From the alignment of magnets to the melting of ice, the transition between different phases of matter underpins our exploitation of materials. Both a quantum and a classical phase can undergo an instability into another state. In this thesis, we study the stability of matter in both contexts: topological states and crystalline solids. We start with the stability of fractional quantum Hall states on a lattice, known as fractional Chern insulators. We investigate, using exact diagonalization, fractional Chern insulators in higher Chern bands of the Harper-Hofstadter model, and examine the robustness of their many-body energy gap in the effective continuum limit. We report evidence of stable states in this regime; comment on two cases associated with a bosonic integer quantum Hall effect; and find a modulation of the correlation function in higher Chern bands. We next examine the stability of molecules using variational and diffusion Monte Carlo. By incorporating the matrix of force constants directly into the algorithms, we find that we are able to improve the efficiency and accuracy of atomic relaxation and eigenfrequency calculation. We test the performance on a diverse selection of case studies, with varying symmetries and mass distributions, and show that the proposed formalism outperforms existing restricted Hartree-Fock and density functional theory methods. Finally, we analyze the stability of three-dimensional crystals. We note that for repulsive Coulomb crystals of point nuclei, cubic systems have a zero matrix of force constants at second order. We investigate this by constructing an analytical model in the tight-binding approximation, and present a phase diagram of the most stable crystal structures, as we tune core and valence orbital radii. We reconcile our results with calculations in the nearly free electron regime, as well as current research in condensed matter and plasma physics.

Despite of enormous progress in experimental nanophysics theoretical studies of low-dimensional electron systems still remains a challenging task. Indeed, most of the structures are strongly correlated, so that an effective perturbative treatment is impossible due to the lack of a small parameter. The problem can be partly solved within the dynamical mean-field theory (DMFT) paradigm, nevertheless the correlations in physically relevant high-temperature superconductors are of purely non-local nature. The recently developed dual fermion approximation, combining field-theoretical diagram technique and numerical methods, allows for explicit account of spatial correlations. The approximation was shown to be of fastest convergence compared with standard DMFT extensions, and along with renormalization group is used here to study Fermi condensation on a triangular lattice near van Hove singularities. The still debated phenomenon of Fermi condensation is believed to be a precursor to strongly correlated low-temperature instability and is found in this thesis to be robust even at high temperature, making its experimental verification feasible. Unlike homogeneous ferromagnetic ordering a variety of non-collinear ground state configurations emerge as a result of competition among exchange, anisotropy, and dipole-dipole interaction. These particle-like states, e.g. magnetic soliton, skyrmion, domain wall, form a spatially localized clot of magnetic energy. Consistent study of spin, which essentially is a quantum mechanical entity, led to the emergence of spintronics (spin-based electronics) and magnonics (photonics with spin waves), in the meanwhile topologically protected magnetic solitons and skyrmions might potentially be applied for data processing and information storage in next generation of electronic technology (rapidly advancing solitonics and skyrmionics). An ability to easily create, address, and manipulate such structures is among the prerequisite forming a basis of "-onics" technology. It is shown here that spins on a kagome lattice, interacting via Heisenberg exchange and Dzyaloshinskii-Moriya coupling, allow the formation of topologically protected edge states through which a skyrmion can propagate. Not only can chemical methods be used to design novel functionality, but also geometric structuring. It is demonstrated that for graphene sandwiched between two insulating media external circularly-polarized light serves as an effective magnetic field. The direct practical implication permits to control light polarization and induce spin-waves propagating on the surface of e.g. a topological insulator. The newly discovered Dirac materials, graphene and three-dimensional topological insulators, are not easy to handle. In fact, the quasiparticle band function is gapless preventing them from being used in integrated circuits, nevertheless the problem is shown here to be partially relaxed by placing a vacancy on top of it.

