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Quantum coherences are phase relations between distinct quantum states responsible for quantum interference. Many emerging technologies in computing, metrology, and energy generation, share the goal of exploiting macroscopic control of quantum coherence to de- sign high-efficiency devices. A central issue in quantum technology is sustaining coherence at scale. While accurate quantum dynamics methods are essential to the design of these materials and the nonlinear spectroscopic techniques used to characterize them, exact quantum solutions for these systems are intractable. What is needed is a quantum dynamics method which approximates coherences accurately, scales sensibly, and has control over the extent of quantumness assumed in the equations of motion. To address this problem, a new phase space quantum dynamics method is developed. Working in the Wigner-Moyal representation, exact solutions to the coherence dynamics of a two-state displaced oscillator and model conical intersection are derived using a Thawed Gaussian ansatz. This Thawed Moyal solution corrects the lower order semiclassical approach traditionally used in time-domain spectroscopy. Using the kinematic insights of the Thawed Moyal theory, a new formalism called the Star Coherence Representation is derived. The Star Coherence Representation solves the unitary evolution of a pure state quantum density entirely in terms of its populations and relative phases, with explicit dependence on the off-diagonal coherences eliminated. This representation instead constructs quantum coherences on-the-fly in terms of the instantaneous values of the populations and phases. By recasting evolution in terms of population distribution functions, quantum equations of motion to be solved by linearly scaling classical trajectory ensembles which are parallelizable and amenable to statistical estimation through standard techniques of machine learning.

This thesis presents new directions for cosmological data analysis using Bayesian techniques and machine learning. First, I introduce a novel machine learning spectroscopic analysis technique to detect absorption systems in the Lyman-α forest. Using Gaussian processes, I build data-driven models for the quasar emission and apply Bayesian model selection to classify the damped Lyman alpha absorbers (DLAs), which are high column density absorption systems found in quasar spectra. The Gaussian process DLA finder (GP-DLA) is applied to the Sloan Digital Sky Survey (SDSS) quasar spectra and and now adopted by the Dark Energy Spectroscopic Instrument (DESI) collaboration. This GP-DLA technique allows us to construct probabilistic catalogs of damped Lyman-α absorbers, offering a new approach to studying the intergalactic medium and cosmology at z = 2 - 5.Next, I present a new method to infer cosmological parameters using Bayesian surrogate modeling with multi-fidelity emulators. Multi-fidelity emulators are a type of surrogate model that use information from multiple levels of fidelity to improve the accuracy of the surrogate model. This approach accelerates both the analysis of cosmological simulations and the inference of cosmological parameters, providing a probabilistic method to quantify and correct the resolution in cosmological simulations. Multi-fidelity emulators make it possible to perform fast and accurate parameter inference on large-scale structure data, such as the matter power spectrum, using computationally expensive simulations in high-dimensional parameter spaces.Finally, I discuss population inference of gravitational wave (GW) data using a mixture model approach. The population statistics of GW events can provide insights into the formation and evolution of binary black holes (BBHs). I present a data-driven method to infer the mixing fraction between BH populations, along with a Bayesian hierarchical approach to correct for selection effects. The results of the mixing fraction analysis suggest that the population of 35 M⊙ BHs is likely separate from the rest of the population, indicating that current formation channels for this mass bump need to be revised to include explanations for the separation of these massive 35 M⊙ binaries.

In quantum materials, the interactions between the bulk lattice, free electrons, and lattice vibrations give rise to interesting phases of matter. In recent years, new materials have been discovered that have nontrivial topological physics, allowing them to host topologically protected surface states. Much work has been done in exploring whether or not such materials can be rendered superconducting via the proximity effect. In this work, I will explore interesting phases of two topological materials: Weyl semimetals and transition metal dichalcogenides. In the first section, I explore the properties of proximity induced superconductivity within these materials. Using a numerical technique known as the spectral method, I determine what, if any, types of superconductivity can be induced within these materials, and whether or not the superconducting state retains their topological properties. Ultimately, I show that, in the absence of a non-trivial tunneling interaction at the interface, no interesting superconducting properties can be induced. In the second section, I will explore the optical absorption properties of a semiconducting materials in the presence of electron-phonon interactions, which typically give rise to phonon side-bands in optical spectra. I demonstrate how, with an appropriate transformation, an expression for the optical spectra to arbitrary phonon side-band order can be calculated for a general interacting system. To demonstrate its validity, I fit the model to the measured optical absorption spectrum of a device consisting of layered transition metal dichalcogenides.

