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To investigate the effects due to proximity between a three-dimensional topological insulator (TI) and an insulating ferromagnet (IF), TI-IF thin film bilayers were fabricated with pulsed laser deposition. Either bismuth(III) selenide (Bi2Se3) or bismuth-antimony(III) telluride (BST) was used for the TI layer, whereas the IF layer was formed by the Heisenberg ferromagnet EuS. While a positive magnetoresistance was observed above the Curie temperature of EuS, as ubiquitously observed in high-quality TI thin films, an unusual negative magnetoresistance was observed below the Curie temperature in the variable-range hopping regime. The angular dependence of such negative magnetoresistance indicates an orbit origin. Specific to BST-EuS bilayers, when the bulk conduction is minimized, magnetic anomalies in AC susceptibility were observed concurrently with resistive anomalies at the same temperatures, suggesting an interface magnetic order. These phenomena together suggest two-stage proximity effects between the topological insulators and the insulating ferromagnet, and provide first steps to realize the half-integer quantum anomalous Hall effect.

The behavior of the strongly correlated strange and bad metals have eluded theoretical understanding for some time. They exhibit anomalous transport and thermodynamic properties that defy typical predictions according to the quasiparticle picture of Fermi liquid theory. I will first discuss a theory for the widely studied strange metal, strontium ruthenate. Specifically, strontium ruthenate contains "hot" electrons that are less quantum mechanical than the other "cold" electrons. A scattering process in which a cold electron becomes hot after colliding with a second cold electron is unusually strong. This fact is seen to underpin the strange metallicity of strontium ruthenate. I will also discuss a tractable microscopic model for bad metals, by supplementing the well-known Hubbard model with a screened Coulomb interaction. Throughout the phase diagram, we observe T-linear resistivity above the Mott-Ioffe-Regel limit. At intermediate temperatures, this arises from a cancellation between the nontrivial temperature dependence of both diffusivities and thermodynamic susceptibilities

Topological states of matter exhibit properties that are protected by certain topological invariants and are invariant under perturbations. Depending on whether they have an excitation gap or not, these topological states can be classified into gapped and gapless phases. In this thesis, we first discuss the novel properties and recent developments of two kinds of gapped topological states in two-dimensions (2D): the quantum anomalous Hall (QAH) insulator and the chiral topological superconductor (TSC), which can be realized in thin films of 3D topological insulators. In particular, we identify the presence of a zero Hall conductance plateau and a unique half plateau of two-terminal conductance in the QAH state and the chiral TSC state, respectively, both of which are confirmed by experiments. In the second part, we present two novel kinds of gapless topological states of matter characterized by linking numbers: the 5D Weyl semimetal and the 3D time reversal invariant superconductor with linked nodal lines, which show interesting bulk-boundary correspondences and topological responses.

Topological semimetals are characterized by topologically protected band-structure nodes. One prominent example is the Weyl semimetal, characterized by Weyl points carrying topological Chern numbers. In this dissertation, we explore the topology of band-structure nodes with three ingredients: crystal symmetry, non-Hermiticity and periodic driving. First, we show that point group symmetry facilitates a new type of topological invariant from relative homotopy theory, which determines the rules for converting generic band nodes to nodes at high-symmetry momenta (and vice versa) as we tune the lattice Hamiltonian. Secondly, we show that exceptional lines in non-Hermitian bands act as Alice strings. This is manifested by the reversal of the topological charge of a node, if it is braided around an exceptional line. Finally, we discuss the Nielsen-Ninomiya no-go theorem in Floquet bands and the possibility of simulating chiral Weyl particles in the adiabatic limit

Holographic duality is a proposed correspondence between quantum many-body systems living on the conformal boundary of an asymptotically anti-de Sitter space and the quantum gravity living in one higher dimension in the bulk. A particularly interesting aspect of this duality is played by the quantum entanglement. According to the Ryu-Takayanagi formula, the entanglement entropy of a boundary region corresponds to the area of the minimal surface bounding this region, which agrees with the entropy properties of the tensor network states. This dissertation aims to build concrete relations between the holographic duality and the tensor networks. We first propose the concept of the bidirectional holographic code, which means all the boundary states can be represented in the bulk while the states within a constrained code subspace play the role of "classical geometries". We implement the bidirectional holographic code explicitly using the random tensors, which generates a class of solvable tensor network models that reproduce many properties of the holographic duality. We further generalize the random tensor network approach to allow quantum superposition of di erent spatial geometries, which moves one step towards the quantum description of the gravity theory. Lastly, we study the dynamical properties in the holographic duality using the tensor networks and find that the quantum error correction properties bridge the light-cone structure in the bulk and the buttefly velocity (i.e. the speed of chaos propagation) on the boundary.

Strongly correlated electronic systems have stimulated numerous developments in theoretical frameworks, numerical tools, and experimental techniques. One group of materials that have been attracting meticulous efforts is the high-temperature (high-Tc) superconductors, not merely due to its large transition temperature, but also because it offers a well characterized laboratory for the study of exotic phenomena such as quantum criticality and non-Fermi liquid behavior. One pressing question in the field is the role played by disorder: inevitable in real materials, disorder is able to fundamentally alter the properties of system under certain circumstances. This thesis studies, from a theoretical point of view, how disorder affects various electronic orders in several model high-Tc superconductors, in particular copper-based high-Tc materials (cuprates). Various recent experiments in these materials are readily interpreted in light of our results.

Condensed matter systems are complex yet simple. Amidst their complexity, one often find order specified by not more than a few parameters. Key to such a reductionistic description is an appropriate choice of basis, two of which I shall describe in this thesis. The first, an exact mapping known as the Wannier State Representation (WSR), provides an exact Hilbert space correspondence between two intensely-studied topological systems, the Fractional Quantum Hall (FQH) and Fractional Chern Insulator (FCI) systems. FQH states exist within the partially filled Landau levels of interacting 2D electron gases under strong magnetic fields, where quasiparticles exhibit topologically nontrivial braiding statistics. FCI systems, which are novel lattice realizations of FQH systems without orbital magnetic field, are still not completely understood and will benefit from a basis that explicitly connects them to the much better understood FQH systems. The second basis mapping, the Exact Holographic Mapping (EHM), maps any lattice system to a holographic 'bulk' with an additional emergent dimension representing scale. Devised in the spirit of the highly popular AdS-CFT correspondence, it attempts to understand the relationship between a given system and its equivalent dual geometry. In particular, I found excellent theoretical agreement of certain dual geometries from EHM with those expected from the Ryu-Takanayagi formula relating bulk geometry with entanglement entropy. Additionally, the EHM also proves useful in providing a link between different topological quantities, such as relating the Chern number of the abovementioned Chern Insulators with the Axion angle of 3D topological insulators.

Electrons in solid state materials display a remarkable diversity of strong correlation effects, many of which are poorly understood theoretically. In this regard, the cuprate high temperature superconductors are the shining example, remaining enigmatic despite decades of study. This dissertation examines basic questions in condensed matter physics inspired by these remarkable materials. In the first part, I consider aspects of the phenomenology of the cuprate and ruthenate unconventional superconductors. Emphasis is placed on the interpretation of existing experiments, and the proposal of new ones, to clarify the electronic structure of these materials and permit a more thorough microscopic understanding. In the second part, I focus on the abstract problem of nematic quantum criticality in metals, which recent experiments suggest may play a crucial role in the physics of both cuprate and iron-based superconductors. I attack this problem using both a field theoretic approach, valid at weak coupling, as well as fully nonperturbative, numerically exact Monte Carlo simulations in the strong coupling regime, yielding simple, unexpected results that are reminiscent of experiment.