This thesis contains a study of topological phases of matter, with a strong emphasis on symmetry as a unifying theme. We take the point of view that the "topology" in many examples of what is loosely termed "topological matter", has its origin in the symmetry data of the system in question. From the fundamental work of Wigner, we know that topology resides not only in the group of symmetries, but also in the cohomological data of projective unitary-antiunitary representations. Furthermore, recent ideas from condensed matter physics highlight the fundamental role of charge-conjugation symmetry. With these as physical motivation, we propose to study the topological features of gapped phases of free fermions through a Z₂-graded C*-algebra encoding the symmetry data of their dynamics. In particular, each combination of time reversal and charge conjugation symmetries can be associated with a Clifford algebra. K-theory is intimately related to topology, representation theory, Clifford algebras, and Z₂-gradings, so it presents itself as a powerful tool for studying gapped topological phases. Our basic strategy is to use various K</em-theoretic invariants of the symmetry algebra to classify symmetry-compatible gapped phases. The super-representation group of the algebra classifies such gapped phases, while its K-theoretic difference-group classifies the obstructions in passing between two such phases. Our approach is a noncommutative version of the twisted K-theory approach of Freed--Moore, and generalises the K-theoretic classification first suggested by Kitaev. It has the advantage of conceptual simplicity in its uniform treatment of all symmetries. Physically, it encompasses phenomena which require noncommutative algebras in their description; mathematically, it clarifies and provides rigour to the meaning of "homotopic phases", and easily explains the salient features of Kitaev's Periodic Table.

Quantum computers have the potential to solve certain problems in minutes that would otherwise take classical computers thousands of years due to the exponential speed-up certain quantum algorithms have over classical algorithms. In order to leverage such quantum algorithms, it is necessary for them to run on quantum devices. Examples of such devices include, but are not limited to, semiconductor and superconducting qubits, and semiconductor single and entangled photon emitters. The conventional method of constructing a semiconductor qubit is to apply gates on a semiconductor surface to localize electrons, where the electronic spin states are mapped to a qubit basis. Examples of this include the spin qubit where the spin-1/2 states of a single electron is the qubit basis and the gated singlet-triplet qubit where the states of two coupled electrons are mapped to a qubit basis. In general, gated semiconductor spin qubits are subject to decoherence from the environment which alters the electronic wavefunction by entanglement with the nuclear spins and phonons in the lattice compromising the stability of the qubit. Semiconductor nanostructures can also be designed as photon emitters. Self-assembled quantum dots are an example of such nanostructures and have been shown to emit single photons through exciton recombination and entangled photons through biexciton-exciton cascade. The difficulty in designing photon sources using self-assembled quantum dots is that the size and shape varies from dot to dot, implying that the electronic and magnetic properties also vary. In this thesis, I present the design of a single photon emitter using an InAsP quantum dot embedded in an InP nanowire and the design of a singlet-triplet qubit that is topologically protected from decoherence using an array of such quantum dots embedded in an InP nanowire. The advantage of using quantum dot nanowires over self-assembled quantum dots as photon emitters is that the quantum dot thickness, radius and composition can be controlled deterministically using a technique known as vapour-liquid-solid epitaxy which allows the emission spectrum to be engineered. Using a microscopic model, I simulated an InAsP quantum dot embedded in a nanowire with upwards of millions of atoms and showed that the emission spectrum came in agreement with the actual InAsP/InP quantum dot nanowires that were fabricated at the National Research Council of Canada. Moreover, I showed that altering the distribution of As atoms in the quantum dot can cause dramatic change in the emission spectrum. For the design of the topologically protected singlet-triplet qubit, I demonstrated that the ground state of an array of such quantum dots embedded in an InP nanowire, with four electrons in each dot, is four-fold degenerate and is topologically protected from higher energy states, making the ground state robust against perturbations. This state is known as the Haldane phase and can be understood in terms of two spin-1/2 quasiparticles at each edge of the array. Though the spectral gap in my simulation was of the order of 1 meV, this work provides insight into the potential design of a room temperature operating Haldane qubit where the spectral gap is of the order of room temperature.