This dissertation explores numerous applications of concepts from quantum information theory to quantum gravity, the AdS/CFT correspondence, as well as holographic dualities more generally. A recurring theme will be the notion that geometric quantities in the bulk are dual to information-theoretic quantities in the boundary.The Ryu-Takayanagi formula and related results tell us that entanglement entropies in the CFT are given in terms of the areas of minimal surface in the bulk. This geometric formula for entanglement entropy imposes inequalities on the entanglement entropies of various CFT regions for static spacetimes, known as the holographic entropy cone. We will numerically investigate the validity of the five-region inequalities for a specific dynamical spacetime, a collapsing black hole spacetime. We find that, for all cases considered, all of the inequalities are satisfied when the null energy condition is satisfied in the bulk, while all of the inequalities are violated if the null energy condition is not satisfied in the bulk. This provides some evidence for the validity of the five-region inequalities in general settings in AdS/CFT with a dynamical bulk. We then discuss the complexity-action and complexity-volume conjectures in the setting of holographic Friedmann–Robertson–Walker universes, which are more realistic cosmological models than AdS space. These conjectures pass some non-trivial consistency checks in these settings. Following this, operator complexity is studied numerically for one- and two-qubit systems using Nielsen complexity geometry. Even though these are relatively simple systems, some interesting similarities are found with the behavior of complexity for more interesting, larger systems. Finally, we explore the use of sandwiched Renyi relative entropy in holography, and in finite-dimensional models of quantum error correction.

Molecular polaritons offer a promising avenue for manipulating light and matter properties through both single-molecule and collective strong light-matter coupling within optical cavities. Over the past decade, numerous theoretical and experimental studies have reported changes in optical and chemical properties as a result of this strong interaction. However, the field is fraught with inconsistent findings. Some experimental results cannot be reproduced or are later given non-polaritonic explanations, while theoretical models often fail to account for observed changes and make correct predictions. This disconnect between theory and experiment arises from the use of overly simplistic models to explain the highly complex nature of polaritonic systems in general, and organic molecules in particular.Specifically, in the field of polariton chemistry, which aims to exploit collective strong coupling to modify chemical reactivity, there has been a tendency to interpret experiments conducted in the collective regime using single-molecule strong coupling models. In the case of single-molecule strong coupling, the excited states of individual molecules hybridize with cavity modes to create vibronic-polariton states, altering the energy levels of the molecules and hence their reactivity. In contrast, in the collective regime, polaritons are excitations delocalized over the entire ensemble of molecules, and it is unclear how they influence the local vibronic dynamics of individual molecules.This thesis presents our efforts to unveil the novel photochemical and photophysical phenomena in organic exciton polaritons. Our findings can be summarized as follows: while collective strong light-matter coupling can significantly alter optical properties, such as the photonic density of states, it has negligible direct effects on the internal degrees of freedom of individual molecules, which are involved in chemical reactivity. Nevertheless, we conclude that polaritonic modifications to optical properties can influence molecular processes in a weak coupling manner, leading to long-range resonance energy transfer, and changes in absorption, emission, and Raman scattering rates. Further advancements require identifying the missing elements in our theories. Effects such as temperature and the multimode nature of optical microcavities may be crucial for understanding the experimental observations that remain unexplained to this day, and for definitively determining novel applications of collective strong light-matter coupling with organic molecules.

Due to recent technological advances in the generation, manipulation, and detection of non-classical light, quantum light spectroscopy has gained attention as a candidate for expanding the current capabilities of classical laser light spectroscopy. In this dissertation, I develop an input-output formulation of quantum light spectroscopy by combining the input-output theory, traditionally used in the quantum optics community, with the perturbative expansion method for nonliear spectroscopy, traditionally used in the chemical physics community. Using this new spectroscopic formalism, we show that the optical signal in a class of quantum light spectroscopy experiments can be emulated by classical laser spectroscopy experiments. This class of quantum light spectroscopy experiments uses n = 0, 1, 2, · · · classical light pulses and an entangled photon pair (a biphoton state) where one photon acts as a reference without interacting with the matter sample.To model the interaction between non-classical light and photosynthetic light harvesting systems, we develop a method to simulate the excitonic dynamics coupled to non-Markovian phonon degrees of freedom and to an N-photon Fock state pulse. This method combines the input-output and the hierarchical equations of motion (HEOM) formalisms into a double hierarchy of density matrix equations. We show analytically that, under weak field excitation relevant to natural photosynthesis conditions, an N-photon Fock state input and a corresponding coherent state input give rise to equal density matrices in the excited manifold. However, an N-photon Fock state input induces no off-diagonal coherence between the ground and excited subspaces, in contrast with the coherences created by a coherent state input. Detailed analysis of the absorption and emission behavior are discussed.

This thesis reports on aspects of semiclassical gravity with an eye towards holography.Chapter 2 introduces a perturbatively traversable wormhole in a particular four-dimensional quotient spacetime, where traversability is ensured by the Casimir energy of bulk fermions. We compute the fermionic contribution to the integrated null stress-energy tensor and find hints that traversability holds at all times.Chapters 3 and 4 report on computations of entanglement negativity, a multipartite entanglement measure which distinguishes between classical and quantum correlations. We compute holographic entanglement negativity in a toy model of Jackiw-Teitelboim gravity with end-of-the-world branes, finding a rich phase structure which includes replica symmetry breaking in a large region of phase space. We also compute entanglement negativity in a toy model of chaotic eigenstates, finding some qualitative agreement with phase transitions in the holographic model.Chapter 5 describes a modification to random tensor networks to incorporate bulk gauge symmetries which we term the "gauged random tensor network." We find an area operator valued in the center of the gauged random tensor network bulk algebra which more closely resembles the area operator provided by the quantum-corrected Ryu-Takayanagi formula.

