High critical-temperature (Tc) superconductivity in the cuprates has been a defining challenge of condensed matter physics for the 35 years since their discovery. One strategy to address this challenge has been to look for "cuprate analog'' materials: alternative transition metal oxides that exhibit ingredients that are considered proxies for cuprate physics. These key ingredients include a quasi-2D structure based on the CuO2 planes, a nominal oxidation state for Cu2+: 3d9 with a single hole in the uppermost dx2-y2 orbital, and a strong O(2p) and Cu(3d) hybridization. Nickelates have been an obvious choice of study in this context due to the proximity of Ni to Cu on the periodic table. After a 30 year wait, superconductivity in nickelates was realized for the first time in 2019 in hole-doped NdNiO2 (Li et al, 2019). This material contains NiO2 planes (analog to the CuO2 planes of the cuprates), and realizes a Ni1+ oxidation state (analog to Cu2+). NdNiO2 is simply the infinite-layer member of a larger family of materials represented by the chemical formula Rn+1NinO2n+2 (R= La, Pr, Nd; n >= 2), where n refers to the number of NiO2 planes along the c axis. In this thesis, a comprehensive description of the electronic structure of the Rn+1NinO2n+2 family of layered rare-earth nickelates (for n= oo and n=2-6) using state-of-the-art first-principles methods is presented. Specifically, different levels of theory are used to describe the electronic structure of this family of materials: from density-functional band theory (DFT) to incorporating correlation effects at the mean-field level via DFT+U, and finally including dynamical many-body effects via DFT+dynamical mean-field theory (DFT+DMFT). It is shown that the cuprate-like character of the layered nickelate series increases from the n=oo to the n=3 members. Namely, as n decreases the electronic structure becomes more single-band-like, and the degree of p-d hybridization increases while correlations are dominated by the dx2-y2 orbitals. Insights from these calculations allowed for the prediction of the n=4-6 nickelates as ideal candidates for nickelate superconductivity. Indeed, superconductivity was subsequently observed in the quintuple layer nickelate Nd6Ni5O12 (Pan et al, 2021). That superconductivity arises in this layered rare-earth nickelate series, suggests that a new family of superconductors has been uncovered, currently with two members, n=oo and n=5.

In this thesis, applications of sparsity, specifically sparse-tensors are motivated in physics.An algorithm is introduced to natively compute sparse-tensor's partial-traces, along with direct implementations in popular python libraries for immediate use. These applications include the infamous exponentially-scaling (with system size) Quantum-Many-Body problems (both Heisenberg/spin-chain-like and Chemical Hamiltonian models). This sparsity aspect is stressed as an important and essential feature in solving many real-world physical problems approximately-and-numerically. These include the original motivation of solving radiation-damage questions for ultrafast light and electron sources.

Rare-earth tritellurides (RTe3) are two-dimensional materials with unique quantum properties, ideal for investigating quantum phenomena and applications in supercapacitors, spintronics, and twistronics. This dissertation examines the electronic, magnetic, and phononic properties of the RTe3 family, exploring how these can be controlled using chemical pressure, cationic alloying, and external pressure.The impact of chemical pressure on RTe3 phononic properties was investigated through noninvasive micro-Raman spectroscopy, demonstrating the potential of optical measurements for determining charge density wave (CDW) transition temperatures. Cationic alloying studies showed seamless tuning of CDW transition temperatures by modifying lattice constants and revealed complex magnetism in alloyed RTe3 with multiple magnetic transitions. A comprehensive external pressure study examined the influence of spacing between RTe3 layers on phononic and CDW properties across the RTe3 family. Comparisons between different RTe3 materials showed LaTe3, with the largest thermodynamic equilibrium interlayer spacing (smallest chemical pressure), has the most stable CDW phases at high pressures. Conversely, CDW phases in late RTe3 systems with larger internal chemical pressures were more easily suppressed by applied pressure. The dissertation also investigated Schottky barrier realignment at RTe3/semiconductor interfaces induced by CDW transitions, revealing changes in Schottky barrier height and ideality factor around the CDW transition temperature. This indicates that chemical potential changes of RTe3 below the CDW transition temperature influence Schottky junction properties, enabling CDW state probing through interface property measurements. A detailed experimental and theoretical analysis of the oxidation process of RTe3 compounds was performed, which revealed faster degradation in late RTe3 systems. Electronic property changes, like CDW transition temperature and chemical potential, are observed as degradation progresses. Quantum mechanical simulations suggested that degradation primarily results from strong oxidizing reactions with O2 molecules, while humidity (H2O) plays a negligible role unless Te vacancies exist. Lastly, the dissertation establishes a large-area thin film deposition at relatively low temperatures using a soft sputtering technique. While focused on MoTe2 deposition, this technique may also apply to RTe3 thin film deposition. Overall, this dissertation expands the understanding of the fundamental properties of RTe3 materials and lays the groundwork for potential device applications.

