The subject of the present thesis are some aspects of impurities affecting mesoscopic systems with regard to their topological properties and related effects like Majorana fermions and quantized conductance. A focus is on two-dimensional systems including both topological insulators and superconductors. First, the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal invariant superconductors from Altland-Zirnbauer (AZ) symmetry class DIII is addressed, and a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength is defined. These general results are applied to the time-reversal invariant p-wave phase of the doped Kitaev-Heisenberg model, where it is also demonstrated how a lattice of impurities can drive a topologically trivial system into the nontrivial phase. Second, the result about the existence of zero-energy impurity states is generalized to all AZ symmetry classes. This is achieved by considering, for general Hamiltonians H from the respective symmetry classes, the “generalized roots of det H”, which subsequently are used to further explore the opportunities that lattices of nonmagnetic impurities provide for the realization of topologically nontrivial phases. The 1d Kitaev chain model, the 2d px + ipy superconductor, and the 2d Chern insulator are considered to show that impurity lattices generically enable topological phase transitions and, in the case of the 2d models, even provide access to a number of phases with large Chern numbers. Third, elastic backscattering in helical edge modes caused by a magnetic impurity with spin S and random Rashba spin-orbit coupling is investigated. In a finite bias steady state, the impurity induced resistance is found to slightly increase with decreasing temperature for S > 1/2. Since the underlying backscattering mechanism is elastic, interference between different scatterers can explain reproducible conductance fluctuations. Thus, the model is in agreement with central experimental results on edge transport in 2d topological insulators.

Recent advances in experimental condensed matter physics suggest a powerful new paradigm for the realization of exotic phases of quantum matter in the laboratory. Rather than conducting an exhaustive search for materials that realize these phases at low temperatures, it may be possible to design quantum systems that exhibit the desired properties. With the numerous advances made recently in the fields of cold atomic gases, superconducting qubits, trapped ions, and nitrogen-vacancy centers in diamond, it appears that we will soon have a host of platforms that can be used to put exotic theoretical predictions to the test. In this dissertation, I will highlight two ways in which theorists can interact productively with this fast-emerging field. First, there is a growing interest in driving quantum systems out of equilibrium in order to induce novel topological phases where they would otherwise never appear. In particular, systems driven by time-periodic perturbations—known as “Floquet systems”—offer fertile ground for theoretical investigation. This approach to designer quantum matter brings its own unique set of challenges. In particular, Floquet systems explicitly violate conservation of energy, providing no notion of a ground state. In the first part of my dissertation, I will present research that addresses this problem in two ways. First, I will present studies of open Floquet systems, where coupling to an external reservoir drives the system into a steady state at long times. Second, I will discuss examples of isolated quantum systems that exhibit signatures of topological properties in their finite-time dynamics. The second part of this dissertation presents another way in which theorists can benefit from the designer approach to quantum matter; in particular, one can design analytically tractable theories of exotic phases. I will present an exemplar of this philosophy in the form of coupled-wire constructions. In this approach, one builds a topological state of matter from the ground up by coupling together an array of one-dimensional quantum wires with local interactions. I will demonstrate the power of this technique by showing how to build both Abelian and non-Abelian topological phases in three dimensions by coupling together an array of quantum wires.

Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.

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<p>In this thesis, we present the results for the finite frequency response of a variety of materials. These materials all share the common theme that their low energy excitations are Dirac-like. This coincidence was not by design, and highlights the now-ubiquitous nature of Dirac-quasiparticles in condensed matter physics. We present results for graphene, the high temperature superconducting cuprates, and Weyl semi metals. For graphene, our calculations revolve around a new experimental technique: Near field infrared spectroscopy. Conventionally it is ok to use the $\vec{q}\rightarrow 0$ limit when calculating the low energy optical response. This new technique is able to directly probe the finite $\vec{q}$ response by using an atomic force microscope tip as an antenna. We computed the optical conductivity of graphene at finite wavevector and studied how the quasiparticle peak is altered by disorder and the electron-phonon interaction. The calculations on the high $T_c$ cuprates use a model of the pseudogap phase known as the Yang, Rice and Zhang (YRZ) model. We employed the model to study the resistivity in the pseudogap regime, both in-plane and along the c-axis. We used a coherent tunneling matrix element to describe transport along the c-axis. We found that the model was able to reproduce the metaliclike behavior in the plane while being resistive out of plane. We then extended the model to the finite frequency response, as well as the superconducting phase. We found a pseduogap feature at finite frequency that was previously explained through an interlayer collective mode. We also found that microwave spectroscopy puts strong limits on the form of the scattering rate. Finally, we computed the optical response of Weyl semimetals subjected to an applied magnetic field. Weyl semimetals are a topological phase of matter that have yet to be observed. The form of the conductivity contains a series of asymmetric peaks, whose spacing is a signature of the underlying relativistic dispersion. These peaks remain robust, even with moderate disorder.</p> / Doctor of Philosophy (PhD)

The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.