Symmetry, anomaly and duality put strong constraints on the low-energy physics of the quantum field theory. In this dissertation, we used generalized symmetry and symmetry anomaly to investigate the quantum phases of matter and their transitions. In particular, we studied the generalized exact duality in 1+1d systems and infrared duality for 2+1d gauge theories. We investigated the 't Hooft anomaly matching of non-linear sigma model with the Wess-Zumino-Witten term. We focus on the implications of these non-perturbative methods on 1+1d quantum phase transitions, 2+1d deconfined quantum criticality and Fermi surface symmetric mass generation.

Amplitudes methods have found increasingly varied applications across physics, in particular in the field of gravitational waves. In this manuscript we will present how amplitudes techniques apply to interactions of gravitating spinning bodies. We will analyze the amplitudes of generic spinning bodies and pay special attention to how the structure of these amplitudes simplifies when specializing to spinning black holes. In Chapter 1, we develop how several amplitudes tools such as the Kosower-Maybee-O'Connell formalism, eikonal methods, and effective field theory in the classical limit apply to the dynamics of classically colored particles. Classically colored particles share many of the theoretical features of classical spinning bodies, with the color directly analogous to the spin in many ways, and so serve as an effective proof of concept of how these same tools may be applied to spinning bodies. In Chapter 2, we directly apply these techniques to classical spinning electromagnetically interacting bodies, which share all of the spin related complications of the gravitational problem while being simpler due to the relative simplicity of electromagnetism compared to gravity. In doing so, we find that the effective field theory is capable of carrying an extra vector degree of freedom compared with the previously established worldline formalism, and that that vector degree of freedom allows for spin magnitude change in the theory. We also present a modification of the traditional worldline formalism which perfectly matches the effective field theory. In Chapter 3, we use the worldline formalism to compute generic spinning body Compton amplitudes through the fifth order in spin, at which several interesting complications occur. This Compton amplitude is essential for computing one-loop observables for the spinning binary system. We then use Dixon's multipole moment formalism to identify an effective source energy-momentum tensor for a spinning black hole, which if treated as appropriate in the effective theory determines several previously unconstrained Wilson coefficients which affect black hole observables. In Chapter 4, we extend the electromagnetic methods of Chapter 2 to gravity. We find that the nonminimal degrees of freedom persist in having matching physical effects between the worldline and field theory approaches. As well, we find that those degrees of freedom have observable effects on the gravitational waveform.

While the Standard Model of particle physics and the $\Lambda$CDM model of cosmology have been celebrated for their explanatory prowess, many phenomena remain unexplained. In this dissertation, we explore several methods of inquiry into these enigmas. In Chapters 2 and 3, we scrutinize signals that may hint at the presence of dark matter, and develop models through which these signals can be explained. Next, in Chapter 4, we employ well developed tools such as the path integral to formulate a new method for determining time evolution in cosmology. Finally, in Chapters 5 through 7, we use precise observations of primordial light elements to constrain theories beyond both the Standard Model and $\Lambda$CDM. Together, these investigations shed light on some of the mysteries of the cosmos and will hopefully inspire future avenues of exploration.

In this thesis, we investigate the fundamental aspects of particle physics, specifically in the context of physics beyond the Standard Model. We focus on addressing two of the problems in this context: the flavor problem and the dynamical evolution of chiral symmetries.We explore the flavor problem, specifically concerning the mystery behind the number of generations in the Standard Model (SM) by using the tools in supersymmetric theories to challenge notions of ‘generations’ as an invariant through RG flows from UV to IR. To achieve this, we perform a detailed analysis of non-perturbative field theoretic dynamics in supersymmetric models. This study not only offers profound insights into the peculiarities of QFTs but also helps navigate the string landscape of viable models. Furthermore, we also develop model-building tools to achieve mass gaps in chiral models, a crucial ingredient in generation flows, in full generality for various gauge groups.In the second part of this thesis, we explore the flavor problem using the lens of modular symmetries. Focusing on a torus based approach, the underlying physical parameters in this methodology are derived by adding flux to the two compactified extra dimensions. In chapter 4, we wade through the strengths and challenges of this method. We can understand the flavor parameters by examining the modular-form Yukawas, and the overall symmetry group in terms of the flux parameters. Expanding on the strengths of the modular approach, in chapter 5, we also propose an eclectic scheme, where we draw from the union of modular and traditional discrete flavor groups in string theory, to provide a unified framework forunderstanding flavor physics.Using mathematical analysis in various settings, be it stringy models, or supersymmetric theories, we present this thesis as a minor contribution to the search for an understanding of the puzzles of the Standard Model. It offers several model-building tools and frameworks that provide non-standard insights beyond the Standard Model.

Spintronics investigates the interactions between magnetization dynamics and electron transport, particularly in ferromagnetic (FM) materials. My dissertation expands the realm of spintronics to antiferromagnetic (AF) materials, which are relatively unexplored and lack substantial theoretical frameworks. It specifically examines dc spin pumping in canted easy-plane antiferromagnet α-Fe2O3, facilitated by the Dzyaloshinskii-Moriya interaction.Additionally, the dissertation explores the effects of magnetic fields on NiO below the spin-flop transition, demonstrating robust sub-terahertz spin pumping akin to that in easy-axis AFs by altering the resonance polarization through field application. The study also shows that an external magnetic field can significantly reduce the threshold for Néel vector auto-oscillation via spin-transfer torques, enhancing the potential for innovative device applications in the sub-terahertz spectrum.This comprehensive study not only deepens our understanding of AF spintronics but also paves the way for the development of new spintronic devices leveraging the unique properties of antiferromagnetic materials. Moreover, the dissertation broadens the scope of spintronics by extending its examination to the physical properties of two-sublattice ferrimagnets (FIMs) and antiferromagnetic van der Waals materials within two-dimensional (2D) layered systems.

