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Since the prediction of the quantum spin Hall effect in graphene by Kane and Mele, \(Z_2\) topology in hexagonal monolayers is indissociably linked to high-symmetric honeycomb lattices. This thesis breaks with this paradigm by focusing on topological phases in the fundamental two-dimensional hexagonal crystal, the triangular lattice. In contrast to Kane-Mele-type systems, electrons on the triangular lattice profit from a sizable, since local, spin-orbit coupling (SOC) and feature a non-trivial ground state only in the presence of inversion symmetry breaking. This tends to displace the valence charge form the atomic position. Therefore, all non-trivial phases are real-space obstructed. Inspired by the contemporary conception of topological classification of electronic systems, a comprehensive lattice and band symmetry analysis of insulating phases of a \(p\)-shell on the triangular lattice is presented. This reveals not only the mechanism at the origin of band topology, the competition of SOC and symmetry breaking, but sheds also light on the electric polarization arising from a displacement of the valence charge centers from the nuclei, i. e., real-space obstruction. In particular, the competition of SOC versus horizontal and vertical reflection symmetry breaking gives rise to four topologically distinct insulating phases: two kinds of quantum spin Hall insulators (QSHI), an atomic insulator and a real-space obstructed higher-order topological insulator. The theoretical analysis is complemented with state-of-the-art first principles calculations and experiments on trigonal monolayer adsorbate systems. This comprises the recently discovered triangular QSHI indenene, formed by In atoms, and focuses on its topological classification and real-space obstruction. The analysis reveals Kane-Mele-type valence bands which profit from the atomic SOC of the triangular lattice. The realization of a HOTI is proposed by reducing SOC by considering lighter adsorbates. Further the orbital Rashba effect is analyzed in AgTe, a consequence of mirror symmetry breaking, the formation of local angular momentum polarization and SOC. As an outlook beyond topology, the Fermi surface and electronic susceptibility of Group V adsorbates on silicon carbide are investigated.
In summary, this thesis elucidates the interplay of symmetry breaking and SOC on the triangular lattice, which can promote non-trivial insulating phase.
This thesis investigates the charged moments and the symmetry-resolved
entanglement entropy in the context of the AdS3/CFT2 duality. In the
first part, I focus on the holographic U(1) Chern-Simons-Einstein gravity,
a toy model of AdS3/CFT2 with U(1) Kac-Moody symmetry. I
start with the vacuum background with a single entangling interval. I
show that, apart from a partition function in the grand canonical ensemble,
the charged moments can also be interpreted as the two-point
function of vertex operators on the replica surface. For the holographic
description, I propose a duality between the bulk U(1) Wilson line and
the boundary vertex operators. I verify this duality by deriving the
effective action for the Chern-Simons fields and comparing the result
with the vertex correlator. In the twist field approach, I show that the
charged moments are given by the correlation function of the charged
twist operators and the additional background operators. To solve the
correlation functions involved, I prove the factorization of the U(1) extended
conformal block into a U(1) block and a Virasoro block. The
general expression for the U(1) block is derived by directly summing
over the current descendant states, and the result shows that it takes
an identical form as the vertex correlators. This leads to the conclusion
that the disjoint Wilson lines compute the neutral U(1) block. The final
result for the symmetry-resolved entanglement entropy shows that
it is always charge-independent in this model. In the second part, I
study charged moments in higher spin holography, where the boundary
theory is a CFT with W3 symmetry. I define the notion of the
higher spin charged moments by introducing a spin-3 modular charge
operator. Restricting to the vacuum background with a single entangling
interval, I employ the grand canonical ensemble interpretation
and calculate the charged moments via the known higher spin black
hole solution. On the CFT side, I perform a perturbative expansion for
the higher spin charged moments in terms of the connected correlation
functions of the spin-3 modular charge operators. Using the recursion
relation for the correlation functions of the W3 currents, I evaluate the
charged moments up to the quartic order of the chemical potential. The
final expression matches with the holographic result. My results both
for U(1) Chern-Simons Einstein gravity and W3 higher spin gravity
constitute novel checks of the AdS3/CFT2 correspondence.
This work presents the first ILT observations of high redshift blazars and their study in terms of jet evolution, morphology, and interaction with the surrounding medium. Each of these represents a highly topical area of astronomywith a large number of open questions. To better understand Active Galactic Nuclei (AGN) and their fundamental inner workings, new techniques are needed to exploit the full potential of the next generation of radio interferometers. Some of these tools are presented here and applied to one of the latest generation of software radio telescopes. A major focus of the studies presented is on the unification model, where the observed blazars are discussed for their properties to be rotated counterparts of Fanaroff-Riley Class II (FR-II) radio galaxies, when classified as Flat Spectrum Radio Quasars (FSRQs). In addition, multiwavelength information has been included in the analysis. Both studies are feasibility studies that will serve as a basis for future similar studies. The characteristics discussed and their interpretation do not allow conclusions to be drawn for their respective populations. However, by applying them to a larger number of targets, population studies will be possible. The first chapters introduce the necessary topics, AGN, principles of radio observations and ILT, in the necessary depth to provide the reader with a solid knowledge base. They are particularly important for understanding the current limits and influences of uncertainties in the observation, calibration and imaging process. But they also shed light on realistic future improvements. A particular focus is on the development and evolution of the LOw-Frequency ARray (LOFAR)-Very Long Baseline Interferometry (VLBI) pipeline. With the tools at hand, the first study addresses the high redshift blazar S5 0836+710 $(z=2.218)$, which has been observed at various wavelengths and resolutions. It has a disrupted one-sided jet with an associated extended region further out. Despite the excellent wavelength coverage, only the additional ILT observations provided a complete picture of the source. With the data, the extended region could be classified as a hotspot moving at slightly relativistic speeds.. With the ILT data it was also possible to extract the flux of the core region of the AGN, and in projection to reveal the mixed counter-hotspot behind it. This also allowed constraints on jet parameters and environmental properties to be modelled, which were previously inconclusive. Technically, this study shows that the ILT can be used as an effective VLBI array for compact sources with small angular scales. However, the detection of faint components beyond redshifts of $z=2$ may require the capabilities of the Square Kilometre Array (SKA) to provide a significant number of detections to enable statistical conclusions. The second study uses a much improved calibration pipeline to analyse the high redshift blazar GB1508+5714 $(z=4.30)$. The ILT data revealed a previously unseen component in the eastern direction. A spectral index map was generated from the Karl G. Jansky Very Large Array (VLA) data, showing spectral index values of $-1.2_{-0.2}^{+0.4}$ for the western component, steeper than $-1.1$ for the eastern region, and $0.023 \pm 0.007$ for the core. Using the information provided by the ILT observation, as well as multi-wavelength information from other observations ranging from the long radio wavelengths to the $\gamma$ regime, four models were developed to interpret the observed flux with different emission origins. This also allowed to test a proposed interaction channel of the electrons provided by the jet, to cool off via inverse compton scattering with the Cosmic Microwave Background (CMB) photons, rather than by the usual synchrotron emission. This is referred to as cmb quenching in the literature, which could be shown in the study, to be necessary in any case. Finally, one of the four models was considered in which the hotspots in the detected components are unresolved and mixed by the lobe emission, with the X-ray emission coming from the lobes and partially mixed by the bright core region. The results of this preferred model are consistent with hotspots in a state of equipartition and lobes almost so. The study shows that high redshift blazars can be studied with the ILT, and expanding the sample of high redshift blazars resolved at multiple frequencies will allow a statistical study of the population. Finally, this work successfully demonstrates the powerful capabilities of the ILT to address questions that were previously inaccessible. The current state of the LOFAR-VLBI pipeline, when properly executed, allows work on the most challenging objects and will only improve in the future. In particular, this gives a glimpse of the possibilities that SKA will bring to astronomy.
