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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.

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

Topological states of matter exhibit a range of exceptional properties such as unidirectional wave propagation along topological boundaries and an unparalleled robustness to different types of perturbations. The primary aim of this thesis is to leverage gradient-based optimization techniques for the design of topological matter. Unlike other inverse design approaches for topological systems, this work targets the topological invariant directly. At first, this approach may seem paradoxical, as topological invariants are invariant under continuous deformation of materials, meaning that small perturbations cannot identify the direction of change of the invariant. To solve the paradox, in my first study, I introduce the concept of symmetry relaxation. Since topological invariants are protected by symmetries, relaxing these symmetries converts the quantized topological invariant into a differentiable entity which I call ``topological indicator". This indicator matches the invariant when symmetries are intact and interpolates between various topological phases when symmetries are relaxed. By following the gradient of this indicator with respect to design or model parameters, a material can be guided through a topological phase transition. This technique is applied to both discrete and continuous, as well as higher-order, topological insulators. In a subsequent project, I employ the adjoint method from PDE-constrained optimization to compute these gradients. This method enables the efficient and exact computation of gradients, particularly when faced with a vast design space. Collectively, I believe these studies lay the groundwork for the systematic design of topological information processing devices. An emerging domain exploiting topological effects in the classical realm involves time-modulated metamaterials. Here, topological effects often draw parallels to adiabatic processes observed in quantum systems, such as those of Majorana fermions and braiding operations. I have dedicated a study to contrast classical and quantum adiabatic evolutions. It is conventionally assumed that they are largely equivalent, especially concerning the conditions for adiabaticity and the emergence of geometric entities. However, my results challenge this perception. I discern that mode coupling in classical systems diverges fundamentally from its quantum counterpart, implying certain quantum adiabatic processes are not replicable in classical settings. When addressing the general multiband setting, the manifestation of the non-Abelian gauge potential is also distinct, requiring a classical ``correction'' term. Finally, classical adiabatic evolution requires an extra condition compared to quantum adiabatic evolution. Only under this extra condition adiabaticity is ensured, and geometric quantities like Berry phases or the Wilczek-Zee matrix emerge. The fourth study provides an overview of an ongoing collaborative project with the Bertoldi Lab at Harvard. This project aims to outsource the sensory system and control unit of a mobile robot into a mechanical metamaterial. To achieve this goal the metamaterial is mounted onto a wheeled mobile basis. The task of the robot is to navigate through a course of obstacles. As the robot collides with an obstacle the metamaterial deforms nonlinearly because of the contact. This deformation is exploited to compute the motor command voltages to free the robot from the obstacle using Physical Reservoir Computing. In the fifth study, a framework is established for mapping a 1-D discretized non-homogeneous elastic wave equation to a Schrödinger equation which can be solved using gate-based quantum computers. We validate feasibility by solving a small-scale problem on the quantum computer IBM Brisbane and discuss the potential for exponential speedup in comparison to classical wave equation integrators. Lastly, a work is presented that employs random field interferometry for ultrasound computed tomography for medical imaging.

This dissertation comprises two parts, each of which details an investigation within topological matter.In the first part, we investigate the real space localization properties of static topological insulators in two dimensions. We develop three numerical procedures each of which removes an extensive fraction of the local degrees of freedom. We probe the real space properties of the residual projection operator, showing that the maximally localized length scale ξ supported in the residual Hilbert space diverges as the fraction of states remaining approaches 0. The power-law exponent ν = 1/2 characterizing the singular behavior of ξ appears universal across class A insulators with non-zero Chern number.In the second part, we explore a class of Floquet topological phases in three dimensions. These phases have translational invariance but no other symmetry, and exhibit anomalous transport at a boundary surface. We show that the boundary behavior of such phases falls into equivalence classes up to local 2D unitary evolution. This provides a classification of the 3D bulk, which we conjecture to be complete. We demonstrate that such phases may be generated by exactly solvable “exchange drives” in the bulk. The edge behavior of a general exchange drive in two or three dimensions is shown to derive from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.

This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption.