This thesis includes two parts discussing two fascinating areas of fundamental physics. \textbf{Part (I)} explores the enigma of quark confinement within the intricate domain of quantum field theory, addressing a foundational puzzle in particle physics. Quark confinement dictates that quarks are bound within hadrons, so we cannot observe quarks as free, isolated particles. Despite extensive efforts over the past fifty years, the theoretical foundation of quark confinement in quantum chromodynamics (QCD) is still unclear. Studying the Schwinger effect (particle-antiparticle pair creation in vacuum in the presence of a strong external electromagnetic field) can pave the way to explore the behavior of quarks within hadrons. We utilize an indirect method, in particular the so-called AdS/QCD correspondence, to investigate the response of a QCD-like gauge theory to a static electromagnetic field. AdS/QCD correspondence is a form of gauge/gravity duality. This duality connects a gauge theory (representing particle physics) to a gravitational theory (specifically, string theory with an Anti-de Sitter background metric). Leveraging this method enables us to study the Schwinger effect for quark-antiquark pairs through potential analysis and calculation of the pair-production rate $\Gamma$. Our findings show that both the potential analysis and the calculation of the pair-production rate yield consistent results. We identify two critical electric fields $E_{s}$ and $E_{c}$ as lower and upper bounds of a range in which pair production can occur only by tunneling through a potential barrier. Below $E_{s}$, the potential barrier is insurmountable, and pair production cannot happen. Above $E_{c}$, there is no potential barrier to restrict the pair production. While previous studies have explored various aspects of the Schwinger effect using AdS/QCD duality, further investigation is required for scenarios involving QCD-like gauge theories with simultaneous electric and magnetic fields. Addressing this research gap, our findings reveal that a magnetic field perpendicular to the electric field suppresses $\Gamma$ and increases $E_{s}$. Conversely, a purely parallel magnetic field does not influence the system's response to an external electric field but enhances $\Gamma$ in the presence of a perpendicular magnetic field. \textbf{Part (II)} ventures into the cosmos, examining the formation of young stellar groups (both bound star clusters and unbound stellar associations) within a full cosmological context. A young stellar group is a collection of newly born stars moving together as a relatively coherent unit through a galaxy. Recent observational advancements, driven by improved precision of Gaia DR3, along with complementary near-field studies like PHANGS-HST and PHANGS-JWST, have significantly enhanced our understanding of stellar groups in the Milky Way and local universe galaxies, enabling us to study the conditions and environment that set clustered star formation across a statistically significant sample. Yet despite the exciting progress in observations, there is still a notable absence of robust theoretical cosmological models to interpret the data. Recent advancements in generating galaxy zoom-in simulations enable the comprehensive study of the formation and evolution of giant molecular clouds and stellar groups within a cosmological galactic framework and, in turn, facilitate filling the gap between observations and theoretical models. We present the fundamental properties of young massive stellar groups, both bound and unbound, formed within or near the galactic disk at the present time (at redshifts $z < 0.008$) in the \Latte suite of FIRE-2 Milky Way-like galaxy simulations. Our analysis encompasses the measurement of various characteristics for each stellar group, including its boundedness, mass, size (stellar group’s radius), 1D velocity dispersion, dispersion in age, and dispersion in metallicity [Fe/H]. We find the properties of simulated stellar groups with ages between 0 and 3 million years are within the range of values reported in observational studies. Our results depict the capability of \Latte simulations to generate reasonably realistic star clusters and associations and set the stage for the forthcoming project that will focus on generating synthetic images of the simulated stellar groups and measuring their properties by utilizing the conventional pipelines used in observational studies. This approach will allow a more consistent comparison between simulations and observations, aiding in the establishment of benchmarks for interpreting observations and advancing our understanding of various aspects of galaxy formation, such as stellar evolution, the impact of feedback on galactic dynamics, and the processes involved in star and planetary system formation.