In this thesis I explore the interplay of geometry and quantum information theory via the holographic principle, with a specific focus on geometric phases in quantum systems like two interacting qubits, and how they relate to entanglement measures and Hilbert space factorisation. I establish geometric phases as an indicator for Hilbert space factorsiation, both in an abstract sense using von Neumann operator algebras as well as applied to the eternal black hole within the AdS/CFT correspondence. For the latter case I show that geometric phases allow to diagnose non-factorisation from a boundary point of view. I also introduce geometric quantum discord as a second geometric measure for non-factorisation and reveals its potential implications for the study of black hole microstates.
The present thesis is concerned with the automated computation of integrated and differential
cross sections of diboson production in proton–proton and electron–positron collisions at very
high energies, including a resummation of electroweak Sudakov logarithms to all orders in the
fine-structure constant using soft–collinear effective theory.
The search for new physics at future colliders such as the FCC–hh or the CLIC requires
precise predictions for scattering cross sections from the theoretical high-energy physics com-
munity. Electroweak Sudakov logarithms, which currently limit the accuracy of predictions in
the high-energy tails of differential distributions for LHC-like energies, are known to destroy the
convergence behaviour of the fixed-order perturbative series, once sufficiently high energies are
considered.
To resum these large corrections, soft–collinear effective theory has been applied to simple
processes, which permits analytic calculations. Within this work, we present an automated
computation within a Monte Carlo integration framework, thus facilitating the computation of
fully differential cross section to complicated processes. This requires the use of the Catani–
Seymour subtraction algorithm to treat the occurring infrared divergences. The machinery is
applied to all diboson processes with intermediate weak gauge bosons, including the photon-
induced W+ W− -production channel.
To this end we carefully study the validity of the necessary assumptions such as the double-
pole approximation and estimate the order of magnitude of neglected effects. Especially the
non-doubly-resonant contributions turn out to be sizeable in several interesting phase-space
regions.
For lepton collisions at 3 TeV we obtain the integrated cross sections of W-pair and Z-pair
production to be shifted by more than 20% with respect to the Born value, owing to the resum-
mation of the leading-logarithmic corrections These effects are partly cancelled by subleading
effects. For proton–proton collisions at √
s = 100 TeV we observe sizeable resummation effects
in the high-energy tails, while the integrated cross sections are dominated by interactions, for
which soft–collinear effective theory is not applicable.
This thesis is dedicated to construct a non-abelian holographic dynamical minimal composite Higgs model. We first build a non-abelian bottom-up AdS/YM model that can explain the QCD meson spectrum well. The model is made non-abelian by considering non-abelian DBI action in the top-down model. We then change the dual theory from the QCD to the minimal composite Higgs model U (4)/Sp(4). By adding a second explicit U (4) → Sp(4) breaking through the NJL interaction at the boundary, we managed to construct a composite Higgs phase and a technicolor phase in this model. The transition between the two phases is also realized, which is controlled by the NJL coupling. This thesis is based on the works [1, 2].
In this thesis we examine the vector boson scattering (VBS) process \(\mathrm p \mathrm p \to \mathrm e^+ \nu_\mathrm e\mu^-\bar\nu_\mu\mathrm j\mathrm j +X\) (short: \(\mathrm W^+\mathrm W^-\) scattering) at NLO accuracy in two experimental setups by performing a Monte Carlo analysis of a \(13\,\mathrm{TeV}\) LHC run. \(\mathrm W^+\mathrm W^-\) scattering shows similarities and differences compared to the scattering of other vector bosons. We present a detailed description of the types of appearing subprocesses and background processes. We give insight into our code which solves the problems we are faced within \(\mathrm W^+\mathrm W^-\) scattering. This is especially the presence of the Higgs-boson resonance in the fiducial phase-space region. Particular attention is dedicated to the permutation of resonances. The integrated signal cross section at LO \(\mathcal O(\alpha^6)\) amounts to \(2.6988(3)\,\mathrm{fb}\) and \(1.5322(2)\,\mathrm{fb}\), respectively, in the two experimental setups. The LO QCD-induced background of \(\mathcal O(\alpha_\mathrm s^2\alpha^4)\) amounts to \(6.9115(9)\,\mathrm{fb}\) and \(1.6923(3)\,\mathrm{fb}\). The EW corrections to the signal are \(-11.4\%\) and \(-6.7\%\), the QCD corrections amount to \(-5.2\%\) and \(-23.0\%\). The EW corrections to the background are \(-8.3\%\) and \(-5.3\%\), the QCD corrections amount to \(-30.3\%\) and \(-77.6\%\). Our results for the QCD corrections and the QCD-induced background include a large uncertainty from varying the renormalisation and factorisation scale, and we discuss improvements for future calculations. We show the differential cross sections with unique features of \(\mathrm W^+\mathrm W^-\) scattering compared to other VBS processes and investigate in particular the subprocess of Higgs-boson production by using a modified version of our setups.
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.
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.
Strong correlations caused by interaction in systems of electrons can bring about unusual physical phenomena due to many-body quantum effects that cannot properly be captured by standard electronic structure methods like density functional theory. In this thesis, we apply the state-of-the-art continuous-time quantum Monte Carlo algorithm in hybridization expansion (CT-HYB) for the strongly correlated multi-orbital Anderson impurity model (AIM) to the solution of models of magnetic impurities on metallic surfaces and, via dynamical mean-field theory (DMFT), to the solution of a lattice model, the multi-orbital Hubbard model with Hund's coupling.
A concise introduction to the theoretical background focuses on information directly relevant to the understanding of applied models, methods, and the interpretation of results. It starts with a discussion of the AIM with its parameters and its solution in the path integral formalism, the basis of the CT-HYB algorithm. We consider its derivation and implementation in some detail before reviewing the DMFT approach to correlated lattice models and the interpretation of the single-particle Green's function.
We review two algorithmic developments for the CT-HYB algorithm that help to increase the performance of calculations especially in case of a complex structure of the interaction matrix and allow the precise calculation of self-energies and vertex functions also at intermediate and higher frequencies.
Our comparative analysis of Kondo screening in the cobalt on copper impurity system points out the importance of an accurate interaction matrix for qualitatively correct Kondo temperatures and the relevance of all d-orbitals in that case. Theoretical modeling of cobalt impurities in copper "atomic wires" fails to reproduce variations and partial absence of Kondo resonances depending on the wire size. We analyze the dependence of results on parameters and consider possible reasons for the discrepancy. Different Kondo temperatures of iron adatoms adsorbed on clean or oxygen-reconstructed niobium in the normal state are qualitatively reproduced, with the adsorption distance identified as major factor and implications for the superconducting state pointed out.
Moving on to lattice problems, we demonstrate the connection between Hund's coupling, shown to cause first-order character of the interaction-driven Mott transition at half-filling in the two-orbital Hubbard model, and a phase separation zone ending in a quantum critical point at finite doping. We touch on similarities in realistic models of iron-pnictide superconductors. We analyze the manifestation of the compressibility divergence at the finite-temperature critical points away from half-filling in the eigenbasis of the two-particle generalized susceptibility. A threshold for impurity susceptibility eigenvalues that indicates divergence of the DMFT lattice compressibility and distinguishes thermodynamic stability and instability of DMFT solutions is determined.
