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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.
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.
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.
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.
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.