This dissertation is a synthesis of work completed during my Ph.D., loosely connected by the appearance of higher spin particles.In Chapter 2, we explicitly construct an affine generalization of the Dirac action employing infinite dimensional spinorial representations of the group. This implies that it is built from an infinite number of spinor Lorentz multiplets with all possible half integer spins. We introduce a systematic procedure for constructing $GL(d,\mathbb{R})$ and $SL(d,\mathbb{R})$ invariant interaction terms to obtain quite general interacting models. Such models have order operators whose expectation value can break affine symmetry to Poincar\'{e} symmetry. This symmetry breaking pattern is known to exhibit a graviton as its Goldstone boson. We discuss possible interactions and mechanisms for this symmetry breaking to occur, which would provide a dynamical explanation of the Lorentzian signature of spacetime.In Chapter 3, we study consistent deformations of Veneziano and Virasoro amplitudes. Under some physical assumptions, we find that their spectra must satisfy an over-determined set of non-linear recursion relations. The recursion relation for the generalized Veneziano amplitudes can be solved analytically and yields a two-parameter family which includes the Veneziano amplitude, the one-parameter family of Coon amplitudes, and a larger two-parameter family of amplitudes with an infinite tower of spins at each mass level. In the generalized Virasoro case, the only consistent solution is the string spectrum.In Chapter 4, we construct a new formalism for free massive particles of any spin, including integers and half integers, in any spacetime dimension. Massive particles are realized in this formalism as ``dimensionally reduced" massless particles. Explicit propagators are found for any spin and dimension, including the propagators for all auxiliary fields necessary in the formalism. For spin $s=n,n+1/2$, the propagators may be expressed as degree $n$ Gegenbauer polynomials, whose asymptotic $n\to\infty$ properties are known. This enables one to take an appropriate classical limit for particles with classically relevant spin $S\sim \hbar n\gg\hbar$ without extrapolation, all while using dimensional regularization.

This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``$SO(n)$ AKLT chains'' and the phase they occupy the ``$SO(n)$ Haldane phase''. We present new results describing their ground state structure and, when $n$ is even, their peculiar $O(n)$-to-$SO(n)$ symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing $n$, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge.We extend a definition of Ogata's of an SPT index for a split state for a finite symmetry group $G$ to an SPT index for a compact Lie group $G$. We then compute this index, which takes values in the second Borel group cohomology $H^2(SO(n),U(1))$, at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.

The holographic duality, often coined the AdS/CFT correspondence, conjectures a relation between strongly coupled quantum systems and quantum gravity in higher-dimensional spacetimes. Gravitational theories in two and three dimensions are meaningful examples for classical and quantum exploration due to their unique characteristics, notably the absence of propagating bulk degrees of freedom and the presence of only boundary degrees of freedom, distinguishing them from higher-dimensional counterparts. These gravitational theories exhibit complex interactions when the bulk spacetime has a finite size, regulated by Zamolodchikov's double-trace irrelevant $T\overline{T}$ operator. This thesis aims to gain a holographic understanding of pure three-dimensional AdS$_3$ gravity and JT gravity under the influence of the $T\overline{T}$ deformation. Under a finite radial cutoff, these theories exhibit perturbative behavior that implies the emergence of the Nambu-Goto action for the corresponding boundary graviton action. We also conducted semi-classical calculations of observables related to finite-cutoff gravity and its dual $T\overline{T}$-deformed CFT description, including correlation functions involving stress tensors and gravitational Wilson lines, along with an analysis of their supersymmetric extensions. Additionally, we explored the implications of general stress tensor deformations within field-theoretic and holographic settings.

In this thesis, we study the relaxation dynamics and phase transformations in supercooled liquids where microscopic motion dramatically slows down below an onset temperature To. To this end, we use theoretical and computational tools from statistical mechanics and compare our predictions to the available experimental and simulation data.In the first chapter, we focus on the impact of glassy dynamics on crystallization. We develop a coarse-grained lattice model, the Arrow-Potts model, which can simulate crystallization and vitrification. The Arrow-Potts model produces phenomena observed in experiments and molecular dynamics (MD) simulations, including poly-crystalline microstructures near the melting temperature $T_m$ and ramified crystals at low temperatures. To explain these phenomena, we combine several theories that account for the nucleation and growth of crystals, e.g., the Kolmogorov-Johnson-Mehl-Avrami theory, with dynamical facilitation (DF) theory of glassy dynamics. As a result, we derive a formula that predicts the energy barrier for crystallization without fitting to simulation data.In the last two chapters, we delve deeper into understanding relaxation dynamics at a microscopic level by studying the open questions surrounding DF theory. DF theory explains glassy dynamics by assuming the existence of spatially localized excitations, which drive motion and facilitate the formation of nearby excitations. Despite its success, open questions remain within DF theory regarding the microscopic origin of these assumptions. To this end, we develop a theory of excitations and facilitation in supercooled liquids, where we find that excitations are localized pure-shear events whose characteristics and activation energy can be explained using the theory of linear elastic solids. Furthermore, linear elasticity theory can explain how excitations facilitate new excitations via elastic interactions. By constructing a Markov state model based on elastically interacting excitations, we can reproduce key elements of glassy dynamics that agree with experimental observations and MD simulations.

The eternal quest to reconcile quantum mechanics and general relativity has recently led to new insights at the interface between gravitational physics, condensed matter, and quantum information science. Much of this progress owes to the development of the holographic principle, a conjectured duality between gravitational systems and ordinary quantum mechanical systems. This opens up the possibility of translating gravitational phenomena into the language of microscopic quantum physics, and vice versa. This thesis explores several surprising connections that have emerged from a sharper understanding of this duality, with a particular emphasis on questions regarding the dynamics of quantum information. What general lessons can we draw from the physics of black holes about the propagation of quantum information in quantum many-body systems? How can we characterize this propagation and leverage it as a resource for performing tasks on large-scale quantum technologies? From a complementary perspective, how can we utilize quantum simulators to probe features of quantum gravity? By investigating these questions, this thesis sheds light on a rich variety of long-standing topics—including quantum chaos, teleportation, and metrology. More broadly, it demonstrates holography as a powerful theoretical framework for guiding our exploration of quantum many-body physics.