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 next-generation Event Horizon Telescope (ngEHT) will be a significant enhancement of the Event Horizon Telescope (EHT) array, with ∼10 new antennas and instrumental upgrades of existing antennas. The increased uv-coverage, sensitivity, and frequency coverage allow a wide range of new science opportunities to be explored. The ngEHT Analysis Challenges have been launched to inform the development of the ngEHT array design, science objectives, and analysis pathways. For each challenge, synthetic EHT and ngEHT datasets are generated from theoretical source models and released to the challenge participants, who analyze the datasets using image reconstruction and other methods. The submitted analysis results are evaluated with quantitative metrics. In this work, we report on the first two ngEHT Analysis Challenges. These have focused on static and dynamical models of M87* and Sgr A* and shown that high-quality movies of the extended jet structure of M87* and near-horizon hourly timescale variability of Sgr A* can be reconstructed by the reference ngEHT array in realistic observing conditions using current analysis algorithms. We identify areas where there is still room for improvement of these algorithms and analysis strategies. Other science cases and arrays will be explored in future challenges.
The Event Horizon Telescope (EHT) has led to the first images of a supermassive black hole, revealing the central compact objects in the elliptical galaxy M87 and the Milky Way. Proposed upgrades to this array through the next-generation EHT (ngEHT) program would sharply improve the angular resolution, dynamic range, and temporal coverage of the existing EHT observations. These improvements will uniquely enable a wealth of transformative new discoveries related to black hole science, extending from event-horizon-scale studies of strong gravity to studies of explosive transients to the cosmological growth and influence of supermassive black holes. Here, we present the key science goals for the ngEHT and their associated instrument requirements, both of which have been formulated through a multi-year international effort involving hundreds of scientists worldwide.
In the past few years, the Event Horizon Telescope (EHT) has provided the first-ever event horizon-scale images of the supermassive black holes (BHs) M87* and Sagittarius A* (Sgr A*). The next-generation EHT project is an extension of the EHT array that promises larger angular resolution and higher sensitivity to the dim, extended flux around the central ring-like structure, possibly connecting the accretion flow and the jet. The ngEHT Analysis Challenges aim to understand the science extractability from synthetic images and movies to inform the ngEHT array design and analysis algorithm development. In this work, we compare the accretion flow structure and dynamics in numerical fluid simulations that specifically target M87* and Sgr A*, and were used to construct the source models in the challenge set. We consider (1) a steady-state axisymmetric radiatively inefficient accretion flow model with a time-dependent shearing hotspot, (2) two time-dependent single fluid general relativistic magnetohydrodynamic (GRMHD) simulations from the H-AMR code, (3) a two-temperature GRMHD simulation from the BHAC code, and (4) a two-temperature radiative GRMHD simulation from the KORAL code. We find that the different models exhibit remarkably similar temporal and spatial properties, except for the electron temperature, since radiative losses substantially cool down electrons near the BH and the jet sheath, signaling the importance of radiative cooling even for slowly accreting BHs such as M87*. We restrict ourselves to standard torus accretion flows, and leave larger explorations of alternate accretion models to future work.
Josephson junctions (JJs) in the presence of a magnetic field exhibit qualitatively different interference patterns depending on the spatial distribution of the supercurrent through the junction. In JJs based on two-dimensional topological insulators (2DTIs), the electrons/holes forming a Cooper pair (CP) can either propagate along the same edge or be split into the two edges. The former leads to a SQUID-like interference pattern, with the superconducting flux quantum ϕ\(_0\) (where ϕ\(_0\)=h/2e) as a fundamental period. If CPs’ splitting is additionally included, the resultant periodicity doubles. Since the edge states are typically considered to be strongly localized, the critical current does not decay as a function of the magnetic field. The present paper goes beyond this approach and inspects a topological JJ in the tunneling regime featuring extended edge states. It is here considered the possibility that the two electrons of a CP propagate and explore the junction independently over length scales comparable to the superconducting coherence length. As a consequence of the spatial extension, a decaying pattern with different possible periods is obtained. In particular, it is shown that, if crossed Andreev reflections (CARs) are dominant and the edge states overlap, the resulting interference pattern features oscillations whose periodicity approaches 2ϕ\(_0\).
A well-known result for the interference of two single-mode fields is that the degree of coherence and the degree of indistinguishability are the same when we consider the detection of a single photon. In this article, we present the relation between the degree of coherence, path indistinguishability and the fringe visibility considering interference of multiple numbers of single-mode fields while being interested in the detection of a single photon only. We will also mention how Born’s rule of interference for multiple sources is reflected in these results.
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.
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, 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.
The topological classification of electronic band structures is based on symmetry properties of Bloch eigenstates of single-particle Hamiltonians. In parallel, topological field theory has opened the doors to the formulation and characterization of non-trivial phases of matter driven by strong electron-electron interaction. Even though important examples of topological Mott insulators have been constructed, the relevance of the underlying non-interacting band topology to the physics of the Mott phase has remained unexplored. Here, we show that the momentum structure of the Green’s function zeros defining the “Luttinger surface" provides a topological characterization of the Mott phase related, in the simplest description, to the one of the single-particle electronic dispersion. Considerations on the zeros lead to the prediction of new phenomena: a topological Mott insulator with an inverted gap for the bulk zeros must possess gapless zeros at the boundary, which behave as a form of “topological antimatter” annihilating conventional edge states. Placing band and Mott topological insulators in contact produces distinctive observable signatures at the interface, revealing the otherwise spectroscopically elusive Green’s function zeros.
The last years have witnessed an exciting scientific quest for intriguing topological phenomena in time-dependent quantum systems. A key to many manifestations of topology in dynamical systems relies on the effective dimensional extension by time-periodic drives. An archetypal example is provided by the Thouless pump in one spatial dimension, where a robust and quantized charge transport can be described in terms of an integer quantum Hall effect upon interpreting time as an extra dimension. Generalizing this fundamental concept to multifrequency driving, a variety of higher-dimensional topological models can be engineered in dynamical synthetic dimensions, where the underlying topological classification leads to quantized pumping effects in the associated lower-dimensional time-dependent systems.
In this Thesis, we explore how correlations profoundly impact the topological features of dynamical synthetic quantum materials. More precisely, we demonstrate that the interplay of interaction and dynamical synthetic dimension gives rise to striking topological phenomena that go beyond noninteracting implementations. As a starting point, we exploit the Floquet counterpart of an integer quantum Hall scenario, namely a two-level system driven by two incommensurate frequencies. In this model, the topologically quantized response translates into a process in which photons of different frequencies are exchanged between the external modes, referred to as topological frequency conversion. We extend this prototypical setup to an interacting version, focusing on the minimal case of two correlated spins equally exposed to the external drives. We show that the topological invariant determining the frequency conversion can be changed by odd integers, something explicitly forbidden in the noninteracting limit of two identical spins. This correlated topological feature may, in turn, result in an enhancement of the quantized response.
Robust response signals, such as those predicted for the topological frequency converter, are of fundamental interest for potential technological applications of topological quantum matter. Based on an open quantum system implementation of the frequency converter, we propose a novel mechanism of topological quantization coined ''topological burning glass effect''. Remarkably, this mechanism amplifies the local response of the driven two-level system by an integer that is proportional to the number of environmental degrees of freedom to which the system is strongly coupled. Specifically, our findings are illustrated by the extension of the frequency converter to a central spin model. There, the local energy transfer mediated exclusively by the central spin is significantly enhanced by the collective motion of the surrounding spins. In this sense, the central spin adopts the topological nature of the total system in its non-unitary dynamics, taking into account the correlations with the environment.
Emergent phenomena in condensed matter physics like, e.g., magnetism, superconductivity, or non-trivial topology often come along with a surprise and exert great fascination to researchers up to this day. Within this thesis, we are concerned with the analysis of associated types of order that arise due to strong electronic interactions and focus on the high-\(T_c\) cuprates and Kondo systems as two prime candidates. The underlying many-body problem cannot be solved analytically and has given rise to the development of various approximation techniques to tackle the problem.
