered, a precise determination of their properties is necessary to differentiate between different models, in particular the MSSM. A muon collider, running at center of mass energies around the neutral Higgs boson resonances, would allow precise measurements of masses and widths, as well as the couplings to their decay products. In particular their couplings to supersymmetric particles are essential to probe SUSY. Therefore, we study the decays of the heavier CP-even and CP-odd Higgs bosons into lighter chargino or neutralino pairs. In this thesis we have analyzed the polarization effects of the beams and the charginos and neutralinos produced in mu+ mu- annihilation around the center of mass energies of the Higgs boson resonances H and A. For the production of equal charginos we have shown that the ratio of H-chargino and A-chargino couplings can be precisely determined independently of the chargino decay mechanism. This method avoids reference to other experiments and makes only a few model-dependent assumptions. Here we have analyzed the effect of the energy spread and of the error from the non-resonant channels, including an irreducible standard model background contribution. For small tan(beta) the process yields large cross sections of up to a pb. For the production of two different charginos we have shown that the H-A interference can be analyzed using asymmetries of the charge conjugated processes. The asymmetries depend on the muon longitudinal beam polarizations and vanish for unpolarized beams. For the chargino pair production with subsequent two-body decay of one of the charginos we have shown that charge and beam polarization asymmetries in the energy distributions of the decay particles are sensitive to the interference of scalar exchange channels with different CP quantum numbers. This process provides unique information on the interference of overlapping Higgs boson resonances. The effect is larger for regions of parameter space with intermediate values of tan(beta) and light sleptons or LSP neutralinos. For the chargino pair production with subsequent two-body decays of both charginos we have defined energy distribution and angular asymmetries in the final particles, in order to analyze the spin-spin correlations of the charginos. The transverse polarizations of the charginos are sensitive to the CP quantum number of the exchanged Higgs bosons and can thus be used to separate overlapping resonances, as well as to determine the CP quantum number of a single resonance. For equal charginos, these asymmetries are not sensitive to the interference of CP-even and CP-odd Higgs exchange channels. For the neutralino pair production in mu+ mu- annihilation we study similar processes as for chargino production. Line shape measurements of neutralino pair production allow to precisely determine the ratio of H-neutralino and A-neutralino couplings. Neutralino pair production with subsequent two-body decay of one of the neutralinos in the intermediate tan(beta) region is sensitive to the interference of H and A and may be measured with a large statistical significance. The Majorana nature of the neutralinos implies that the beam polarization asymmetries vanish for the remaining production channels. For neutralino pair production with subsequent two-body decays of both neutralinos we analyze similar observables as in chargino production. The main difference consists in the intrinsic relative CP quantum number of the neutralino pair, which depends on the chosen scenario. We have thus shown that the interaction of the Higgs bosons to the gaugino-higgsino sector can be probed at a muon collider in chargino and neutralino pair production, both analyzing the production line-shape around the resonances as well as studying the chargino and neutralino polarizations via their decays.
We consider the prospects for a neutrino factory measuring mixing angles, the CP violating phase and mass-squared differences by detecting wrong-charge muons arising from the chain $\mu^+\to\nu_e\to\nu_\mu\to\mu^-$ and the right-charge muons coming from the chain $\mu^+\to\bar{\nu}_\mu\to\bar{\nu}_\mu\to\mu^+$ (similar to $\mu^-$ chains), where $\nu_e\to\nu_\mu$ and $\bar{\nu}_\mu\to\bar{\nu}_\mu$ are neutrino oscillation channels through a long baseline. First, we study physics with near detectors and consider the treatment of systematic errors including cross section errors, flux errors, and background uncertainties. We illustrate for which measurements near detectors are required, discuss how many are needed, and what the role of the flux monitoring is. We demonstrate that near detectors are mandatory for the leading atmospheric parameter measurements if the neutrino factory has only one baseline, whereas systematic errors partially cancel if the neutrino factory complex includes the magic baseline. Second, we perform the baseline and energy optimization of the neutrino factory including the latest simulation results from the magnetized iron neutrino detector (MIND). We also consider the impact of $\tau$ decays, generated by appearance channels $\nu_\mu \rightarrow \nu_\tau$ and $\nu_e \rightarrow \nu_\tau$, on the discovery reaches of the mass orderings, the leptonic CP violation, and the non-zero $\theta_{13}$, which we find to be negligible for the considered detector. Third, we make a comparison of a high energy neutrino factory to a low energy neutrino factory and find that they are just two versions of the same experiment optimized for different regions of the parameter space. In addition, we briefly comment on whether it is useful to build the bi-magic baseline at the low energy neutrino factory. Finally, the effects of one additional massive sterile neutrino are discussed in the context of a combined short and long baseline setup. It is found that near detectors can provide the required sensitivity at the LSND-motivated $\Delta m_{41}^2$-range, while some sensitivity can also be obtained in the region of the atmospheric mass splitting introduced by the sterile neutrino from the long baselines.