Classically, the ground states of $\mathcal{N}=4$ Super Yang-Mills Theory (SYM) on $\mathbb{R}\times S^3/\Gamma$ where $\Gamma$ is one of the ADE subgroup of $SU(2)$ are flat Wilson lines winding around the ADE singularity. S-duality acts on this finite-dimensional ground state Hilbert space and its action is the same as the $S$ operator in a certain dual Chern-Simons theory on $T^2$. The dual Chern-Simons theory arises out of the only long-range interaction in a string/M theory construction by considering a stack of D3(M5) branes on ADE singularity. This SYM/Chern-Simons duality is verified by matching the ground state Hilbert spaces of both theories and by comparing the S-duality operators of both theories.To one-loop order, the SYM ground state degeneracy is exact. A detailed computation using the superconformal index shows that each classical SYM ground state acquires the same supersymmetric Casimir energy. S-duality maps the SYM ground state Wilson lines to ground state t' Hooft lines taking values in the Langlands dual group. The number of t' Hooft lines are shown to agree with that of the Wilson lines. In addition, the t' Hooft lines have the same supersymmetric Casimir energy as the corresponding Wilson lines. These two facts provide a ground state test of S-duality. The SYM/Chern-Simons duality has an important extension to the class S theory obtained from compactifying M5 branes on a Riemann surface $\mathcal{R}$. The ground states of class S theory on $\mathbb{R}\times S^3/\Gamma$ are dual to the states of the dual Chern-Simons theory on $\mathcal{R}$. In particular, we uncover a surprising result that there is only one unique ground state for the conformal $\mathcal{N}=2$ $SU(2)$ four-flavor theory on $\mathbb{R}\times S^3/\Gamma$. Finally, we apply the SYM/Chern-Simons duality to a non-Lagrangian class S theory and find that its ground states obey the fusion rule of the current algebra of the dual Chern-Simons theory.

Motivated by finding models that admit a solution to the current Hubble constant tension, we implement a general extension to Lambda-CDM. We first review the needed machinery to model the constituents of the universe to linear order in perturbation theory before introducing generalized dark matter. We then present our numerical implementation of high-dimensional generalized dark matter (HDGDM) with a redshift-dependent equation of state parameter. Following statistical analysis of the model with CMB, BAO, and supernova data, we demonstrate that marginalized posterior distributions can easily lead to misleading conclusions on the viability of a high-dimensional model such as this one. We highlight problems with standard techniques and corresponding mitigation strategies. The subsequent results find an observationally favorable parameter space region, making near-future testable predictions. We finally present a technique of dimension reduction for high-dimensional models that focuses on changing statistical predictions of a target parameter.

Quantum field theory plays a central role in the study of theoretical physics, where a satisfying understanding of general field theories is usually absent. In this thesis, the path integral approach will be revisited, and it will be shown how to organize the computation of path integral into the framework of (fully) extended field theory. Several aspects of quantum field theory will be explored with this new viewpoint, including symmetries and anomalies.

In dark matter direct detection experiments, the free nuclear recoil description breaks down as the de Broglie wavelength of dark matter approaches the typical atomic spacing of the target material. In this work, we investigate the particular case of scattering off of crystal targets, whose collective excitations are well understood as phonons. As experimental energy thresholds decrease, it becomes increasingly important to understand the response of targets to energy depositions below the nuclear recoil scale. For dark matter masses lighter than 1 MeV, the scattering rate is dominated by single phonons, while at masses larger than 100 MeV, we expect the scattering to approach the free nuclear recoil result. Starting from the phonon formalism in the harmonic crystal approximation, we perform the first calculations of scattering rates in the intermediate 1-100 MeV mass regime where multiphonon process dominate and demonstrate how the multiphonon response smoothly approaches free nuclear recoil. We then drop the harmonic approximation and calculate possible corrections to the scattering rates due to crystal anharmonicity. We find these anharmonic corrections to be large at small dark matter masses ∼ MeV and large experimental thresholds ∼ 100 MeV, but approach the harmonic result as the DM mass increases.

In many systems, both physical and mathematical, internal boundaries play an outsized role in dictating the large-scale behavior of the system. This theme is explored here in two parts: by identifying natural boundaries to diffusion on undirected graphs for the improvement of spectral graph methods for clustering and classification, and by imposing boundaries in two dimensional PEPS tensor networks to reduce their computational complexity for image generation. In the graph diffusion work, an algorithm is developed that identifies and removes vertices serving as bridges between well-connected clusters. With the use of this algorithm, the performance of unsupervised spectral clustering on graphs derived from synthetic point cloud data shows excellent cluster separation down to approximately two-thirds the point cloud gap size that standard spectral clustering alone tolerates. The same vertex-removal scheme applied to a diffusion-informed active learning classification algorithm shows an approximately order-of-magnitude best-case reduction in the classification error rate on benchmark hyperspectral imagery data in the low-label-query-limit versus the same active learning algorithm applied without vertex removal. The PEPS work proposes a scheme whereby such networks are cut into overlapping patch networks, whose individual contraction complexity is comparatively low. Feasibility testing is performed showing that in principle, this scheme could be used for image in-painting, and has intuitively appealing model features that are directly reflective of the data.