L'électronique moléculaire est l'un des domaines les plus intrigants de la recherche moderne. Ce domaine pourrait produire un système de construction modulaire et évolutif pour des applications spintroniques à l'échelle nanométrique. Un exemple particulièrement prometteur est celui des aimants à une seule molécule, qui se sont déjà avérés être appropriés pour des la réalisation de spin valve et de qubit de spin. L'un des plus grands défis du domaine est l'intégration de ces objets de taille nanométrique dans des circuits complexes afin de permettre la détection et la manipulation d'états de spin moléculaires. Comme l'ont montré ces dernières années le groupe NanoSpin, les nanotubes de carbone (CNTs) peuvent servir de support pour les aimants à une seule molécule, en combinant les caractéristiques des deux constituants.Une pierre angulaire de ce projet de thèse a donc été le développement d'une technique de fabrication fiable pour des dispositifs de CNTs de haute qualité, contrôlables par de multiples électrodes de grille locales afin de permettre le contrôle local des systèmes hybrides moléculaires. Un procédé basé sur la fabrication conventionnelle à un substrat a été développé à partir de zéro, pour lequel l'optimisation de la conception des échantillons, les techniques de lithographie et de dépôt ainsi que les choix de matériaux ont dû être soigneusement incorporés afin de respecter les restrictions imposées par les conditions de croissance. Nous avons d'abord réussi à produire des échantillons CNT propres, permettant de mettre en évidence une configuration à double boite quantique, tout en ajustant des caractéristiques de type p à n. Les segments créés de cette manière peuvent être contrôlés de manière stable sur toute la longueur du dispositif et devraient donc constituer une base appropriée pour l'étude de la physique moléculaire.La matière topologique non triviale constitue une plate-forme séduisante pour étudier à la fois les principes fondamentaux et les applications possibles de la spintronique au calcul quantique. Les isolants cristallins topologiques, avec tellurure d'étain (SnTe) comme exemple principal, représentent un nouvel état au sein de ce zoo des matériaux topologiques 3D. Peu de temps après les premières réalisations expérimentales, des suggestions ont été faites sur la possibilité d’un type de supraconductivité non conventionnelle hébergé à l'interface entre la matière topologique et les supraconducteurs classiques. Les implications possibles de ces systèmes comprennent l'appariement de Cooper avec une quantité de mouvement finie dans la phase FFLO ou l’ordinateur quantique topologique, basé sur des excitations particulières, appelé quasi-particule Majorana.Ce projet de thèse visait à participer à l'enquête sur les signes de supraconductivité non conventionnelle dans SnTe. Les expériences de transport sur des couches pures dans les géométries de la barre de Hall et des dispositifs hybrides supraconducteurs, réalisés à la fois comme jonctions Josephson et SQUID, sont discutés. Un couplage étonnamment fort de SnTe au supraconducteur a été trouvé et dépendances de la supraconductivité sur les géométries des échantillons, la température et le champ magnétique ont été étudiées. La relation courant-phase a été analysée dans la limite d’effets cinétiques forts. Le couplage électrostatique et l'exposition à des micro-ondes ont été explorée, mais la physique prédominante dans de telles configurations s'est avéré être de type purement conventionnel, soulignant l’importance des améliorations sur le côté matériaux.Des mesures de champ magnétique dans le plan ont donné lieu à la signature d’un φ0-SQUID avec des transitions 0-π accordables, fournissant des preuves de possibles de transitions contrôlées de la supraconductivité triviale aux régimes de couplage non conventionnels dans SnTe. / Molecular electronics is one of the most intriguing fields of modern research, which could bring forth a modular and scalable building system for nanoscale spintronics applications. A particularly promising example are single-molecule magnets, which have already successfully shown to be suitable for spin valve or spin qubit operations. One of the biggest challenges of the field is the integration of these nanometer-sized objects in complex circuits in order to allow for detection and manipulation of moleculear spin states. As shown in recent years by the NanoSpin group, carbon nanotubes (CNTs) can serve as such type of carrier for the single-molecule magnets, combining features of both constituents.A corner stone of this thesis project was hence the development of a dependable fabrication technique for high-quality CNT devices, controllable by multiple local gate electrodes in order to enable local control of molecular hybrid systems. A process based on conventional one-chip fabrication was developed from scratch, for which optimization of sample design, lithography and deposition techniques as well as material choices had to be carefully incorporated, in order to accomodate the restrictions imposed by the CNT growth conditions on the prevention of leakage currents. We succeeded in producing clean CNT devices, which could support a double dot configuration, tunable from p- to n-type characteristics. The segments created in this way can be stabily controlled over the entire device length and should hence provide a suitable backbone to study molecular physics.Topological matter constitutes an enticing platform to investigate both fundamental principles as well as possible applications from spintronics to quantum computation. Topological crystalline insulators, with tin telluride ( SnTe ) as a prime example, represent a new state of matter within this zoo of 3D topological materials. Soon after first experimental realizations, suggestions were made about the possibility of an unconventional type of superconductivity hosted at the interface between topological matter and conventional superconductors. Possible implications of such systems include Cooper pairing with finite momentum, the FFLO phase, or topological quantum computing, based on peculiar excitations, called Majorana bound states.This thesis project aimed to participate in the investigation of signs of unconventional superconductivity in SnTe . Transport experiments on bare films in Hall bar geometries and superconducting hybrid devices, realized as both Josephson junctions and SQUIDs, are discussed. A surprisingly strong coupling of SnTe to Ta superconductor was found and dependencies of superconductivity on sample geometries, temperature and magnetic field were investigated. The current-phase relation was analyzed in the limit of strong kinetic effects. Electrostatic gating and rf exposure was explored, but predominant physics in such configurations turned out to be of purely conventional type, pointing out the importance of improvements on the material side.In-plane magnetic field measurements gave rise to the manifestation of ϕ0-SQUIDs with tunable 0−π-transitions, providing evidence for possible controlled transitions from trivial superconductivity to unconventional coupling regimes in SnTe.