Topological insulator (TI) is a new class of quantum material with inverted band structure caused by strong spin-orbit coupling (SOC), resulting an insulating bulk and topologically protected surface states. By introducing magnetism to break the time-reversal symmetry, it can host emerging topological quantum phases such as quantum anomalous Hall (QAH) effect, axion insulator quantum phase, and can host chiral Majorana fermion when coupled to superconductors. The goal of this dissertation is to study these quantum phases by material growth and transport measurements.First, we studied the growth condition by molecular beam epitaxy (MBE) and achieved high quality magnetically doped topological insulator (MTI) film growth that can host QAH effect. To understand the different origins of zero Hall conductance plateau in QAH insulators and axion insulators, we carried out the thickness dependent study on MTI sandwiched structures and observed a dimension crossover between 2D QAH and 3D axion insulators by minor loops measurements and scaling analysis. Then, we investigated the QAH insulator and antiferromagnetic heterostructures which offer an additional degree of freedom to manipulate the QAH state in MTI. By proximity coupling the QAH insulator to antiferromagnetic insulator Al-Cr2O3, we observed an exchange biased QAH effect, and the exchange bias can be effectively controlled by a field training process. In addition, chiral Majorana edge modes are reported to exist in QAH insulator and superconductors heterostructures. Aiming to understand the discrepancy of recent experimental reports, we fabricated QAH insulator with superconductor and normal metals hybrid devices, and we found the details of material characteristics, interface conditions play important roles on the experiment observations. Finally, two collaborative works closely related to the main scope of this dissertation are introduced. To probe the size limit and to understand the inter-channel scattering of QAH effect, we fabricated mesoscopic scale sub-micron size QAH insulator devices and carried out size dependent studies. Enabled by the successful growth of QAH film on insulating 2D van-der-Waals substrate MICA, we were able to fabricate dual-gate QAH device by exfoliation and transfer techniques. By applying a vertical electric field, we observe a topological phase transition from QAH to a trivial insulator state. This research will greatly improve the understanding of emerging topological quantum physics, and lead to the new revolution of quantum electronic devices.

Typical thermodynamic systems have a small number of minimum-energy ground states into which the system strives to arrange itself when the temperature is sufficiently low. In some cases, however, competition between interactions leads to the suppression of conventional long-ranged magnetic ordering, through a mechanism known as frustration. One of the typical features is the appearance of a large number of degenerate (or nearly degenerate) minimal energy states, that can exhibit topological order. The dynamical properties of topological systems, in particular those that host fractionalised excitations, is currently a topic of intense research activity. This thesis focuses on one specific instance of a topological frustrated magnetic system: spin ice. Spin ice is one of the most thoroughly researched frustrated models, exhibiting exciting phenomena typical of topological matter - including an emergent gauge field description with fractionalised excitations that take the form of emergent magnetic monopoles. Highly accurate numerical methods and analytical techniques have been developed to model spin ice materials, and have facilitated a detailed understanding of their static properties. The primary aim of this thesis is to address some of the outstanding questions about the dynamics of classical spin ice materials. The focus will be on two dynamical properties: the magnetic relaxation time and the magnetic noise spectrum. Several independent experiments, some dating back over two decades, have found that the magnetic relaxation time in these materials diverges rapidly upon cooling - more rapidly than what any previous theory can explain without invoking extrinsic contributions such as surface effects, disorder, and temperature dependent microscopic time scales. Previous theories also predict that the magnetic noise spectrum of spin ice should be Lorentzian, with a power spectral density that decays as ν^(-2) with the frequency, ν. Here, noise spectra measured on a single crystal of Dy₂Ti₂O₇ are presented and shown to instead decay as ν^(−α), with an anomalous exponent α ≈ 1.5. Through a combination of extensive numerical modelling and fundamental arguments about the motion of the magnetic monopoles, I establish that the conventional model of spin ice dynamics cannot explain the anomalous magnetic noise. I resolve these issues by introducing a new model for the dynamics, which includes the effects of local symmetry on the flip rate of spins. The main insight the new model leads to is the emergence of dynamical fractal structures on which the magnetic monopoles are constrained to move. It is by hosting the monopole motion that the fractals bequeath the magnetic noise with an anomalous exponent and cause a slow-down of the relaxation. The spatial properties of the fractals thus manifest themselves in the time dependence of the magnetisation - and the anomalous dynamics observed in Dy₂Ti₂O₇ are evidence of fractal objects in a disorder-free, bulk crystal. Indeed, the new model dynamics quantitatively captures the experimentally observed noise spectrum and relaxation times, without the introduction of any additional fitting parameters. The dynamical fractals cannot be imaged directly, but do influence other dynamical properties, besides the already discussed magnetic noise and relaxation time. Using simulations and simplified models of analogous systems, I investigate how the fractals affect both equilibrium and out-of- equilibrium transport properties. The response of spin ice to an oscillating magnetic field is identified as one particularly promising area to explore in future experiments, with suggestions made for the field and temperature regimes in which fractals are likely to manifest themselves. An alternative spin ice model capable of reproducing the experimentally observed relaxation time and anomalous magnetic noise, but incompatible with the thermodynamics of spin ice materials, is also introduced. The model includes a specific choice of third-nearest-neighbour exchange interactions, which are frustrated independently of the usual spin ice interactions. These additional interactions are found to induce a phase transition between the normal spin ice phase and a spin nematic phase - a phase displaying rotational symmetry breaking without long-ranged order. In the "nematic spin ice" phase, low energy paths are formed along which magnetic monopoles move freely. These paths form anisotropic fractal clusters with properties akin to the dynamical fractals mentioned previously. Thus, the anomalous noise spectrum is again found to originate from the motion of magnetic monopoles on fractals.