In concrete terms, we apply the auxiliary particle approach to investigate tight-binding Hamiltonians subject to a Hubbard interaction term to account for the screened Coulomb repulsion. Thereby, we adopt the so-called Kotliar-Ruckenstein slave-boson representation that reduces the problem to non-interacting quasiparticles within a mean-field approximation. Part I provides a pedagogical review of the theory and generalizes the established formalism to encompass Gaussian fluctuations around magnetic ground states as a crucial step to obtaining novel results.
Part II addresses the two-dimensional one-band Hubbard model, which is known to approximately describe the physics of the high-\(T_c\) cuprates that feature high-temperature superconductivity and various other exotic quantum phases that are not yet fully understood. First, we provide a comprehensive slave-boson analysis of the model, including the discussion of incommensurate magnetic phases, collective modes, and a comparison to other theoretical methods that shows that our results can be massively improved through the newly implemented fluctuation corrections. Afterward, we focus on the underdoped regime and find an intertwining of spin and charge order signaled by divergences of the static charge susceptibility within the antiferromagnetic domain. There is experimental evidence for such inhomogeneous phases in various cuprate materials, which has recently aroused interest because such correlations are believed to impact the formation of Cooper pairs. Our analysis identifies two distinct charge-ordering vectors, one of which can be attributed to a Fermi-surface nesting effect and quantitatively fits experimental data in \(\mathrm{Nd}_{2-\mathrm{x}}\mathrm{Ce}_\mathrm{x}\mathrm{CuO}_4\) (NCCO), an electron-doped cuprate compound. The other resembles the so-called Yamada relation implying the formation of periodic, double-occupied domain walls with a crossover to phase separation for small dopings.
Part III investigates Kondo systems by analyzing the periodic Anderson model and its generalizations. First, we consider Kondo metals and detect weakly magnetized ferromagnetic order in qualitative agreement with experimental observations, which hinders the formation of heavy fermions. Nevertheless, we suggest two different parameter regimes that could host a possible Kondo regime in the context of one or two conduction bands. The part is concluded with the study of topological order in Kondo insulators based on a three-dimensional model with centrosymmetric spin-orbit coupling. Thereby, we classify topologically distinct phases through appropriate \(\mathbb{Z}_2\) invariants and consider paramagnetic and antiferromagnetic mean-field ground states. Our model parameters are chosen to specifically describe samarium hexaboride (\(\mbox{SmB}_6\)), which is widely believed to be a topological Kondo insulator, and we identify topologically protected surface states in agreement with experimental evidence in that material. Moreover, our theory predicts the emergence of an antiferromagnetic topological insulator featuring one-dimensional hinge-states as the signature of higher-order topology in the strong coupling regime. While the nature of the true ground state is still under debate, corresponding long-range magnetic order has been observed in pressurized or alloyed \(\mbox{SmB}_6\), and recent experimental findings point towards non-trivial topology under these circumstances. The ability to understand and control topological systems brings forth promising applications in the context of spintronics and quantum computing.
Atomically thin semiconductors from the transition metal dichalcogenide family are materials in which the optical response is dominated by strongly bound excitonic complexes. Here, we present a theory of excitons in two-dimensional semiconductors using a tight-binding model of the electronic structure. In the first part, we review extensive literature on 2D van der Waals materials, with particular focus on their optical response from both experimental and theoretical points of view. In the second part, we discuss our ab initio calculations of the electronic structure of MoS\(_2\), representative of a wide class of materials, and review our minimal tight-binding model, which reproduces low-energy physics around the Fermi level and, at the same time, allows for the understanding of their electronic structure. Next, we describe how electron-hole pair excitations from the mean-field-level ground state are constructed. The electron–electron interactions mix the electron-hole pair excitations, resulting in excitonic wave functions and energies obtained by solving the Bethe–Salpeter equation. This is enabled by the efficient computation of the Coulomb matrix elements optimized for two-dimensional crystals. Next, we discuss non-local screening in various geometries usually used in experiments. We conclude with a discussion of the fine structure and excited excitonic spectra. In particular, we discuss the effect of band nesting on the exciton fine structure; Coulomb interactions; and the topology of the wave functions, screening and dielectric environment. Finally, we follow by adding another layer and discuss excitons in heterostructures built from two-dimensional semiconductors.
Context. In active galaxies, matter is accreted onto super massive black holes (SMBH). This accretion process causes a region roughly the size of our solar system to outshine the entire host galaxy, forming an active galactic nucleus (AGN). In some of these active galaxies, highly relativistic particle jets are formed parallel to the rotation axis of the super massive black hole. A fraction of these sources is observed under a small inclination angle between the pointing direction of the jet and the observing line of sight. These sources are called blazars. Due to the small inclination angle and the highly relativistic speeds of the particles in the jet, beaming effects occur in the radiation of these particles. Blazars can be subdivided into the high luminosity flat spectrum radio quasars (FSRQs) and the low luminosity BL Lacertae objects (BL Lacs). As all AGN, blazars are broadband emitters and therefore observable from the longest wavelengths in the radio regime to the shortest wavelengths in the gamma-ray regime. In this thesis I will analyze blazars at these two extremes with respect to their parsec-scale properties in the radio and their time evolution properties in gamma-ray flux.
Method. In the radio regime the technique of very long baseline interferometry (VLBI) can be used in order to spatially resolve the synchrotron radiation coming from those objects down to sub-parsec scales. This information can be used to observe the time evolution of the structure of such sources. This is done in large monitoring programs such as the MOJAVE (15 GHz) and the Boston University blazar monitoring program (43 GHz). In this thesis I utilize data of 28 sources from these monitoring programs spanning 10 years of observation from 2003 to 2013, resulting in over 1800 observed epochs, to study the brightness temperature and diameter gradients of these jets. I conduct a search for systematic geometry transitions in the radio jets. The synchrotron cooling time in the radio core of the jets is used to determine the magnetic field strength in the radio core. Considering the jet geometry, these magnetic field strengths are scaled to the ergosphere of the SMBH in order to obtain the distance of the radio core to the SMBH.
In the gamma-regime these blazars cannot be spatially resolved. Due to this, it is hard to put strong constrains onto where the gamma-ray emitting region is. Blazars have shown to be variable at high energies on time scales down to minutes. The nature of this variability can be studied in order to put constrains on the particle acceleration mechanism and possibly the region and size of the gamma-ray emitting region. The variability of blazars in the energy range between 0.1 GeV and 1 GeV can for example be observed with the pair-conversion telescope on board the Fermi satellite. I use 10 years of data from the Fermi-LAT (LAT: Large Area Telescope) satellite in order to study the variability of a large sample of blazars (300-800, depending on the used significance filters for data points). I quantify this variability with the Ornstein-Uhlenbeck (OU) parameters and the power spectral density (PSD) slopes. The same procedure is applied to 20 light curves available for the radio sample.
Results. The diameter evolution along the jet axis of the radio sources suggests, that FSRQs feature flatter gradients than BL Lacs. Fitting these gradients, it is revealed that BL Lacs are systematically better described by a simple single power law than FSRQs. I found 9 sources with a strongly constrained geometry transition. The sources are 0219+421, 0336-019, 0415+379, 0528+134, 0836+710, 1101+384, 1156+295, 1253-055 and 2200+420. In all of these sources, the geometry transition regions are further out in the jet than the Bondi sphere. The magnetic field strengths of BL Lacs is systematically larger than that of FSRQs. However the scaling of these fields suggest that the radio cores of BL Lac objects are closer to the SMBHs than the radio cores of FSRQs. Analyzing the variability of Fermi-LAT light curves yields consistent results for all samples. FSRQs show systematically steeper PSD slopes and feature OU parameters more favorable to strong variability than BL Lacs. The Fermi-LAT light curves of the sub-sample of radio jets, suggest an anticorrelation between the jet complexity from the radio observations and the OU-parameters as well as the PSD slopes from the gamma-ray observations.