In this dissertation, I describe my work investigating quantum entanglement in a variety of contexts, with the aim of making concrete technical progress towards answering deep physical questions about the quantum nature of the world around us. Part One discusses work that broadly falls under quantum information science—with some motivations fromcosmology—while the work in Part Two is mainly cosmological.The research presented in Part One of this dissertation involves developing and utilizing the adapted Caldeira-Leggett (ACL) model, a toy model that describes a harmonic oscillator coupled to an idealized environment. The model has allowed my collaborators and me to explore novel quantum entanglement phenomena. In Part One I introduce the model and demonstrate its robustness (as done in [1]). I then present a derivation and investigation of behavior that occurs in the early stages of entanglement between a system initially in a quantum superposition and its environment (as done in [2]). Part One concludes with an investigation of entanglement phenomena in equilibrium, to see if the emergence of classicality requires an arrow of time, and what that might imply for deep questions about our Universe (as done in [3]).Part Two considers entanglement in another cosmological context. In a series of papers [4, 5, 6] my collaborators and I investigated the influence entanglement may have on the period of inflation in the very early universe. Specifically, we calculated how entanglement can be generated between the quantum perturbations in the inflaton field—which drives inflation—and those in another spectator field. In Part Two I begin by deriving and demonstrating how entanglement is naturally and inevitably generated during inflation (as done in [4]). I then present results from a full parameter estimation using Monte Carlo techniques to determine what amount of entanglement is allowed by current cosmological data, and what that might imply for our Universe’s quantum origins (as done in [5]). Finally, I present results that explore whether signatures of entanglement during inflation can be used to answer other questions about the history of the early universe—via distinguishing features of phase transitions and/or the inflationary energy scale that may be imprinted on cosmological observables due to entanglement (as explored in [6]).

Warm Dense Matter (WDM) can be found in a myriad of useful and worthwhile applications, whether it be in the ongoing study of inertial confinement fusion, novel shock experiments, or planetary interiors. One crucial tool that has revolutionized our understanding of WDM over the last few decades has been Finite-Temperature Density Functional Theory (FT-DFT), which has enabled various groundbreaking simulations in the field (often when experimental studies were too difficult and costly to be performed). Although successful, modern implementations of FT-DFT suffer from significant drawbacks, namely the large computational burden inherent to studying high-temperature materials, and the lack of sophisticated approximations describing temperature-dependent exchange-correlation interactions. The research contained in this dissertation works to address these fundamental challenges both directly and indirectly.Chapter 2 serves as an introduction to the underlying theorems of ground-state DFT, all of which extend to nonzero temperatures, and are depicted through illustrative examples using the Hubbard dimer. Chapter 3 introduces Density-Corrected DFT, a novel theory studying the origin of errors in functional approximations, and a means of analysis for approximations in both the ground state and at finite temperatures.Chapter 4 derives exact conditions for Ensemble Density Functional Theory, all of which provide insight into the behavior of temperature-dependent exact conditions for FT-DFT. Chapter 5 proposes a novel temperature-dependent generalized gradient approximation, the locally thermal Perdew–Burke–Ernzerhof (ltPBE) approximation, for FT-DFT calculations of WDM. This temperature-dependent approximation is the first of its kind, with its origin rooted entirely in a model of the PBE gradient-dependent exchange-correlation hole density at nonzero temperatures.Lastly, Chapter 6 discusses how Tensor Processing Units (hardware that was originally designed to perform machine-learning tasks) has been repurposed to perform highly efficient Kohn-Sham DFT calculations. This in particular has significant impact on future studies of WDM, where high temperatures greatly increase the computational cost of calculations.

Due to phenomenal progress in trapping and cooling of cold atoms, ultracold atoms has been a successful platform for studying fundamental physics and simulating quantum many-body systems with the advantage of tuning the parameters in real space. In this dissertation, we theoretically investigated inhomogeneous structures of different ultracold atomic mixtures in quasi-one-dimensional box potentials to explore the rich phases of atomic mixtures. The realization of uniform box potentials in experiments allows us to study inhomogeneous structures, phase diagrams and interface structures in various atomic mixtures which arise from the competition between interaction energy and kinetic energy. The presence of real space constraints like box potentials or quenching of interactions permits the extraction of characteristic length scale of the systems. We examined the characteristic lengths of spatial change such as, the healing lengths in the case of boson-fermion mixtures and correlation lengths in the case of superfluid-normal phase transition. The analyses of the healing lengths at the boson-fermion interface confirms the energy competition mechanism behind the phase-separation structures. We modelled spatially tunable inhomogeneous interactions for attractive Fermi gases to study analogs of the proximity effect and the spatial Kibble-Zurek mechanism (KZM) in a unified framework. The introduction of inhomogeneity lead to the distortion of the order parameter which is characterized by the correlation length. We extracted the correlation lengths from the pair wavefunction and correlation function to determine the penetration of the order parameter in the region where interaction vanishes. In the setup resembling the proximity effect, we found that the correlation lengths follow the BCS coherence length while only the exponent from the pair correlation function agrees to the Kibble-Zurek scaling in the case of spatial KZM setup. We also extract the critical exponent by adding a uniform bosonic background to the attractive Fermi gases and found that it does not alter the scaling behavior in the miscible phase. With recent experimental progress in the field of atomic mixtures and box potentials, our results will characterize the density profiles, correlation lengths and scaling behavior of those multi-component quantum systems.

