## Institut für Theoretische Physik und Astrophysik

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In this thesis we discuss the potential of nanodevices based on topological insulators. This novel class of matter is characterized by an insulating bulk with simultaneously conducting boundaries. To lowest order, the states that are evoking the conducting behavior in TIs are typically described by a Dirac theory. In the two-dimensional case, together with time- reversal symmetry, this implies a helical nature of respective states. Then, interesting physics appears when two such helical edge state pairs are brought close together in a two-dimensional topological insulator quantum constriction. This has several advantages. Inside the constriction, the system obeys essentially the same number of fermionic fields as a conventional quantum wire, however, it possesses more symmetries. Moreover, such a constriction can be naturally contacted by helical probes, which eventually allows spin- resolved transport measurements.
We use these intriguing properties of such devices to predict the formation and detection of several profound physical effects. We demonstrate that narrow trenches in quantum spin Hall materials – a structure we coin anti-wire – are able to show a topological super- conducting phase, hosting isolated non-Abelian Majorana modes. They can be detected by means of a simple conductance experiment using a weak coupling to passing by helical edge states. The presence of Majorana modes implies the formation of unconventional odd-frequency superconductivity. Interestingly, however, we find that regardless of the presence or absence of Majoranas, related (superconducting) devices possess an uncon- ventional odd-frequency superconducting pairing component, which can be associated to a particular transport channel. Eventually, this enables us to prove the existence of odd- frequency pairing in superconducting quantum spin Hall quantum constrictions. The symmetries that are present in quantum spin Hall quantum constrictions play an essen- tial role for many physical effects. As distinguished from quantum wires, quantum spin Hall quantum constrictions additionally possess an inbuilt charge-conjugation symmetry. This can be used to form a non-equilibrium Floquet topological phase in the presence of a time-periodic electro-magnetic field. This non-equilibrium phase is accompanied by topological bound states that are detectable in transport characteristics of the system. Despite single-particle effects, symmetries are particularly important when electronic in- teractions are considered. As such, charge-conjugation symmetry implies the presence of a Dirac point, which in turn enables the formation of interaction induced gaps. Unlike single-particle gaps, interaction induced gaps can lead to large ground state manifolds. In combination with ordinary superconductivity, this eventually evokes exotic non-Abelian anyons beyond the Majorana. In the present case, these interactions gaps can even form in the weakly interacting regime (which is rather untypical), so that the coexistence with superconductivity is no longer contradictory. Eventually this leads to the simultaneous presence of a Z4 parafermion and a Majorana mode bound at interfaces between quantum constrictions and superconducting regions.

The AdS/CFT correspondence is an explicit realization of the holographic principle. It describes a field theory living on the boundary of a volume by a gravitational theory living in the interior and vice-versa. With its origins in string theory, the correspondence incorporates an explicit relationship between the degrees of freedom of both theories: the AdS/CFT dictionary. One astonishing aspect of the AdS/CFT correspondence is the emergence of geometry from field theory.
On the gravity side, a natural way to probe the geometry is to study boundary-anchored extremal surfaces of different dimensionality. While there is no unified way to determine the field theory dual for such non-local quantities, the AdS/CFT dictionary contains entries for surfaces of certain dimensionality: it relates two-point functions to geodesics, the Wilson loop expectation value to two-dimensional surfaces and the entanglement entropy, i.e. a measure for entanglement between states in a region and in its complement, to co-dimension two surfaces in the bulk.
In this dissertation, we calculate these observables for gravity setups dual to thermal states in the field theory. The geometric dual is given by AdS Schwarzschild black holes in general dimensions. We find analytic results for minimal areas in this setup. One focus of our analysis is the high-temperature limit. The leading and subleading term in this limit have diverse interpretation for the different observables. For example, the subleading term of the entanglement entropy satisfies a c-theorem for renormalization flows and gives insights into the number of effective degrees of freedom.
The entanglement entropy emerged as the favorable way to probe the geometric dual. In addition to the extremal bulk surface, the holographic entanglement entropy associates a bulk region to the considered boundary region. The volume of this region is conjectured to be a measure of complexity, i.e. a measure of how difficult it is to obtain the corresponding field-theory state. Building on our aforementioned results for the entanglement entropy, we study this complexity for AdS Schwarzschild black holes in general dimensions.
In particular, we draw conclusions on how efficient holography encodes the field theory and compare these results to MERA tensor networks, a numerical tool to study quantum many-body systems.
Moreover, we holographically study the complexity of pure states. This sheds light on the notion of complexity in field theories. We calculate the complexity for a simple, calculable example: states obtained by conformal transformations of the vacuum state in AdS3/CFT2. In this lower-dimensional realization of AdS/CFT, the conformal group is infinite dimensional. We construct a continuous space of states with the same complexity as the vacuum state. Furthermore, we determine the change of complexity caused by small conformal transformation. The field-theory operator implementing this transformation is known and allows to compare the holographic results to field theory expectations.