We study a class of three dimensional exactly solvable models of topological matter first put forward by Walker and Wang. While these are not models of interacting bosons or fermions, they may well capture the topological behaviour of some strongly correlated systems. In this work we give a full pedagogical treatment of a special simple case of these models, which we call the 3D semion model: We calculate its ground state degeneracies for a variety of boundary conditions, and classify its low-lying excitations. While point defects in the bulk are confined in pairs connected by energetic strings, the surface excitations are more interesting: the model has deconfined point defects pinned to the boundary of the lattice, and these exhibit semionic braiding statistics. The surface physics is reminiscent of a υ=1/2 bosonic fractional quantum Hall effect in its topological limit. Our special example of the 3D semion model captures much of the behaviour of more general `confined Walker-Wang models'. We contrast the 3D semion model with the closely related 3D version of the toric code (a lattice gauge theory) which has deconfined point excitations in the bulk. We discuss how more general Walker-Wang models can have a combination of fermionic or bosonic deconfined excitations in the bulk, and excitations which are deconfined on the surface but confined in the bulk. Having classified the spectra and surface properties of a number of these models, we construct a topological field theory which describes a subset of them exactly. Moreover, we argue that some Walker-Wang models capture the low-energy long-wavelength behaviour of certain gapped phases of axion electrodynamics which, in turn, are believed to be germane to certain condensed matter systems with strong spin-orbit coupling.

Out-of-equilibrium many-body physics is the fascinating study of complex systems that are subjected to driving, dissipation, or both. Through such procedures it is possible to generate exciting new dynamical phases of matter that have no static counterpart. In this thesis, we investigate the interplay between driving and dissipation in several paradigmatic many-body quantum systems. In particular, we analyze the impact of nonequilibrium phenomena on the stability of topological matter and light-matter interactions. We begin by addressing how to characterize the topological criticality of lower-dimensional systems whose parameters are periodically driven in time. We first introduce a curvature renormalization group method that extends the notions of Landau criticality to the realm of topological phase transitions. We show how this framework presents a simple and efficient way to map out topological phase diagrams of static and Floquet systems in any dimension. Furthermore, we illustrate its ability to precisely capture topological critical phenomena. We then apply it to prototypical static and Floquet topological systems to extract correlation lengths, critical exponents, scaling laws, and universality classes. In one dimension, we apply this methodology to the periodically driven Kitaev chain hosting Majorana end modes. As an interlude, we also examine the consequences of multifrequency driving, and reveal that it can be used as a valuable tool to dynamically control the number and the stability of the topological modes. In two dimensions, we analyze a Floquet-Chern insulator and the periodically driven Kitaev model on the honeycomb lattice. We also establish that periodic driving can induce very exotic topological semimetal phases hosting nodal loop gap closures. We demonstrate that the corresponding phase transitions belong to a different universality class than the ones described by Dirac low-energy theories, but both can coexist at multicritical points. We then discuss the physical origin of nodal loop gap closures and establish that they emerge as a consequence of enhanced symmetries introduced by the driving scheme. We separately analyze symmetry protection and provide concrete examples of how to induce it in both static and driven settings. In realistic settings, one has to often deal with open systems. To incorporate dissipation in our analysis, we re-examine the periodically driven Kitaev chain additionally subjected to a coupling with end reservoirs. In this context we discover that the Floquet-Majorana edge modes survive despite dissipation, and have strong signatures in stroboscopic observables that can reliably track their appearance and disappearance in different phases. Another challenge of contemporary physics is to understand the role of many-body interactions in driven-dissipative settings. As a first step in this direction, we present a very efficient numerical algorithm — the MultiConfigurational Time-Dependent Hartree method — tailored at reliably simulating the time-evolution of interacting many-body quantum systems. We illustrate its applications to several systems of ultracold atoms coupled with the radiation field of optical cavities that host intriguing light-matter phases stabilized by dissipation, such as superradiant phases, Mott insulating phases, and Tonks-Girardeau phases. We highlight how to calculate density distributions, correlation functions, and various order parameters for such setups, and how to simulate realistic single-shot experiments. We conclude this thesis by presenting a brief outlook on unresolved issues and new intriguing questions that have surfaced throughout our study.