In this thesis, a phenomenological phase-fluctuation model for the pseudogap regime of the underdoped cuprates was discussed. The key idea of the phase-fluctuation scenario in the high-T_c superconductors is the notion that the pseudogap observed in a wide variety of experiments arises from phase fluctuations of the superconducting gap. In this scenario, below a mean-field temperature scale T_c^{MF}, a d_{x^2-y^2}-wave gap amplitude is assumed to develop. However, the superconducting transition is suppressed to a considerably lower transition temperature T_c by phase fluctuations. In the intermediate temperature regime between T_c^{MF} and T_c, phase fluctuations of the superconducting order parameter give rise to the pseudogap phenomena. The phenomenological phase-fluctuation model discussed in this thesis consists of a two-dimensional BCS-like Hamiltonian where the phase of the pairing-amplitude is free to fluctuate. The fluctuations of the phase were treated by a Monte Carlo simulation of a classical XY model. First, the density of states was calculated. The quasiparticle tunneling conductance (dI/dV) obtained from our phenomenological phase fluctuation model was able to reproduce characteristic and salient features of recent scanning-tunneling studies of Bi2212 and Bi2201 suggesting that the pseudogap behavior observed in these experiments arises from phase fluctuations of the d_{x^2-y^2}-wave pairing gap. In calculating the single-particle spectral weight, we were further able to show how phase fluctuations influence the experimentally observed quasiparticle spectra in detail. In particular the disappearance of the BCS-Bogoliubov quasiparticle band at T_c and the change from a more V-like superconducting gap to a rather U-like pseudogap above T_c can be explained in a consistent way by assuming that the low-energy pseudogap in the underdoped cuprates is due to phase fluctuations of a local d_{x^2-y^2}-wave pairing gap with fixed magnitude. Furthermore, phase fluctuations can explain why the pseudogap starts closing from the nodal points, whereas it rather fills in along the anti-nodal directions and they can also account for the characteristic temperature dependence of the superconducting (pi,0)-photoemission-peak. Next, we have shown that the "violation" of the low-frequency optical sum rule recently observed in the SC state of underdoped Bi2212, which is associated with a reduction of kinetic energy, can be related to the role of phase fluctuations. The decrease in kinetic energy is due to the sharpening of the quasiparticle peaks close to the superconducting transition at T_c == T_{KT}, where the phase correlation length xi diverges. A detailed analysis of the temperature and frequency dependence of the optical conductivity sigma(omega)=sigma_1(omega)+i sigma_2(omega) revealed a superconducting scaling of sigma_2(omega), which starts already above T_c, exactly as observed in high-frequency microwave conductivity experiments on Bi2212. On the other hand, our model was only able to account for the characteristic peak, which is observed in sigma_1(omega) close to the superconducting transition, after the inclusion of an additional marginal-Fermi-liquid scattering-rate in the optical conductivity formula. Finally, we calculated the static uniform diamagnetic susceptibility. It turned out that the precursor effects of the fluctuating diamagnetism above T_c are very small and limited to temperatures close to T_c in a phase-fluctuation scenario of the pseudogap. Instead, the temperature dependence of the uniform static magnetic susceptibility is dominated by the Pauli spin susceptibility, which displayed a very characteristic temperature dependence, independent of the details of the gap function used in our model. This temperature dependence is qualitatively very similar to the experimentally observed change of the Knight-shift as a function of temperature in underdoped Bi2212.