Conclusion.
The flatter diameter gradients of FSRQs suggest that these sources are more collimated further down the jet than BL Lacs. The systematically better description of the diameter and brightness temperature gradient by a single power law of BL Lacs, suggest that FSRQs are more complex with respect to the diameter evolution along the jet and the surface brightness distribution than BL Lac objects. FSRQs often feature regions where recollimation can occur in distinct knots within the jets. For the sources where a geometry transition could be constrained, the Bondi radius, being systematically smaller than the position of the transition region along the jet axis, suggest that changing pressure gradients are not the sole cause for these systematic geometry transitions. Nevertheless they may be responsible for recollimation regions, found typically downstream the jet, beyond the Bondi radius and the transition zone. The difference in the distance of the radio cores between FSRQs and BL Lacs is most likely due to the combination of differences in SMBH masses and systematically smaller jet powers in BL Lacs. The variability in energy ranges above 100 MeV and above 1 GeV-regime suggest that many light curves of BL Lac objects are more likely to be white noise while the PSD slopes and the OU parameters of FSRQ gamma-ray light curves favor stronger variability on larger time scales with respect to the time binning of the analyzed light curve. Although the anticorrelation of the jet complexity acquired from the radio observations and the PSD slopes and OU parameters from the gamma-observations suggest that more complex sources favor OU parameters and PSD slopes resulting in more variability (not white noise) it is beyond the scope of this thesis to pinpoint whether this correlation results from causation. The question whether a complex jet causes more gamma-ray variability or more gamma-ray variability causes more complex jets cannot be answered at this point. Nevertheless the computed correlation measures suggest that this dependence is most likely not linear and therefore an indication that these effects might even interact.
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.
We study the topological properties of the generalized two-dimensional (2D) Su-Schrieffer-Heeger (SSH) models. We show that a pair of Dirac points appear in the Brillouin zone (BZ), consisting a semimetallic phase. Interestingly, the locations of these Dirac points are not pinned to any high-symmetry points of the BZ but tunable by model parameters. Moreover, the merging of two Dirac points undergoes a novel topological phase transition, which leads to either a weak topological insulator or a nodal-line metallic phase. We demonstrate these properties by constructing two specific models, which we referred as type-I and type-II 2D SSH models. The feasible experimental platforms to realize our models are also discussed.
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.
We develop a joint formalism and numerical framework for analyzing the superconducting instability of metals from a weak coupling perspective. This encompasses the Kohn–Luttinger formulation of weak coupling renormalization group for superconductivity as well as the random phase approximation imposed on the diagrammatic expansion of the two-particle Green’s function. The central quantity to resolve is the effective interaction in the Cooper channel, for which we develop an optimized numerical framework. Our code is capable of treating generic multi-orbital models in two as well as three spatial dimensions and, in particular, arbitrary avenues of spin-orbit coupling.
We analyze a variety of integration schemes for the momentum space functional renormalization group calculation with the goal of finding an optimized scheme. Using the square lattice t-t' Hubbard model as a testbed we define and benchmark the quality. Most notably we define an error estimate of the solution for the ordinary differential equation circumventing the issues introduced by the divergences at the end of the FRG flow. Using this measure to control for accuracy we find a threefold reduction in number of required integration steps achievable by choice of integrator. We herewith publish a set of recommended choices for the functional renormalization group, shown to decrease the computational cost for FRG calculations and representing a valuable basis for further investigations.
Effective lifting of the topological protection of quantum spin Hall edge states by edge coupling
(2022)
The scientific interest in two-dimensional topological insulators (2D TIs) is currently shifting from a more fundamental perspective to the exploration and design of novel functionalities. Key concepts for the use of 2D TIs in spintronics are based on the topological protection and spin-momentum locking of their helical edge states. In this study we present experimental evidence that topological protection can be (partially) lifted by pairwise coupling of 2D TI edges in close proximity. Using direct wave function mapping via scanning tunneling microscopy/spectroscopy (STM/STS) we compare isolated and coupled topological edges in the 2D TI bismuthene. The latter situation is realized by natural lattice line defects and reveals distinct quasi-particle interference (QPI) patterns, identified as electronic Fabry-Pérot resonator modes. In contrast, free edges show no sign of any single-particle backscattering. These results pave the way for novel device concepts based on active control of topological protection through inter-edge hybridization for, e.g., electronic Fabry-Pérot interferometry.
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.
Hard X-ray Properties of Relativistically Beamed Jets from Radio- and Gamma-Ray-Bright Blazars
(2022)
In this work I characterize the hard X-ray properties of blazars, active galactic nuclei with highly beamed emission, which are notoriously hard to detect in this energy range. I employ pre-defined samples of beamed AGN: the radio-selected MOJAVE and TANAMI samples, as well as the most recent gamma-ray-selected Fermi/LAT 4LAC catalog. The hard X-ray data is extracted from the 105-month all-sky survey maps of the Swift/BAT (Burst Alert Telescope) in the energy band of 20 keV to 100 keV. A great majority of both the MOJAVE and TANAMI samples are significantly detected, with signal-to noise ratios of the sources often just below the X-ray catalog signal thresholds. All blazar sub-types (FSRQs, BL Lacs) and radio galaxies show characteristic ranges of X-ray flux, luminosity, and photon index. Their properties are correlated with the corresponding SED's shape / peak frequency. The LogN-LogS distributions of the samples show a scarcity of blazars in the middle and lower X-ray flux range, indicating differing evolutionary paths between radio and X-ray emission, which is also suggested by the corresponding luminosity functions. Compared to the radio samples, the 4LAC sources are on average significantly less bright in the BAT band since this range often coincides with the spectral gap region between the two big SED emission bumps. Also, the spectral shapes differ notably, especially for the sub-type of BL Lacs. Using the parameter space of X-ray and gamma-ray photon indices, 35 blazar candidate sources can be assigned to either the FSRQ or BL Lac type with high certainty. The reason why many blazars are weak in this energy band can be traced back to a number of factors: the selection bias of the initial sample, differential evolution of the X-rays and the wavelengths in which the sample is defined, and the limited sensitivity of the observing instruments.
In the last decade continuous-time quantum Monte Carlo in the hybridization expansion (CTHYB) was one of the most successful Monte Carlo techniques to describe correlated quantum phenomena in conjunction with dynamical mean field theory (DMFT). The first part of the thesis consists of algorithmical developments regarding CTHYB and DMFT. I provide a complete derivation and an extensive discussion of the expansion formula. We generalized it to treat spin-orbit coupling, and invented the superstate sampling algorithm to make it efficient enough for describing systems with general interactions, crystal fields and spin-orbit coupling at low temperatures. But CTHYB is known to fail in the standard implementation for equal-time correlators, certain higher-order Green’s functions and the atomic limit; we discovered that its estimator for the Greens function is also inconsistent for Anderson impurities with finite, discrete baths. I focus then on further improvements of CTHYB that we have conceived and worked on, in particular for f-orbitals and for taking physical symmetries into account in the calculation of the Monte Carlo observables. The second part of the thesis presents selected physical applications of these methods. I show DMFT calculations of highest accuracy for elemental iron and nickel and discover a new mechanism of magnetic ordering in nickel: the ordering of band structure-induced local moments. Then we analyze the stability of this phenomenon under pressure and temperatures, that characterize in the Earth’s core. We find, that the mechanism survives these conditions and may give a significant contribution to the generation of the Earth’s magnetic field. The next topic is the stability of double Dirac fermions against electronic correlations. We find, that the Coulomb interaction in the corresponding material Bi2 CuO4 are strong enough to destroy the double Dirac cone, and substantial uniform pressure is necessary to restore them. In the last chapter I derive the properties of Higgs and Goldstone bosons from Ginzburg-Landau theory, and identify these excitations in a model of an excitonic magnet.