The subject of the present thesis are some aspects of impurities affecting mesoscopic systems with regard to their topological properties and related effects like Majorana fermions and quantized conductance. A focus is on two-dimensional systems including both topological insulators and superconductors. First, the question of whether individual nonmagnetic impurities can induce zero-energy states in time-reversal invariant superconductors from Altland-Zirnbauer (AZ) symmetry class DIII is addressed, and a class of symmetries which guarantee the existence of such states for a specific value of the impurity strength is defined. These general results are applied to the time-reversal invariant p-wave phase of the doped Kitaev-Heisenberg model, where it is also demonstrated how a lattice of impurities can drive a topologically trivial system into the nontrivial phase. Second, the result about the existence of zero-energy impurity states is generalized to all AZ symmetry classes. This is achieved by considering, for general Hamiltonians H from the respective symmetry classes, the “generalized roots of det H”, which subsequently are used to further explore the opportunities that lattices of nonmagnetic impurities provide for the realization of topologically nontrivial phases. The 1d Kitaev chain model, the 2d px + ipy superconductor, and the 2d Chern insulator are considered to show that impurity lattices generically enable topological phase transitions and, in the case of the 2d models, even provide access to a number of phases with large Chern numbers. Third, elastic backscattering in helical edge modes caused by a magnetic impurity with spin S and random Rashba spin-orbit coupling is investigated. In a finite bias steady state, the impurity induced resistance is found to slightly increase with decreasing temperature for S > 1/2. Since the underlying backscattering mechanism is elastic, interference between different scatterers can explain reproducible conductance fluctuations. Thus, the model is in agreement with central experimental results on edge transport in 2d topological insulators.

Dans cette thèse nous abordons des aspects topologiques de la matière condensée. Les états topologiques sont insensibles à un large spectre des perturbations externes et au désordre – une propriété indispensable dans le domaine d'information quantique. L’effet des interactions dans des systèmes topologiques est pourtant loin d’être bien maîtrisé à ce jour. Dans ce travail, nous étudions la corrélation entre la description topologique et l'effet des interactions. Afin d'accomplir notre but, nous utilisons des méthodes analytiques et numériques. Nous nous intéressons aussi à des sondes expérimentales qui peuvent être utilisées pour vérifier nos prédictions théoriques. Tout d’abord, nous étudions la version bosonique en interactions du modèle de Haldane – le modèle célèbre qui décrit l’effet Hall anomal. Nous proposons son implémentation expérimentale dans des circuits quantiques, basée sur l’application de perturbation périodique dépendantes du temps – méthodologie qui s’appelle l’ingénierie de Floquet. En poursuivant ces idées, nous étudions la version bosonique du modèle de Kane-Mele d’un isolant topologique. Ce modèle possède un diagramme de phase très riche. En particulier, lorsque les interactions sont fortes, nous observons l’apparition d’un modèle de magnétisme frustrée présentant une variété d'états exotiques. La mise en œuvre de ces modèles dans des réseaux d'atomes ultra-froids ou des circuits quantiques permettra de sonder expérimentalement les propriétés exotiques que nous avons observées. Ensuite, nous abordons d’une manière plus détaillée la réalisation expérimentale des modèles topologiques dans des circuits quantiques, en considérant le cas particulier du modèle de Su-Schrieffer-Heeger en couplage fort. Nous testons aussi des nouvelles sondes qui peuvent être utilisées afin de mesurer la phase de Zak et en déduire la topologie du système. Finalement, nous nous intéressons aux sondes hors d’équilibre et des méthodes pour tester les propriétés spectrales de systèmes quantiques, en utilisant l’approche de purification, pertinent pour le numérique et les expériences. / In this thesis we study the topological aspects of condensed matter physics, that received a revolutionary development in the last decades. Topological states of matter are protected against perturbations and disorder, making them very promising in the context of quantum information. The interplay between topology and interactions in such systems is however far from being well understood, while the experimental realization is challenging. Thus, in this work we investigate analytically such strongly correlated states of matter and explore new protocols to probe experimentally their properties. In order to do this, we use various analytical and numerical techniques. First, we analyze the properties of an interacting bosonic version of the celebrated Haldane model – the model for the quantum anomalous Hall effect. We propose its quantum circuit implementation based on the application of periodic time-dependent perturbations – Floquet engineering. Continuing these ideas, we study the interacting bosonic version of the Kane-Mele model – the first model of a topological insulator. This model has a very rich phase diagram with an emergence of an effective frustrated magnetic model and a variety of symmetry broken spin states in the strongly interacting regime. Ultra-cold atoms or quantum circuits implementation of both Haldane and Kane-Mele bosonic models would allow for experimental probes of the exotic states we observed. Second, in order to deepen the perspectives of quantum circuit simulations of topological phases we analyze the strong coupling limit of the Su-Schrieffer-Heeger model and we test new experimental probes of its topology associated with the Zak phase. We also work on the out-of-equilibrium protocols to study bulk spectral properties of quantum systems and quantum phase transitions using a purification scheme which could be implemented both numerically and experimentally.

The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.