Flux distribution is an important tool to understand the variability processes in activegalactic nuclei. We now have available a great deal of observational evidences pointing towards thepresence of log-normal components in the high energy light curves, and different models have beenproposed to explain these data. Here, we collect some of the recent developments on this topic usingthe well-known blazar Mrk 501 as example of complex and interesting aspects coming from its fluxdistribution in different energy ranges and at different timescales. The observational data we refer toare those collected in a complementary manner by Fermi-LAT over multiple years, and by the FirstG-APD Cherenkov Telescope (FACT) telescope and the H.E.S.S. array in correspondence of the brightflare of June 2014

The main objectives of the KM3NeT Collaboration are (i) the discovery and subsequent observation of high-energy neutrino sources in the Universe and (ii) the determination of the mass hierarchy of neutrinos. These objectives are strongly motivated by two recent important discoveries, namely: (1) the high-energy astrophysical neutrino signal reported by IceCube and (2) the sizable contribution of electron neutrinos to the third neutrino mass eigenstate as reported by Daya Bay, Reno and others. To meet these objectives, the KM3NeT Collaboration plans to build a new Research Infrastructure consisting of a network of deep-sea neutrino telescopes in the Mediterranean Sea. A phased and distributed implementation is pursued which maximises the access to regional funds, the availability of human resources and the synergistic opportunities for the Earth and sea sciences community. Three suitable deep-sea sites are selected, namely off-shore Toulon (France), Capo Passero (Sicily, Italy) and Pylos (Peloponnese, Greece). The infrastructure will consist of three so-called building blocks. A building block comprises 115 strings, each string comprises 18 optical modules and each optical module comprises 31 photo-multiplier tubes. Each building block thus constitutes a three-dimensional array of photo sensors that can be used to detect the Cherenkov light produced by relativistic particles emerging from neutrino interactions. Two building blocks will be sparsely configured to fully explore the IceCube signal with similar instrumented volume, different methodology, improved resolution and

It has been proposed that the observed diphoton excess at 750 GeV could be explained within the constrained minimal supersymmetric standard model via resonantly produced stop bound states. We reanalyze this scenario critically and extend previous work to include the constraints from the stability of the electroweak vacuum and from the decays of the stoponium into a pair of Higgs bosons. It is shown that the interesting regions of parameter space with a light stop and Higgs of the desired mass are ruled out by these constraints. This conclusion is not affected by the presence of the bound states because the binding energy is usually very small in the regions of parameter space which can explain the Higgs mass. Thus, this also leads to strong constraints on the diphoton production cross section which is in general too small.

