Marcel Franz

When a continuous symmetry is spontaneously broken, collective modes emerge. Usually, their spectrum is dominated by the low-energy physics of massless Goldstone modes. Superconductors, that break U(1) symmetry, are different. Here, the Goldstone boson is gapped out due to the Anderson-Higgs mechanism. The superconducting condensate can therefore host a zoo of massive collective excitations that are stable for lack of a gapless decay channel. The most prominent of them is the Higgs mode. Spectroscopy of collective modes can serve as a probe to reveal the nature of the superconducting state. In this thesis, we study the signatures of collective modes in nonlinear optical experiments. We explore the theoretical description of a new spectroscopic excitation scheme. We show how impurity scattering significantly enhances the optical Higgs mode response. We apply group theoretical methods to multi-order-parameter theories and investigate microscopic signatures of coupled modes in third harmonic generation experiments. We study the phenomenology and collective mode spectrum of an exotic system of twisted cuprate bilayers that supports topological superconductivity. Finally, we propose a novel device implementation of the superconducting diode effect.These results contribute to the emerging field of collective mode spectroscopy.

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The notion of twisting and stacking two-dimensional van der Waals materials has emerged as a paradigm for realizing novel electronic states. With the goal of engineering topological superconductivity, we go beyond the archetypal example of twisted bilayer graphene and consider structures composed of proximitized quantum wires and high-Tc cuprate superconductors. On the basis of symmetry arguments, backed by microscopic modeling, these setups are predicted to spontaneously break time-reversal symmetry and give rise to topological p- and d-wave phases with chiral Majorana edge modes. Given the experimental pertinence, we investigate a suite of Josephson transport phenomena in twisted cuprate bilayers and identify signatures of the topological phase. Further, the analysis is extended to twisted multilayers where the occurrence of higher-order band crossings makes the groundstate susceptible to secondary instabilities and topological superconductivity appears for an extended range of twists.

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In recent years a new paradigm has emerged to investigate non-Fermi liquids without quasiparticles, based on the exactly-solvable Sachdev-Ye-Kitaev model which consists of a large number of fermions with all-to-all, random Gaussian interactions. This model exhibits a rich phenomenology at low energy, including power-law decaying spectral functions, maximal chaos and connections to black hole horizons in anti-de Sitter spacetime. These intriguing properties have inspired a broad research program at the intersection of condensed matter physics, quantum information and quantum gravity.In this thesis we explore the phases and phase transitions occurring in Sachdev-Ye-Kitaev models that are coupled in various ways, with an eye on physical platforms that could enable their realization in condensed matter systems. This journey takes us from the investigation of quantum chaos in many-body systems and revival dynamics in traversable wormholes, all the way to the physics of disordered graphene flakes and unconventional superconductivity.

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Recent decades have seen a proliferation of unconventional quasiparticles in condensed matter systems. Majorana fermions, theoretically predicted in several setups and of great interest in topological quantum computation, are a focus of intense research efforts. Higher-spin moments, relevant for both solid-state compounds and cold atoms, are of great theoretical interest as they interpolate between the quantum and classical limits but sometimes show surprising behavior. In the meantime, successful fabrication and characterization of low-dimensional systems have brought new phenomenology and physical insights. In this dissertation I will theoretically explore a few low-dimensional models using Majorana fermions and higher-spin moments as building blocks. I will first discuss a generalized family of the celebrated Sachdev-Ye-Kitaev model, a zero-dimensional all-to-all Majorana model that exhibits non-Fermi liquid behavior and is holographically dual to a black hole. The generalized model has a phase transition between a non-Fermi liquid and a disordered Fermi liquid. Then I will discuss the Heisenberg model with higher spins, with a focus on chaos and information scrambling. Using matrix-product-state-based methods, we are able to obtain numerical results for spin up to 4 and characterize the Lyapunov growth. After that I will discuss a generalization of the Hubbard model to Majorana fermions on the honeycomb lattice. Unlike previous similar models, we find topological phases with (anti-)chiral edge modes for weak interaction. Finally I will show a construction of explicit supersymmetric Majorana model on the kagome lattice, where a family of exact solutions is found and the nature of supersymmetry breaking is explored.

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Dirac materials have formed a thriving and prosperous direction in modern condensed matter physics. Their bulk bands can linearly attach at discrete points or along curves, leading to arc or drumhead surface states. The candidate Dirac materials are exemplified by Dirac/Weyl semimetals, Dirac/Weyl superconductors, and Dirac/Weyl magnets. Owing to the relativistic band structure, these materials have unique responses to the applied elastic crystalline lattice deformation, which can induce pseudo-magnetic and pseudo-electric fields near the band crossings and produce transport distinguished from that caused by ordinary magnetic and electric fields. In this dissertation, I will demonstrate the exotic transport due to the strain-induced gauge field in Weyl semimetals, Weyl superconductors, and Weyl ferromagnets. I will first elucidate that a simple bend deformation can induce a pseudo-magnetic field that can give rise to the Shubnikov-de Haas oscillation in Weyl semimetals. Then I will elaborate that strain can Landau quantize charge neutral Bogoliubov quasiparticles as well and result in thermal conductivity quantum oscillation in Weyl superconductors. Lastly, I will consider the strain-induced gauge field beyond the fermionic paradigm and explain various quantum anomalies of magnons in Weyl ferromagnets.

