## Institut für Theoretische Physik und Astrophysik

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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 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.

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.

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.

We consider a scenario inspired by natural supersymmetry, where neutrino data is explained within a low-scale seesaw scenario. We extend the Minimal Supersymmetric Standard Model by adding light right-handed neutrinos and their superpartners, the R-sneutrinos, and consider the lightest neutralinos to be higgsino-like. We consider the possibilities of having either an R-sneutrino or a higgsino as lightest supersymmetric particle. Assuming that squarks and gauginos are heavy, we systematically evaluate the bounds on slepton masses due to existing LHC data.

We calculate the next-to-leading order electroweak corrections to the production of a photon pair in association with zero, one and two jets at the LHC. We use GoSam and Sherpa to obtain the results in a fully automated way. For a typical set of fiducial cuts the electroweak corrections lead to a modification of the total cross section of up to 3%, depending on the jet multiplicity. We find substantial contributions in differential distributions, leading to tens of per cent corrections for phase space regions within the reach of the LHC. Furthermore we investigate the importance of photon induced processes as well as subleading contributions. Photon induced processes are found to be negligible, subleading contributions can have a sizeable impact however they can be removed by appropriate phase space cuts.

Chromium dioxide CrO\(_2\) belongs to a class of materials called ferromagnetic half-metals, whose peculiar aspect is that they act as a metal in one spin orientation and as a semiconductor or insulator in the opposite one. Despite numerous experimental and theoretical studies motivated by technologically important applications of this material in spintronics, its fundamental properties such as momentumresolved electron dispersions and the Fermi surface have so far remained experimentally inaccessible because of metastability of its surface, which instantly reduces to amorphous Cr\(_2\)O\(_3\). In this work, we demonstrate that direct access to the native electronic structure of CrO\(_2\) can be achieved with soft-x-ray angle-resolved photoemission spectroscopy whose large probing depth penetrates through the Cr\(_2\)O\(_3\) layer. For the first time, the electronic dispersions and Fermi surface of CrO\(_2\) are measured, which are fundamental prerequisites to solve the long debate on the nature of electronic correlations in this material. Since density functional theory augmented by a relatively weak local Coulomb repulsion gives an exhaustive description of our spectroscopic data, we rule out strong-coupling theories of CrO\(_2\). Crucial for the correct interpretation of our experimental data in terms of the valence-band dispersions is the understanding of a nontrivial spectral response of CrO\(_2\) caused by interference effects in the photoemission process originating from the nonsymmorphic space group of the rutile crystal structure of CrO\(_2\).

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.

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.