This thesis is dedicated to a theoretical study of the 1-band Hubbard model in the strong coupling limit. The investigation is based on the Dynamical Cluster Approximation (DCA) which systematically restores non-local corrections to the Dynamical Mean Field approximation (DMFA). The DCA is formulated in momentum space and is characterised by a patching of the Brillouin zone where momentum conservation is only recovered between two patches. The approximation works well if k-space correlation functions show a weak momentum dependence. In order to study the temperature and doping dependence of the spin- and charge excitation spectra, we explicitly extend the Dynamical Cluster Approximation to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitations. The effective vertex is calculated by using the Quantum Monte Carlo method on the finite cluster whereas the analytical continuation of dynamical quantities is performed by a stochastic version of the maximum entropy method. A comparison with high temperature auxiliary field quantum Monte Carlo data serves as a benchmark for our approach to two-particle correlation functions. Our method can reproduce basic characteristics of the spin- and charge excitation spectrum. Near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations: a collective spin mode emerges at optimal doping and sufficiently low temperatures in the spin response spectrum and exhibits the energy scale of the magnetic exchange interaction J. Simultaneously, the low energy single-particle excitations are characterised by a coherent quasiparticle with bandwidth J. The origin of the quasiparticle can be quite well understood in a picture of a more or less antiferromagnetic ordered background in which holes are dressed by spin-excitations to allow for a coherent motion. By increasing doping, all features which are linked to the spin-polaron vanish in the single-particle as well as two-particle spin response spectrum. In the second part of the thesis an analysis of superconductivity in the Hubbard model is presented. The superconducting instability is implemented within the Dynamical Cluster Approximation by essentially allowing U(1) symmetry breaking baths in the QMC calculations for the cluster. The superconducting transition temperature T_c is derived from the d-wave order parameter which is directly estimated on the Monte Carlo cluster. The critical temperature T_c is in astonishing agreement with the temperature scale estimated by the divergence of the pair-field susceptibility in the paramagnetic phase. A detailed study of the pseudo and superconducting gap is continued by the investigation of the local and angle-resolved spectral function.
In this thesis, I study entanglement in quantum field theory, using methods from operator algebra theory. More precisely, the thesis covers original research on the entanglement properties of the free fermionic field. After giving a pedagogical introduction to algebraic methods in quantum field theory, as well as the modular theory of Tomita-Takesaki and its relation to entanglement, I present a coherent framework that allows to solve Tomita-Takesaki theory for free fermionic fields in any number of dimensions. Subsequently, I use the derived machinery on the free massless fermion in two dimensions, where the formulae can be evaluated analytically. In particular, this entails the derivation of the resolvent of restrictions of the propagator, by means of solving singular integral equations. In this way, I derive the modular flow, modular Hamiltonian, modular correlation function, R\'enyi entanglement entropy, von-Neumann entanglement entropy, relative entanglement entropy, and mutual information for multi-component regions. All of this is done for the vacuum and thermal states, both on the infinite line and the circle with (anti-)periodic boundary conditions. Some of these results confirm previous results from the literature, such as the modular Hamiltonian and entanglement entropy in the vacuum state. The non-universal solutions for modular flow, modular correlation function, and R\'enyi entropy, however are new, in particular at finite temperature on the circle. Additionally, I show how boundaries of spacetime affect entanglement, as well as how one can define relative (entanglement) entropy and mutual information in theories with superselection rules. The findings regarding modular flow in multi-component regions can be summarised as follows: In the non-degenerate vacuum state, modular flow is multi-local, in the sense that it mixes the field operators along multiple trajectories, with one trajectory per component. This was already known from previous literature but is presented here in a more explicit form. In particular, I present the exact solution for the dynamics of the mixing process. What was not previously known at all, is that the modular flow of the thermal state on the circle is infinitely multi-local even for a connected region, in the sense that it mixes the field along an infinite, discretely distributed set, of trajectories. In the limit of high temperatures, all trajectories but the local one are pushed towards the boundary of the region, where their amplitude is damped exponentially, leaving only the local result. At low temperatures, on the other hand, these trajectories distribute densely in the region to either---for anti-periodic boundary conditions---cancel, or---for periodic boundary conditions---recover the non-local contribution due to the degenerate vacuum state. Proceeding to spacetimes with boundaries, I show explicitly how the presence of a boundary implies entanglement between the two components of the Dirac spinor. By computing the mutual information between the components inside a connected region, I show quantitatively that this entanglement decreases as an inverse square law at large distances from the boundary. In addition, full conformal symmetry (which is explicitly broken due to the presence of a boundary) is recovered from the exact solution for modular flow, far away from the boundary. As far as I know, all of these results are new, although related results were published by another group during the final stage of this thesis. Finally, regarding relative entanglement entropy in theories with superselection sectors, I introduce charge and flux resolved relative entropies, which are novel measures for the distinguishability of states, incorporating a charge operator, central to the algebra of observables. While charge resolved relative entropy has the interpretation of being a ``distinguishability per charge sector'', I argue that it is physically meaningless without placing a cutoff, due to infinite short-distance entanglement. Flux resolved relative entropy, on the other hand, overcomes this problem by inserting an Aharonov-Bohm flux and thus passing to a variant of the grand canonical ensemble. It takes a well defined value, even without putting a cutoff, and I compute its value between various states of the free massless fermion on the line, the charge operator being the total fermion number.