The quantum Hall (QH) effect, which can be induced in a two-dimensional (2D) electron gas by an external magnetic field, paved the way for topological concepts in condensed matter physics. While the QH effect can for that reason not exist without Landau levels, there is a plethora of topological phases of matter that can exist even in the absence of a magnetic field. For instance, the quantum spin Hall (QSH), the quantum anomalous Hall (QAH), and the three-dimensional (3D) topological insulator (TI) phase are insulating phases of matter that owe their nontrivial topology to an inverted band structure. The latter results from a strong spin-orbit interaction or, generally, from strong relativistic corrections. The main objective of this thesis is to explore the fate of these preexisting topological states of matter, when they are subjected to an external magnetic field, and analyze their connection to quantum anomalies. In particular, the realization of the parity anomaly in solid state systems is discussed. Furthermore, band structure engineering, i.e., changing the quantum well thickness, the strain, and the material composition, is employed to manipulate and investigate various topological properties of the prototype TI HgTe.
Like the QH phase, the QAH phase exhibits unidirectionally propagating metallic edge channels. But in contrast to the QH phase, it can exist without Landau levels. As such, the QAH phase is a condensed matter analog of the parity anomaly. We demonstrate that this connection facilitates a distinction between QH and QAH states in the presence of a magnetic field. We debunk therefore the widespread belief that these two topological phases of matter cannot be distinguished, since they are both described by a $\mathbb{Z}$ topological invariant. To be more precise, we demonstrate that the QAH topology remains encoded in a peculiar topological quantity, the spectral asymmetry, which quantifies the differences in the number of states between the conduction and valence band. Deriving the effective action of QAH insulators in magnetic fields, we show that the spectral asymmetry is thereby linked to a unique Chern-Simons term which contains the information about the QAH edge states. As a consequence, we reveal that counterpropagating QH and QAH edge states can emerge when a QAH insulator is subjected to an external magnetic field. These helical-like states exhibit exotic properties which make it possible to disentangle QH and QAH phases. Our findings are of particular importance for paramagnetic TIs in which an external magnetic field is required to induce the QAH phase.
A byproduct of the band inversion is the formation of additional extrema in the valence band dispersion at large momenta (the `camelback'). We develop a numerical implementation of the $8 \times 8$ Kane model to investigate signatures of the camelback in (Hg,Mn)Te quantum wells. Varying the quantum well thickness, as well as the Mn-concentration, we show that the class of topologically nontrivial quantum wells can be subdivided into direct gap and indirect gap TIs. In direct gap TIs, we show that, in the bulk $p$-regime, pinning of the chemical potential to the camelback can cause an onset to QH plateaus at exceptionally low magnetic fields (tens of mT). In contrast, in indirect gap TIs, the camelback prevents the observation of QH plateaus in the bulk $p$-regime up to large magnetic fields (a few tesla). These findings allowed us to attribute recent experimental observations in (Hg,Mn)Te quantum wells to the camelback. Although our discussion focuses on (Hg,Mn)Te, our model should likewise apply to other topological materials which exhibit a camelback feature in their valence band dispersion.
Furthermore, we employ the numerical implementation of the $8\times 8$ Kane model to explore the crossover from a 2D QSH to a 3D TI phase in strained HgTe quantum wells. The latter exhibit 2D topological surface states at their interfaces which, as we demonstrate, are very sensitive to the local symmetry of the crystal lattice and electrostatic gating. We determine the classical cyclotron frequency of surface electrons and compare our findings with experiments on strained HgTe.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].
Since the genesis of condensed matter physics, strongly correlated fermionic systems have shown a variety of fascinating properties and remain a vital topic in the field.
Such systems arise through electronic interaction, and despite decades of intensive research, no holistic approach to solving this problem has been found.
During that time, physicists have compiled a wealth of individual experimental and theoretical results, which together give an invaluable insight into these materials, and, in some instances, can explain correlated phenomena.
However, there are several systems that stubbornly refuse to fall completely in line with current theoretical descriptions, among them the high-\( T_c{}\) cuprates and heavy fermion compounds.
Although the two material classes have been around for the better part of the last 50 years, large portions of their respective phase diagram are still under intensive debate.
Recent experiments in several electron-doped cuprates compounds, e.g. neodymium cerium copper oxide (Nd\(_{2x}\)Ce\(_x\)CuO\(_4\)), reveal a charge ordering about an antiferromagnetic ground state.
So far, it has not been conclusively clarified how this intertwining of charge and spin polarization comes about and how it can be reconciled with a rigorous theoretical description.
The heavy-fermion semimetals, on the other hand, have enjoyed renewed scientific interest with the discovery of topological Kondo insulators, a new material class offering a unique interface of topology, symmetry breaking, and correlated phenomena. In this context, samarium hexaboride (SmB\(_6\)) has emerged as a prototypical system, which may feature a topological ground state.
In this thesis, we present a spin rotational invariant auxiliary particle approach to investigate the propensities of interacting electrons towards forming new states of order.
In particular, we study the onset of spin and charge order in high-\( T_c{}\) cuprate systems and Kondo lattices, as well as the interplay of magnetism and topology.
To that end, we use a sophisticated mean-field approximation of bosonic auxiliary particles augmented by a stability analysis of the saddle point via Gaussian fluctuations.
The latter enables the derivation of dynamic susceptibilities, which describe the response of the system under external fields and offer a direct comparison to experiments.
Both the mean-field and fluctuation formalisms require a numerical tool that is capable of extremizing the saddle point equations, on the one hand, and reliably solving a loop integral of the susceptibility-type, on the other.
A full, from scratch derivation of the formalism tailored towards a software implementation, is provided and pedagogically reviewed.
The auxiliary particle method allows for a rigorous description of incommensurate magnetic order and compares well to other established numerical and analytical techniques.
Within our analysis, we employ the two-dimensional one-band Hubbard as well as the periodic Anderson model as minimal Hamiltonians for the high-\( T_c{}\) cuprates and Kondo systems, respectively.
For the former, we observe a regime of intertwined charge- and spin-order in the electron-doped regime, which matches recent experimental observations in the cuprate material Nd\(_{2x}\)Ce\(_x\)CuO\(_4\).
Furthermore, we localize the emergence of a Kondo regime in the periodic Anderson model and establish the magnetic phase diagram of the two-band model for topological Kondo insulators.
The emerging antiferromagnetic ground state can be characterized by its topological properties and shows, for a non-trivial phase, topologically protected hinge modes.
The main goal of this thesis is to elucidate the sense in which recent experimental progress in condensed matter physics, namely the verification of two-dimensional Dirac-like materials and their control in ballistic- as well as hydrodynamic transport experiments enables the observation of a well-known 'high-energy' phenomenon: The parity anomaly of planar quantum electrodynamics (QED\(_{2+1}\)). In a nutshell, the low-energy physics of two-dimensional Quantum Anomalous Hall (QAH) insulators like (Hg,Mn)Te quantum wells or magnetically doped (Bi,Sb)Te thin films can be described by the combined response of two 2+1 space-time dimensional Chern insulators with a linear dispersion in momentum. Due to their Dirac-like spectra, each of those Chern insulators is directly related to the parity anomaly of planar quantum electrodynamics. However, in contrast to a pure QED\(_{2+1}\) system, the Lagrangian of each Chern insulator is described by two different mass terms: A conventional momentum-independent Dirac mass \(m\), as well as a momentum-dependent so-called Newtonian mass term \(B \vert \mathbf{k} \vert^2\). According to the parity anomaly it is not possible to well-define a parity- and U(1) gauge invariant quantum system in 2+1 space-time dimensions. More precisely, starting with a parity symmetric theory at the classical level, insisting on gauge-invariance at the quantum level necessarily induces parity-odd terms in the calculation of the quantum effective action. The role of the Dirac mass term in the calculation of the effective QED\(_{2+1}\) action has been initially studied in Phys. Rev. Lett. 51, 2077 (1983). Even in the presence of a Dirac mass, the associated fermion determinant diverges and lacks gauge invariance. This requires a proper regularization/renormalizaiton scheme and, as such, transfers the peculiarities of the parity anomaly to the massive case.