Adding interactions to topological (non-)trivial free fermion systems can in general have four different effects: (i) In symmetry protected topological band insulators, the correlations may lead to the spontaneous breaking of some protecting symmetries by long-range order that gaps the topological boundary modes. (ii) In free fermion (semi-)metal, the interaction could vice versa also generate long-range order that in turn induces a topological mass term and thus generates non-trivial phases dynamically. (iii) Correlation might reduce the topological classification of free fermion systems by allowing adiabatic deformations between states of formerly distinct phases. (iv) Interaction can generate long-range entangled topological order in states such as quantum spin liquids or fractional quantum Hall states that cannot be represented by non-interacting systems. During the course of this thesis, we use numerically exact quantum Monte Carlo algorithms to study various model systems that (potentially) represent one of the four scenarios, respectively.
First, we investigate a two-dimensional $d_{xy}$-wave, spin-singlet superconductor, which is relevant for high-$T_c$ materials such as the cuprates. This model represents nodal topological superconductors and exhibits chiral flat-band edge states that are protected by time-reversal and translational invariance. We introduce the conventional Hubbard interaction along the edge in order to study their stability with respect to correlations and find ferromagnetic order in case of repulsive interaction as well as charge-density-wave order and/or additional $i$s-wave pairing for attractive couplings. A mean-field analysis that, for the first time, is formulated in terms of the Majorana edge modes suggests that any order has normal and superconducting contributions. For example, the ferromagnetic order appears in linear superposition with triplet pairing. This finding is well confirmed by the numerically exact quantum Monte Carlo investigation.
Second, we consider spinless electrons on a two-dimensional Lieb lattice that are subject to nearest-neighbor Coulomb repulsion. The low energy modes of the free fermion part constitute a spin-$1$ Dirac cone that might be gapped by several mass terms. One option breaks time-reversal symmetry and generates a topological Chern insulator, which mainly motivated this study. We employ two flavors of quantum Monte Carlo methods and find instead the formation of charge-density-wave order that breaks particle-hole symmetry. Additionally, due to sublattices of unequal size in Lieb lattices, this induces a finite chemical potential that drives the system away from half-filling. We argue that this mechanism potentially extends the range of solvable models with finite doping by coupling the Lieb lattice to the target system of interest.
Third, we construct a system with four layers of a topological insulators and interlayer correlation that respects one independent time-reversal and a unitary $\mathbb{Z}_2$ symmetry. Previous studies claim a reduced topological classification from $\mathbb{Z}$ to $\mathbb{Z}_4$, for example by gapping out degenerate zero modes in topological defects once the correlation term is designed properly. Our interaction is chosen according to this analysis such that there should exist an adiabatic deformation between states whose topological invariant differs by $\Delta w=\pm4$ in the free fermion classification. We use a projective quantum Monte Carlo algorithm to determine the ground-state phase diagram and find a symmetry breaking regime, in addition to the non-interacting semi-metal, that separates the free fermion insulators. Frustration reduces the size of the long-range ordered region until it is replaced by a first order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free fermion states into each other. We conclude that the prescribed reduction rules, which often use the bulk-boundary correspondence, are necessary but not sufficient and require a more careful investigation.
Fourth, we study conduction electron on a honeycomb lattice that form a Dirac semi-metal Kondo coupled to spin-1/2 degrees of freedom on a Kagome lattice. The local moments are described by a variant of the Balents-Fisher-Girvin model that has been shown to host a ferromagnetic phase and a $\mathbb{Z}_2$ spin liquid at strong frustration. Here, we report the first numerical exact quantum Monte Carlo simulation of the Kondo-coupled system that does not exhibit the negative-sign problem. When the local moments form a ferromagnet, the Kondo coupling induces an anti-ferromagnetic mass term in the conduction-electron system. At large frustration, the Dirac cone remains massless and the spin system forms a $\mathbb{Z}_2$ spin liquid. Owing to the odd number of spins per unit cell, this constitutes a non-Fermi liquid that violates Luttinger's theorem which relates the Fermi volume to the particle density in a Fermi liquid. This phase is a specific realization of the so called 'fractional Fermi liquid` as it has been first introduced in the context of heavy fermion models.

In this thesis we consider the hybrid quantum Monte Carlo method for simulations of the Hubbard and Su-Schrieffer-Heeger model. In the first instance, we discuss the hybrid quantum Monte Carlo method for the Hubbard model on a square lattice. We point out potential ergodicity issues and provide a way to circumvent them by a complexification of the method. Furthermore, we compare the efficiency of the hybrid quantum Monte Carlo method with a well established determinantal quantum Monte Carlo method for simulations of the half-filled Hubbard model on square lattices. One reason why the hybrid quantum Monte Carlo method loses the comparison is that we do not observe the desired sub-quadratic scaling of the numerical effort. Afterwards we present a formulation of the hybrid quantum Monte Carlo method for the Su-Schrieffer-Heeger model in two dimensions. Electron-phonon models like this are in general very hard to simulate using other Monte Carlo methods in more than one dimensions. It turns out that the hybrid quantum Monte Carlo method is much better suited for this model . We achieve favorable scaling properties and provide a proof of concept. Subsequently, we use the hybrid quantum Monte Carlo method to investigate the Su-Schrieffer-Heeger model in detail at half-filling in two dimensions. We present numerical data for staggered valence bond order at small phonon frequencies and an antiferromagnetic order at high frequencies. Due to an O(4) symmetry the antiferromagnetic order is connected to a superconducting charge density wave. Considering the Su-Schrieffer-Heeger model without tight-binding hopping reveals an additional unconstrained Z_2 gauge theory. In this case, we find indications for π-fluxes and a possible Z_2 Dirac deconfined phase as well as for a columnar valence bond ordered state at low phonon energies. In our investigations of the several phase transitions we discuss the different possibilities for the underlying mechanisms and reveal first insights into a rich phase diagram.