Mémoire en recherche-création / Creative dissertation / Cette thèse de maîtrise traite du processus créatif et de la recherche qui y est associée afin de produire deux performances en direct dans le domaine de la musique électroacoustique. À l’aide de ces deux œuvres, mon intention était de concevoir une pratique artistique qui réunit plusieurs modes autour de la gestuelle et du son, influencée par des algorithmes. Une tentative approfondie d’extraire un processus de composition à partir des réactions de la matière en vibration englobe une grande partie de la recherche. Cette recherche découle de ma transition d’une pratique artistique basée sur la représentation (soucieuse des haut-parleurs, manettes et boutons) vers une pratique imprégnée par la performance (soucieuse de la transformation continuelle du son en relation avec les modulations de la matière). Tout au long de cette recherche, j’ai mené un examen approfondi des rythmes au-delà de la pure mesure de leurs expressions musicales, pour considérer les nombreuses notions du rythme qui se dévoilent dans les interactions quotidiennes de l’expérience vécue. Les micro-rythmes perçus par l’oreille comme des textures, les gestes répétitifs perçus par l’œil comme un mouvement linéaire et les rythmes observés lors de circonstances sociales communes, comme la cadence de la conversation sont, parmi les caractéristiques du rythme qui ont suscité mon intérêt. Le tout se situe dans un récit historique éclairé qui étudie l’influence de l’algorithme et de la matière tout au long de la musique et de l’art sonore du XXe siècle. La recherche conceptuelle est enrichie par des expériences exhaustives en composition algorithmique, analyse gestuelle et modélisation gestuelle. Dans chacun de ces domaines, bien que soutenue par des lectures fondamentales en philosophie et en art, une approche primaire de la création s’est faite dans un processus « réfléchir-en-faisant » qui ont généré de nombreuses ex- périences tant avec la matière physique qu’avec la conception d’instruments numériques. Au- delà de la création des performances qui constituent la base des résultats de cette recherche, un vaste ensemble d’outils interopérables d’analyse gestuelle en temps réel, de modélisation, de composition algorithmique et de traitement du son a été développé et publié pour l’environnement Max/MSP. / This master’s thesis concerns the creative process and related research for the production of two live performances in the domain of electroacoustic music. Across the creation of the two works, my intention has been to develop a unified multi-modal gesture, sound, and algorithm influenced performance practise. Encompassing the largest portion of the research is an earnest attempt to derive compositional process from the behavior of vibrating matter. This research is precipitated by my movement from an artistic practice based on representation (concerned with speakers, knobs, and buttons) towards a practice steeped in performance (concerned with the continuous transformation of sound correlated to material modulation). Across this research, an in-depth investigation was conducted into rhythms beyond their purely metric musical manifestations, and into the numerous alternative notions of rhythm which are revealed through daily interactions and lived experience. Rhythmic artifacts of interest have included micro-rhythms perceived by the ear as textures, repetitive gestures perceived by the eye as linear motion, and rhythms observed in ordinary social situations such as the cadence of conversation. This is all situated within an informed historical narrative which considers the influence of the algorithm and material primarily across 20th century music and sound art. The conceptual research is augmented by extensive experiments in algorithmic composition, gesture analysis, and gesture mapping. In each of these areas, though tied to fundamental readings in philosophy and art, a primary approach to creation has been thinking-through-making, which has led to extensive experimentation with both physical materials and digital instrument design. Beyond the performance creations which form the basis of this research output, a large set of interoperable tools for real-time gesture analysis, mapping, algorithmic composition, and sound processing was developed and published for the Max/MSP environment.