Recent developments inspired by string theoretic considerations provide multiple maps between gravitational and non-gravitational degrees of freedom. In this dis- sertation I discuss aspects of three such dualities, the gauge/gravity duality and how it applies to condensed matter systems, the fluid-gravity duality, and the color-kinematics duality.

The first of these, colloquially referred to as holography, in its simplest form posits a mapping of d-dimensional conformal field theory (boundary) partition functions onto d+1 dimensional gravitational(bulk) partition functions, where the space-time carries a negative cosmological constant. In this dissertation I discuss the results of our calculations examining the emergence of Fermi-surface like structures in the bulk spacetime despite the absence of explicit Fermions in the theory.Specifically the 4+1 dimensional Einstein-Maxwell-Chern-Simons theory with scalar degrees of freedom, with and without symmetry breaking is considered. These theories are gravity duals to spatially modulated gauge theories. The results of calculations presented here indicate the existence of a rich phase space, most prominently Fermi shells are seen.

The second set of dualities considered are the color-kinematic duality, also known as the double-copy paradigm and the fluid-gravity duality. The color-kinematic duality involves identifying spin-2 amplitudes as squares of spin-1 gauge amplitudes. This double copy picture is utilized to construct “single copy” representations for space- times where Einstein’s equations reduce to incompressible Navier-Stokes equations. In this dissertation I show how spacetimes that characterize irrotational fluids and constant vorticity fluids each map to distinct algebraically special spacetimes. The Maxwell fields obtained via the double-copy picture for such spacetimes further provide interesting parallels, for instance, the vorticity of the fluid is proportional to the magnetic field of the associated gauge field.

In this dissertation two kinds of strongly interacting fermionic systems were studied: cold atomic gases and nucleon systems. In the first part I report T=0 diffusion Monte Carlo results for the ground-state and vortex excitation of unpolarized spin-1/2 fermions in a two-dimensional disk. I investigate how vortex core structure properties behave over the BEC-BCS crossover. The vortex excitation energy, density profiles, and vortex core properties related to the current are calculated. A density suppression at the vortex core on the BCS side of the crossover and a depleted core on the BEC limit is found. Size-effect dependencies in the disk geometry were carefully studied. In the second part of this dissertation I turn my attention to a very interesting problem in nuclear physics. In most simulations of nonrelativistic nuclear systems, the wave functions are found by solving the many-body Schrödinger equations, and they describe the quantum-mechanical amplitudes of the nucleonic degrees of freedom. In those simulations the pionic contributions are encoded in nuclear potentials and electroweak currents, and they determine the low-momentum behavior. By contrast, in this work I present a novel quantum Monte Carlo formalism in which both relativistic pions and nonrelativistic nucleons are explicitly included in the quantum-mechanical states of the system. I report the renormalization of the nucleon mass as a function of the momentum cutoff, an Euclidean time density correlation function that deals with the short-time nucleon diffusion, and the pion cloud density and momentum distributions. In the two nucleon sector the interaction of two static nucleons at large distances reduces to the one-pion exchange potential, and I fit the low-energy constants of the contact interactions to reproduce the binding energy of the deuteron and two neutrons in finite volumes. I conclude by showing that the method can be readily applied to light-nuclei.