Blazars like Markarian 421 or Markarian 501 are active galactic nuclei (AGN), with their jets orientated towards the observer. They are among the brightest objects in the very high energy (VHE) gamma ray regime (>100 GeV). Their emitted gamma-ray fluxes are extremely variable, with changing activity levels on timescales between minutes, months, and even years. Several questions are part of the current research, such as the question of the emission regions or the engine of the AGN and the particle acceleration. A dedicated longterm monitoring program is necessary to investigate the properties of blazars in detail. A densely sampled and unbiased light curve allows for observation of both high and low states of the sources, and the combination with multi-wavelength observation could contribute to the answer of several questions mentioned above. FACT (First G-APD Cherenkov Telescope) is the first operational telescope using silicon photomultiplier (SiPM, also known as Geigermode—Avalanche Photo Diode, G-APD) as photon detectors. SiPM have a very homogenous and stable longterm performance, and allow operation even during full moon without any filter, leading to a maximal duty cycle for an Imaging Air Cherenkov Telescope (IACT). Hence, FACT is an ideal device for such a longterm monitoring of bright blazars. A small set of sources (e.g., Markarian 421, Markarian 501, 1ES 1959+650, and 1ES 2344+51.4) is currently being monitored. In this contribution, the FACT telescope and the concept of longterm monitoring of bright blazars will be introduced. The results of the monitoring program will be shown, and the advantages of densely sampled and unbiased light curves will be discussed.

A prototype detection unit of the KM3NeT deep-sea neutrino telescope has been installed at 3500m depth 80 km offshore the Italian coast. KM3NeT in its final configuration will contain several hundreds of detection units. Each detection unit is a mechanical structure anchored to the sea floor, held vertical by a submerged buoy and supporting optical modules for the detection of Cherenkov light emitted by charged secondary particles emerging from neutrino interactions. This prototype string implements three optical modules with 31 photomultiplier tubes each. These optical modules were developed by the KM3NeT Collaboration to enhance the detection capability of neutrino interactions. The prototype detection unit was operated since its deployment in May 2014 until its decommissioning in July 2015. Reconstruction of the particle trajectories from the data requires a nanosecond accuracy in the time calibration. A procedure for relative time calibration of the photomultiplier tubes contained in each optical module is described. This procedure is based on the measured coincidences produced in the sea by the 40K background light and can easily be expanded to a detector with several thousands of optical modules. The time offsets between the different optical modules are obtained using LED nanobeacons mounted inside them. A set of data corresponding to 600 h of livetime was analysed. The results show good agreement with Monte Carlo simulations of the expected optical background and the signal from atmospheric muons. An almost background-free sample of muons was selected by filtering the time correlated signals on all the three optical modules. The zenith angle of the selected muons was reconstructed with a precision of about 3∘.

Killing the cMSSM softly
(2016)

We investigate the constrained Minimal Supersymmetric Standard Model (cMSSM) in the light of constraining experimental and observational data from precision measurements, astrophysics, direct supersymmetry searches at the LHC and measurements of the properties of the Higgs boson, by means of a global fit using the program Fittino. As in previous studies, we find rather poor agreement of the best fit point with the global data. We also investigate the stability of the electro-weak vacuum in the preferred region of parameter space around the best fit point. We find that the vacuum is metastable, with a lifetime significantly longer than the age of the Universe. For the first time in a global fit of supersymmetry, we employ a consistent methodology to evaluate the goodness-of-fit of the cMSSM in a frequentist approach by deriving p values from large sets of toy experiments. We analyse analytically and quantitatively the impact of the choice of the observable set on the p value, and in particular its dilution when confronting the model with a large number of barely constraining measurements. Finally, for the preferred sets of observables, we obtain p values for the cMSSM below 10 %, i.e. we exclude the cMSSM as a model at the 90 % confidence level.