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.
Adding interactions to topological (non-)trivial free fermion systems can in general have four different effects: (i) In symmetry protected topological band insulators, the correlations may lead to the spontaneous breaking of some protecting symmetries by long-range order that gaps the topological boundary modes. (ii) In free fermion (semi-)metal, the interaction could vice versa also generate long-range order that in turn induces a topological mass term and thus generates non-trivial phases dynamically. (iii) Correlation might reduce the topological classification of free fermion systems by allowing adiabatic deformations between states of formerly distinct phases. (iv) Interaction can generate long-range entangled topological order in states such as quantum spin liquids or fractional quantum Hall states that cannot be represented by non-interacting systems. During the course of this thesis, we use numerically exact quantum Monte Carlo algorithms to study various model systems that (potentially) represent one of the four scenarios, respectively.
First, we investigate a two-dimensional $d_{xy}$-wave, spin-singlet superconductor, which is relevant for high-$T_c$ materials such as the cuprates. This model represents nodal topological superconductors and exhibits chiral flat-band edge states that are protected by time-reversal and translational invariance. We introduce the conventional Hubbard interaction along the edge in order to study their stability with respect to correlations and find ferromagnetic order in case of repulsive interaction as well as charge-density-wave order and/or additional $i$s-wave pairing for attractive couplings. A mean-field analysis that, for the first time, is formulated in terms of the Majorana edge modes suggests that any order has normal and superconducting contributions. For example, the ferromagnetic order appears in linear superposition with triplet pairing. This finding is well confirmed by the numerically exact quantum Monte Carlo investigation.
Second, we consider spinless electrons on a two-dimensional Lieb lattice that are subject to nearest-neighbor Coulomb repulsion. The low energy modes of the free fermion part constitute a spin-$1$ Dirac cone that might be gapped by several mass terms. One option breaks time-reversal symmetry and generates a topological Chern insulator, which mainly motivated this study. We employ two flavors of quantum Monte Carlo methods and find instead the formation of charge-density-wave order that breaks particle-hole symmetry. Additionally, due to sublattices of unequal size in Lieb lattices, this induces a finite chemical potential that drives the system away from half-filling. We argue that this mechanism potentially extends the range of solvable models with finite doping by coupling the Lieb lattice to the target system of interest.
Third, we construct a system with four layers of a topological insulators and interlayer correlation that respects one independent time-reversal and a unitary $\mathbb{Z}_2$ symmetry. Previous studies claim a reduced topological classification from $\mathbb{Z}$ to $\mathbb{Z}_4$, for example by gapping out degenerate zero modes in topological defects once the correlation term is designed properly. Our interaction is chosen according to this analysis such that there should exist an adiabatic deformation between states whose topological invariant differs by $\Delta w=\pm4$ in the free fermion classification. We use a projective quantum Monte Carlo algorithm to determine the ground-state phase diagram and find a symmetry breaking regime, in addition to the non-interacting semi-metal, that separates the free fermion insulators. Frustration reduces the size of the long-range ordered region until it is replaced by a first order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free fermion states into each other. We conclude that the prescribed reduction rules, which often use the bulk-boundary correspondence, are necessary but not sufficient and require a more careful investigation.
Fourth, we study conduction electron on a honeycomb lattice that form a Dirac semi-metal Kondo coupled to spin-1/2 degrees of freedom on a Kagome lattice. The local moments are described by a variant of the Balents-Fisher-Girvin model that has been shown to host a ferromagnetic phase and a $\mathbb{Z}_2$ spin liquid at strong frustration. Here, we report the first numerical exact quantum Monte Carlo simulation of the Kondo-coupled system that does not exhibit the negative-sign problem. When the local moments form a ferromagnet, the Kondo coupling induces an anti-ferromagnetic mass term in the conduction-electron system. At large frustration, the Dirac cone remains massless and the spin system forms a $\mathbb{Z}_2$ spin liquid. Owing to the odd number of spins per unit cell, this constitutes a non-Fermi liquid that violates Luttinger's theorem which relates the Fermi volume to the particle density in a Fermi liquid. This phase is a specific realization of the so called 'fractional Fermi liquid` as it has been first introduced in the context of heavy fermion models.