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The discovery of topological phases of matter has brought high-energy and condensed matter communities together by giving us shared interests and challenges. One fruitful outcome is the broadened range of possibilities to study high-energy physics in cost-effective table-top experiments. I have investigated scenarios in which influential high-energy ideas emerge in solid-state systems built from topological semimetals – gapless topological phases which have drawn intense research efforts in recent years. My Thesis details three proposals for realizing Majorana fermions, Adler-Bell-Jackiw anomaly, and holographic black holes in superconductor-Weyl-semimetal heterostructures, mechanically strained Weyl semimetal nanowires/films, and graphene flakes subject to strong magnetic fields, respectively. By analyzing the effects of realistic experimental conditions, I wish to demonstrate that these proposals are experimentally tangible with existing technologies.

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Electronic states in band insulators and semimetals can form nontrivial topological structures which can be classified by introducing a set of well defined topological invariants. There are interesting experimentally observable phenomena tied to these topological invariants which are robust as long as the invariants remain well-defined. One important class manifesting these topological phenomena in the bulk and at the edges is the time reversal invariant topological band insulators first discovered in HgTe in 2007. Since then, there have been enormous efforts from both the experimental and the theoretical sides to discover new topological materials and explore their robust physical signatures. In this thesis, we study one important aspect, i.e., the electromagnetic response in the bulk and at the spatial boundaries. First we show how the topological action, which arises in a time reversal invariant three dimensional band insulator with nontrivial topology, is quantized for open and periodic boundary conditions. This confirms the Z2 nature of the strong topological invariant required to classify time-reversal invariant insulators. Next, we introduce an experimentally observable signature in the response of electronic spins on the surface of these materials to the perpendicular magnetic field. We proceed by considering electromagnetic response in the bulk of topological Weyl semimetals in a systematic way by considering a lattice model and we address important questions on the existence or absence of the Chiral anomaly. In the end, we show how a topological phase in a one dimensional system can be an energetically favourable state of matter and introduce the notion of self-organized topological state by proposing an experimentally feasible setup.

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In this work we study four interesting effects in the field of topological insulators: the Witten effect, the Wormhole effect, topological Anderson insulators and the absence of bulk magnetic order in magnetically doped topological insulators. According to the Witten effect, a unit magnetic monopole placed inside a medium with non zero θ is predicted to bind a fractional electric charge. We conduct a first test of the Witten effect based on the recently established fact that the electromagnetic response of a topological insulator is given by the axion term θ(e²/2πh)B·E, and that θ=π for strong topological insulators. We establish the Wormhole effect, in which a strong topological insulator, with an infinitely thin solenoid of magnetic half flux quantum carries protected gapless and spin filtered one-dimensional fermionic modes, which represent a distinct bulk manifestation of the topologically non-trivial insulator.We demonstrate that not only are strong topological insulators robust to disorder but, remarkably, under certain conditions disorder can become fundamentally responsible for their existence. We show that disorder, when sufficiently strong, can transform an ordinary metal with strong spin-orbit coupling into a strong topological ‘Anderson’ insulator, a new topological phase of matter in three dimensions.Finally, we lay out the hypothesis that a temperature window exists in which the surface of magnetically doped topological insulators is magnetically ordered but the bulk is not. We present a simple and intuitive argument why this is so, and a mean-field calculation for two simple tight binding topological insulator models which shows that indeed a sizeable regime such as described above could exist. This indicates a possible physical explanation for the results seen in recent experiments.

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In this work we introduce a new model for a topological insulator in both two and three dimensions, and then discussthe possibility of creating a fractional topological insulator in the three dimensional version. We also explore the possibilityof engineering a quantum spin Hall phase in Graphene through the application of heavy metal adatoms.We show that the two dimensional model on the Lieb lattice, and its three dimensional counterpart, the so called edge centered cubic or Perovskite lattice, possess non-trivial Z₂ invariants, and gapless edge modes, which are the signatures of the topological insulating state.Having established that these lattices can become topological insulators, we then tune several short range hoppings in the model and showthat it is possible to flatten the lowest energy bands in each case. After flattening the bands we add in a Hubbard term and then use a mean field decoupling to show that there is a portion of phase space where the system remains non-magnetic, and then conjecture that the many body ground state in three dimensions could become a fractional topological insulator.For the model on graphene, we start by using density functional theory (DFT) to find a pair of suitable heavy elements that are non-magnetic, have a strong spin orbit coupling and prefer to sit at the center of the hexagonal lattice. We then establish that for adatoms distributed in a periodic configuration, again with DFT, that a gap will open at the Dirac point in the presence of spin orbit coupling. To prove the gap is topologically non-trivial, we show that it is possible to adiabatically connect this model to the original Kane-Mele model, a known topological insulator. Lastly, we show that for adatoms distributed randomly the effect survives.

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