Developing optoelectronic devices with increased efficiency and novel functionality requires an understanding of the electronic structure of exotic materials and their interfaces. Perovskite oxides and two-dimensional (2D) materials have emerged as promising classes of materials that exhibit intriguing properties such as high temperature superconductivity and spin-valley locking. Combining these materials into heterostructures offers even more functionality and is facilitated by a shared crystal structure (in the case of perovskites) or van der Waals bonds that do not require epitaxial relationships for clean interfaces (in the case of 2D materials). In this thesis, we present original studies of four materials within these broader classes. Thin films are fabricated by molecular beam epitaxy (MBE) and chemical vapor deposition (CVD) and studied primarily through angle-resolved photoemission (ARPES) measurements conducted at Cornell and the SOLEIL synchrotron in Gif-sur-Yvette, France. LaxBa1-xSnO3 (LBSO) is a high mobility perovskite oxide with a large band gap, enabling promising applications as a transparent conductor for use in solar energy harvesting or fully transparent electronics and as a channel material in all-oxide transistors. LBSO films are grown by MBE and studied by in situ ARPES. While the valence band structure is found to agree well with bulk density functional theory (DFT) calculations, a La-dependent upward band banding is observed at the surface. Additional exposure to ultraviolet (UV) light induces a reduction in the original band bending, offering a route for controlling carrier concentration and band offsets at LBSO interfaces. Graphene is a 2D semimetal with a host of exotic properties, including an extremely high mobility and tunable carrier concentration. Here, monolayer graphene is grown by CVD and studied by ex situ ARPES. Although the films originate from multiple nucleations, all individual graphene grains share the same crystallographic orientation, providing an ideal building block for van der Waals heterostructures with angle-tunable properties. Twisted graphene bilayers are fabricated from these growths and spatially resolved nano-ARPES reveals angle-dependent gaps in the graphene electronic structure, providing a route for creating devices with tunable optical absorption and exotic electronic states. Cu2Si is a 2D Dirac line node semimetal, a newly appreciated form of topological matter. Here we form van der Waals heterostructures of graphene and Cu2Si on Cu substrates by CVD, representing a unique interface between two atomically thin topological materials. SnSe2 is a layered main-group metal dichalcogenide that has exhibited gate-tunable superconductivity and has promising applications as a component in high efficiency two-dimensional heterojunction interlayer tunneling field effect transistors. However, despite decades of study, basic questions about its electronic structure remain unanswered. Here we synthesize thin films of SnSe2 by MBE and study them with ex situ ARPES. A comparison between ARPES and DFT reveals the importance of spin-orbit coupling and out-of-plane dispersion in the SnSe2 valence band structure, critical information for developing new electronic devices based on SnSe2.

Topological phases of matter is a new and quickly growing field in condensed matter physics. The fundamental ideas of this field are heavily inspired by high-energy physics and cosmology, and it is concerned with topologically protected properties of materials and devices, which are intrinsically robust against small perturbations. Research in this direction promises us a deeper understanding of the connections between the different fields of physics and also alluring practical applications like a fault-tolerant quantum computer based on non-Abelian Majorana zero modes. In this thesis we address problems from the whole spectrum of topological matter: topological insulators, topological semimetals and topological superconductors. In the first part of the thesis we reflect on the well known topological obstruction for describing the ground state of a topological insulator as a smooth and periodic function of the momentum in the whole Brillouin zone, by a so-called smooth gauge. This problem is mathematically equivalent to finding exponentially localized Wannier functions for a group of energy bands that is topologically nontrivial. We show that the obstruction can be avoided by breaking the symmetries that protect the topology of the system and give a generic procedure for obtaining a smooth gauge based on this principle, given that the total Chern number of the system is zero. In the second part of the thesis we find and describe a new topologically protected fermionic quasiparticle, which consists of the crossings of three bands - the triple point fermion. Unlike the Weyl or Dirac fermion, the triple point fermion has no analogue in the relativistic quantum field theory. We find a number of materials realizing these quasiparticles and discuss their unique physical phenomena like Lifshitz transitions, surface states and transport anomalies. In the last two chapters of the thesis we discuss two problems in the field of Majorana zero modes. The first problem is concerned with finding a material with optimal properties, like particularly a large spin-orbit splitting, for realizing Majorana zero modes. Since both the semiconductors InAs and InSb have been successfully used in several Majorana devices we investigate the ternary alloy InAs$_x$Sb$_{1-x}$. For the disordered alloy we find the optimal composition to maximize the structure-inversion-asymmetry induced spin-orbit splitting. Furthermore, we identify the CuPt-ordered alloy InAs$_{0.5}$Sb$_{0.5}$ to have exceedingly large intrinsic bulk spin-orbit splitting, making it very promising for Majorana devices. In the second problem we find an explanation for the anomalously large Land $g$-factors --- several times larger than the corresponding bulk value --- found in recent Majorana nanowire devices. We show that higher subbands in cylindrical nanowires are subject to a $\mathbf{L}\cdot\mathbf{S}$ coupling that can enhance the $g$-factor by an order of magnitude or more for small effective mass semiconductors. Doing numerical simulations of InAs and InSb wires we find that the effect is robust to some degree to the breaking of cylindrical symmetry.