In the scope of this thesis, we connect the momentum-dependent Newtonian mass term of a Chern insulator to the parity anomaly. In particular, we reveal, that in the calculation of the effective action, before renormalization, the Newtonian mass term acts similarly to a parity-breaking element of a high-energy regularization scheme. This calculation allows us to derive the finite frequency correction to the DC Hall conductivity of a QAH insulator. We derive that the leading order AC correction contains a term proportional to the Chern number. This term originates from the Newtonian mass and can be measured via electrical or via magneto-optical experiments. The Newtonian mass, in particular, significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene.
In addition, we study the effective action of the aforementioned Chern insulators in external out-of-plane magnetic fields. We show that as a consequence of the parity anomaly the QAH phase in (Hg,Mn)Te quantum wells or in magnetically doped (Bi,Sb)Te thin films survives in out-of-plane magnetic fields, violates the Onsager relation, and can therefore be distinguished from a conventional quantum Hall (QH) response. As a smoking-gun of the QAH phase in increasing magnetic fields, we predict a transition from a quantized Hall plateau with \(\sigma_\mathrm{xy}= -\mathrm{e}^2/\mathrm{h}\) to a not perfectly quantized plateau which is caused by scattering processes between counter-propagating QH and QAH edge states. This transition is expected to be of significant relevance in paramagnetic QAH insulators like (Hg,Mn)Te/CdTe quantum wells, in which the exchange interaction competes against the out-of-plane magnetic field.
All of the aforementioned results do not incorporate finite temperature effects. In order to shed light on such phenomena, we further analyze the finite temperature Hall response of 2+1 dimensional Chern insulators under the combined influence of a chemical potential and an out-of-plane magnetic field. As we have mentioned above, this non-dissipative transport coefficient is directly related to the parity anomaly of planar quantum electrodynamics. Within the scope of our analysis we show that the parity anomaly itself is not renormalized by finite temperature effects. However, the parity anomaly induces two terms of different physical origin in the effective Chern-Simons action of a QAH insulator, which are directly proportional to its Hall conductivity. The first term is temperature and chemical potential independent and solely encodes the intrinsic topological response. The second term specifies the non-topological thermal response of conduction- and valence band modes, respectively. We show that the relativistic mass \(m\) of a Chern insulator counteracts finite temperature effects, whereas its non-relativistic Newtonian mass \(B \vert \mathbf{k} \vert^2 \) enhances these corrections. In addition, we are extending our associated analysis to finite out-of-plane magnetic fields, and relate the thermal response of a Chern insulator therein to the spectral asymmetry, which is a measure of the parity anomaly in out-of-plane magnetic fields.
In the second part of this thesis, we study the hydrodynamic properties of two-dimensional electron systems with a broken time-reversal and parity symmetry. Within this analysis we are mainly focusing on the non-dissipative transport features originating from a peculiar hydrodynamic transport coefficient: The Hall viscosity \(\eta_\mathrm{H}\). In out-of-plane magnetic fields, the Hall viscous force directly competes with the Lorentz force, as both mechanisms contribute to the overall Hall voltage. In our theoretical considerations, we present a way of uniquely distinguishing these two contributions in a two-dimensional channel geometry by calculating their functional dependencies on all external parameters. We are in particular deriving that the ratio of the Hall viscous contribution to the Lorentz force contribution is negative and that its absolute value decreases with an increasing width, slip-length and carrier density. Instead, it increases with the electron-electron mean free path in the channel geometry considered. We show that in typical materials such as GaAs the Hall viscous contribution can dominate the Lorentz signal up to a few tens of millitesla until the total Hall voltage vanishes and eventually is exceeded by the Lorentz contribution. Last but not least, we derive that the total Hall electric field has a parabolic form originating from Lorentz effects. Most remarkably, the offset of this parabola is directly characterized by the Hall viscosity. Therefore, in summary, our results pave the way to measure and to identify the Hall viscosity via both global and local measurements of the entire Hall voltage.
The tremendous phenomenological success of the Standard Model (SM) suggests that its flavor structure and gauge interactions may not be arbitrary but should have a fundamental first-principle explanation. In this work, we explore how the basic distinctive properties of the SM dynamically emerge from a unified New Physics framework tying together both flavor physics and Grand Unified Theory (GUT) concepts. This framework is suggested by a novel anomaly-free supersymmetric chiral E\(_6\)×SU(2)\(_F\)×U(1)\(_F\) GUT containing the SM. Among the most appealing emergent properties of this theory is the Higgs-matter unification with a highly-constrained massless chiral sector featuring two universal Yukawa couplings close to the GUT scale. At the electroweak scale, the minimal SM-like effective field theory limit of this GUT represents a specific flavored three-Higgs doublet model consistent with the observed large hierarchies in the quark mass spectra and mixing already at tree level.
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.
We investigated emission properties of photonic structures with InAs/InGaAlAs/InP quantum dashes grown by molecular beam epitaxy on a distributed Bragg reflector. In high-spatial-resolution photoluminescence experiment, well-resolved sharp spectral lines are observed and single-photon emission is detected in the third telecommunication window characterized by very low multiphoton events probabilities. The photoluminescence spectra measured on simple photonic structures in the form of cylindrical mesas reveal significant intensity enhancement by a factor of 4 when compared to a planar sample. These results are supported by simulations of the electromagnetic field distribution, which show emission extraction efficiencies even above 18% for optimized designs. When combined with relatively simple and undemanding fabrication approach, it makes this kind of structures competitive with the existing solutions in that spectral range and prospective in the context of efficient and practical single-photon sources for fiber-based quantum networks applications.
The observation of electromagnetic counterparts to both high energy neutrinos and gravitational waves marked the beginning of a new era in astrophysics. The multi-messenger approach allows us to gain new insights into the most energetic events in the Universe such as gamma-ray bursts, supernovas, and black hole mergers. Real-time multi-messenger alerts are the key component of the observational strategies to unravel the transient signals expected from astrophysical sources. Focusing on the high-energy regime, we present a historical perspective of multi-messenger observations, the detectors and observational techniques used to study them, the status of the multi-messenger alerts and the most significant results, together with an overview of the future prospects in the field.
The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.
Single-molecule super-resolution microscopy (SMLM) techniques like dSTORM can reveal biological structures down to the nanometer scale. The achievable resolution is not only defined by the localization precision of individual fluorescent molecules, but also by their density, which becomes a limiting factor e.g., in expansion microscopy. Artificial deep neural networks can learn to reconstruct dense super-resolved structures such as microtubules from a sparse, noisy set of data points. This approach requires a robust method to assess the quality of a predicted density image and to quantitatively compare it to a ground truth image. Such a quality measure needs to be differentiable to be applied as loss function in deep learning. We developed a new trainable quality measure based on Fourier Ring Correlation (FRC) and used it to train deep neural networks to map a small number of sampling points to an underlying density. Smooth ground truth images of microtubules were generated from localization coordinates using an anisotropic Gaussian kernel density estimator. We show that the FRC criterion ideally complements the existing state-of-the-art multiscale structural similarity index, since both are interpretable and there is no trade-off between them during optimization. The TensorFlow implementation of our FRC metric can easily be integrated into existing deep learning workflows.