This thesis contains two major parts: The first part introduces the reader into three independent concepts of treating strongly correlated many body physics. These are, on the analytical side the SO(5)-theory (Chap.3), which poses the general frame. On the numerical side these are the Stochastic Series Expansion (SSE) (Chap.1) and the Contractor Renormalization Group (CORE) approach (Chap. 2}). The central idea of this thesis was to combine these above concepts, in order to achieve a better understanding of the high-T_c superconductors (HTSC). The results obtained by this combination can be found in the second major part of this thesis (chapters 4 and 5). The main idea of this thesis, i.e., to combine the SO(5)-theory with the capabilities of bosonic Quantum-Monte Carlo simulations and those of the CORE approach, has been proven to be a very successful Ansatz. Two different approaches, one based on symmetry and one on renormalization-group arguments, motivate an effective bosonic Hamiltonian. In a subsequent step the effective Hamiltonian has been simulated efficiently using the SSE. The results reproduce salient experiments on high-T_c superconductors. In addition, it has been shown that the model can be extended to capture also charge ordering. These results also form a profound basis for further studies, for example one could address the open question of SO(5)-symmetry restoration at a multicritical point in the extended pSO(5) model, where longer ranged interactions are included.
The astronomical exploration at energies between 30\,GeV and $\lesssim$\,350\,GeV was the main motivation for building the \MAGIC-telescope. With its 17\,m \diameter\ mirror it is the worldwide largest imaging air-Cherenkov telescope. It is located at the Roque de los Muchachos at the Canary island of San Miguel de La Palma at 28.8$^\circ$\,N, 17.8$^\circ$\,W, 2200\,m a.s.l. The telescope detects Cherenkov light produced by relativistic electrons and positrons in air showers initiated by cosmic gamma-rays. The imaging technique is used to powerfully reject the background due to hadronically induced air showers from cosmic rays. Their inverse power-law energy-distribution leads to an increase of the event rate with decreasing energy threshold. For \MAGIC this implies a trigger rate in the order of 250\,Hz, and a correspondingly large data stream to be recorded and analyzed. A robust analysis software package, including the general framework \MARS, was developed and commissioned to allow automation, necessary for data taken under variable observing conditions. Since many of the astronomical sources of high-energy radiation, in particular the enigmatic gamma-ray bursts, are of a transient nature, the telescope was designed to allow repositioning in several tens of seconds, keeping a tracking accuracy of $\lesssim\,$0.01$^\circ$. Employing a starguider, a tracking accuracy of $\lesssim\,$1.3\,minutes of arc was obtained. The main class of sources at very high gamma-ray energies, known from previous imaging air-Cherenkov telescopes, are Active Galactic Nuclei with relativistic jets, the so-called high-peaked Blazars. Their spectrum is entirely dominated by non-thermal emission, spanning more than 15 orders of magnitude in energy, from radio to gamma-ray energies. Predictions based on radiation models invoking a synchrotron self-Compton or hadronic origin of the gamma-rays suggest, that a fairly large number of them should be detectable by \MAGIC. Promising candidates have been chosen from existing compilations, requiring high (synchrotron) X-ray flux, assumed to be related to a high (possibly inverse-Compton) flux at GeV energies, and a low distance, in oder to avoid strong attenuation due to pair-production in interactions with low-energy photons from the extragalactic background radiation along the line of sight. Based on this selection the first \AGN, emitting gamma-rays at 100\,GeV, 1ES\,1218+304 at a redshift of $z=0.182$, was discovered, one of the two farthest known \AGN emitting in the TeV energy region. In this context, the automated analysis chain was successfully demonstrated. The source was observed in January 2005 during six moonless nights for 8.2\,h. At the same time the collaborating \KVA-telescope, located near the \MAGIC site, observed in the optical band. The lightcurve calculated showed no day-to-day variability and is compatible with a constant flux of $F($\,$>$\,$100\,\mbox{GeV})=(8.7\pm1.4) \cdot 10^{-7}\,\mbox{m}^{-2}\,\mbox{s}^{-1}$ within the statistical errors. A differential spectrum between 87\,GeV and 630\,GeV was calculated and is compatible with a power law of $F_E(E) = (8.1\pm 2.1) \cdot 10^{-7}(E/\mbox{250\,GeV})^{-3.0\pm0.4}\,\mbox{TeV}^{-1}\,\mbox{m}^{-2}\,\mbox{s}^{-1}$ within the statistical errors. The spectrum emitted by the source was obtained by taking into account the attenuation due to pair-production with photons of the extragalactic background at low photon energies. A homogeneous, one-zone synchrotron self-Compton model has been fitted to the collected multi-wavelength data. Using the simultaneous optical data, a best fit model could be obtained from which some physical properties of the emitting plasma could be inferred. The result was compared with the so-called {\em Blazar sequence}.