We employ the AdS/CFT correspondence and hydrodynamics to analyze the transport properties of \(2+1\) dimensional electron fluids. In this way, we use theoretical methods from both condensed matter and high-energy physics to derive tangible predictions that are directly verifiable in experiment.
The first research topic we consider is strongly-coupled electron fluids. Motivated by early results by Gurzhi on the transport properties of weakly coupled fluids, we consider whether similar properties are manifest in strongly coupled fluids. More specifically, we focus on the hydrodynamic tail of the Gurzhi effect: A decrease in fluid resistance with increasing temperature due to the formation of a Poiseuille flow of electrons in the sample. We show that the hydrodynamic tail of the Gurzhi effect is also realized in strongly coupled and fully relativistic fluids, but with modified quantitative features. Namely, strongly-coupled fluids always exhibit a smaller resistance than weakly coupled ones and are, thus, far more efficient conductors. We also suggest that the coupling dependence of the resistance can be used to measure the coupling strength of the fluid. In view of these measurements, we provide analytical results for the resistance as a function of the shear viscosity over entropy density \(\eta/s\) of the fluid. \(\eta/s\) is itself a known function of the coupling strength in the weak and infinite coupling limits.
In further analysis for strongly-coupled fluids, we propose a novel strongly coupled Dirac material based on a kagome lattice, Scandium-substituted Herbertsmithite (ScHb). The large coupling strength of this material, as well as its Dirac nature, provides us with theoretical and experimental access to non-perturbative relativistic and quantum critical physics. A highly suitable method for analyzing such a material's transport properties is the AdS/CFT correspondence. Concretely, using AdS/CFT we derive an estimate for ScHb's \(\eta/s\) and show that it takes a value much smaller than that observed in weakly coupled materials. In turn, the smallness of \(\eta/s\) implies that ScHb's Reynolds number, \(Re\), is large. In fact, \(Re\) is large enough for turbulence, the most prevalent feature of fluids in nature, to make its appearance for the first time in electronic fluids.
Switching gears, we proceed to the second research topic considered in this thesis: Weakly coupled parity-breaking electron fluids. More precisely, we analyze the quantitative and qualitative changes to the classical Hall effect, for electrons propagating hydrodynamically in a lead. Apart from the Lorentz force, a parity-breaking fluid's motion is also impacted by the Hall-viscous force; the shear-stress force induced by the Hall-viscosity. We show that the interplay of these two forces leads to a hydrodynamic Hall voltage with non-linear dependence on the magnetic field. More importantly, the Lorentz and Hall-viscous forces become equal at a non-vanishing magnetic field, leading to a trivial hydrodynamic Hall voltage. Moreover, for small magnetic fields we provide analytic results for the dependence of the hydrodynamic Hall voltage on all experimentally-tuned parameters of our simulations, such as temperature and density. These dependences, along with the zero of the hydrodynamic Hall voltage, are distinct features of hydrodynamic transport and can be used to verify our predictions in experiments.
Last but not least, we consider how a distinctly electronic property, spin, can be included into the hydrodynamic framework. In particular, we construct an effective action for non-dissipative spin hydrodynamics up to first order in a suitably defined derivative expansion. We also show that interesting spin-transport effects appear at second order in the derivative expansion. Namely, we show that the fluid's rotation polarizes its spin. This is the hydrodynamic manifestation of the Barnett effect and provides us with an example of hydrodynamic spintronics.
To conclude this thesis, we discuss several possible extensions of our research, as well as proposals for research in related directions.
Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness.
According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables.
Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.
Electron–phonon scatterings in solid-state systems are pivotal processes in determining many key physical quantities such as charge carrier mobilities and thermal conductivities. Here, we report direct probing of phonon mode specific electron–phonon scatterings in layered semiconducting transition metal dichalcogenides WSe2, MoSe2, WS2, and MoS2 through inelastic electron tunneling spectroscopy measurements, quantum transport simulations, and density functional calculation. We experimentally and theoretically characterize momentum-conserving single- and two-phonon electron–phonon scatterings involving up to as many as eight individual phonon modes in mono- and bilayer films, among which transverse, longitudinal acoustic and optical, and flexural optical phonons play significant roles in quantum charge flows. Moreover, the layer-number sensitive higher-order inelastic electron–phonon scatterings, which are confirmed to be generic in all four semiconducting layers, can be attributed to differing electronic structures, symmetry, and quantum interference effects during the scattering processes in the ultrathin semiconducting films.
Analytic integration of soft and collinear radiation in factorised QCD cross sections at NNLO
(2021)
Within the framework of local analytic sector subtraction, we present the full analytic integration of double-real and real-virtual local infrared counterterms that enter NNLO QCD computations with any number of massless final-state partons. We show that a careful choice of phase-space mappings leads to simple analytic results, including non-singular terms, that can be obtained with conventional integration techniques.
We investigate a scenario inspired by natural supersymmetry, where neutrino data is explained within a low-scale seesaw scenario. For this the minimal supersymmetric Standard Model is extended by adding light right-handed neutrinos and their superpartners, the R-sneutrinos. Moreover, we consider the lightest neutralinos to be Higgsino-like. We first update a previous analysis and assess to which extent does existing LHC data constrain the allowed slepton masses. Here we find scenarios where sleptons with masses as low as 175 GeV are consistent with existing data. However, we also show that the upcoming run will either discover or rule out sleptons with masses of 300 GeV, even for these challenging scenarios. We then take a scenario which is on the borderline of observability of the upcoming LHC run assuming a luminosity of 300 fb(-1). We demonstrate that a prospective international e(+)e(-) linear collider with a center of mass energy of 1 TeV will be able to discover sleptons in scenarios which are difficult for the LHC. Moreover, we also show that a measurement of the spectrum will be possible within 1-3 percent accuracy.
Polarized Z bosons from the decay of a Higgs boson produced in association with two jets at the LHC
(2021)
Investigating the polarization of weak bosons provides an important probe of the scalar and gauge sector of the Standard Model. This can be done in the Higgs decay to four leptons, whose Standard-Model leading-order amplitude enables to generate polarized observables from unpolarized ones via a fully-differential reweighting method. We study the Z-boson polarization from the decay of a Higgs boson produced in association with two jets, both in the gluon-fusion and in the vector-boson fusion channel. We also address the possibility of extending the results of this work to higher orders in perturbation theory.
Josephson junctions based on three-dimensional topological insulators offer intriguing possibilities to realize unconventional 𝑝-wave pairing and Majorana modes. Here, we provide a detailed study of the effect of a uniform magnetization in the normal region: We show how the interplay between the spin-momentum locking of the topological insulator and an in-plane magnetization parallel to the direction of phase bias leads to an asymmetry of the Andreev spectrum with respect to transverse momenta. If sufficiently large, this asymmetry induces a transition from a regime of gapless, counterpropagating Majorana modes to a regime with unprotected modes that are unidirectional at small transverse momenta. Intriguingly, the magnetization-induced asymmetry of the Andreev spectrum also gives rise to a Josephson Hall effect, that is, the appearance of a transverse Josephson current. The amplitude and current phase relation of the Josephson Hall current are studied in detail. In particular, we show how magnetic control and gating of the normal region can enable sizable Josephson Hall currents compared to the longitudinal Josephson current. Finally, we also propose in-plane magnetic fields as an alternative to the magnetization in the normal region and discuss how the planar Josephson Hall effect could be observed in experiments.