The planar expansion of large N gauge theories has been a remarkably fruitful idea,inspiring many developments in theoretical physics. Perhaps the most important realization of this idea is the duality between critical string theories in anti de-Sitter spacesand large N conformal field theories, giving a window into otherwise intractable phenomena in quantum field theories. One of the most promising aspects of this duality is thepossibility of studying truly quantum aspects of gravitational systems with conventionaltools from quantum mechanics. A major obstacle to this endeavor lies in the fact thatthe most interesting questions about quantum gravity in anti de-Sitter space are relatedto issues that are beyond current techniques in conformal field theory. This thesis dealswith various semiclassical aspects of large operators in holographic conformal field theories, focusing on the study of very large (near-)BPS operators in $\mathcal{N} = 4$ super Yang-Millstheory and their corresponding gravitational avatars.

Quantum computers, once their fault-tolerant versions are built, are widely believed to revolutionize the world, bringing an exponential quantum advantage to problems in chemistry, physics and material science. In this thesis, we explore various problems in theoretical condensed matter physics that are in one way or another related to quantum computers. The first part of this thesis is devoted to Majorana zero-energy modes and topological quantum computing. We describe how Majorana modes can be synthesized in superconductor-semiconductor heterostructures, and emphasize vulnerabilities of the topological phase in such heterostructures to external magnetic fields. We further demonstrate how and to what degree the addition of a magnetic insulator or spin-orbit scattering to a heterostructure can help alleviating the negative effects of external magnetic fields. Additionally, we analyze quantum dot-based measurements of Majorana qubits. In particular, we identify the optimal regime for such measurements, and estimate to what degree external noise sources and coupling of the quantum dot to quasiparticle modes in superconductors affect the fidelity of the measurements. The other part of this thesis is devoted to the many-body physics in quantum circuits. We describe a Floquet circuit model in which projective measurements can drive a dynamical spectral phase transition in otherwise purely unitary dynamics of the circuit. Using random-matrix theory, we analytically calculate various properties of this transition, and analyze how the transition can be revealed in the long-time dynamics of the system.

The gravitational path integral has long served as a crucial tool in deciphering mysteries within quantum gravity. In recent years, studies of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence have offered many valuable insights into comprehending those mysteries, and many fruitful results have been yielded from utilizing the gravitational path integral within the framework of AdS/CFT.This dissertation is devoted to studying certain aspects of the gravitational path integral, discussing its relation with gravitational entropies, spacetime geometries, and its algebraic aspects. We explore contexts from Euclidean to Lorentz signature, from holographic theories to general theories, with a goal of understanding quantum gravity in the real world. In Part I, we discuss the fixed-(HRT)-area states in the gravitational path integral. The fixed-area states are holographic states where the area of the Hubeny-Rangamani-Takayanagi (HRT) surface, the holographic dual of entanglement entropy for a region in the boundary CFT, is constrained to a small window when prepared by the gravitational path integral. The study of those fixed-area states helps understand quantum gravity beyond the leading semiclassical order. We first show that by decomposing a general holographic state into fixed-area states, an important subleading correction appears to the entanglement entropy near phase transitions. Then we explore the intrinsic spacetime geometries of fixed-area states under Lorentz-signature time evolution.In Part II, we study saddle-point geometries of the real-time gravitational path integral, in the context of computing holographic Renyi entropies. Unlike their Euclidean counterparts, these real-time saddles necessarily have complex metrics, giving an example where the saddle point is off the original contour of integration. We first present the formalism of this setup, illustrating the relevant variational problem, and features of the complex saddles. Then we demonstrate explicitly the structure of those saddles by showing examples in low dimensions by direct calculation. We also find that it is possible to deform the original integration contour to pass through saddles of this kind constructed in two-dimensional Jackiw-Teitelboim gravity. Finally, we show that the existence of these saddles results in a consequence which is necessary for unitarity to hold in quantum gravity.In Part III, we take a step towards explaining the origin of gravitational entropies, by utilizing the mathematical tool of von Neumann algebras. In particular, we give an explanation of the HRT formula purely from the bulk perspective, without making any reference to holography. This is done by constructing Hilbert spaces and von Neumann algebras from boundary conditions of the gravitational path integral with several natural axioms. The von Neumann algebras we find from this construction allows us to define a notion of entropy, which matches the HRT formula in the semiclassical limit. One of the axioms we assume which is crucial for the construction of von Neumann algebras -- the trace inequality, is proven in the semiclassical limit, and it leads to novel positivity conjectures for the gravitational action.

The topological properties of electronic band structures are closely related to the degree of localization possible for the associated wavefunctions. In this dissertation, we investigate certain aspects of this interplay between topology and localization in the context of static as well as driven (Floquet) topological insulators. The first part of this dissertation is motivated by Landau levels, the energy levels of electrons in a two dimensional plane that are subject to a perpendicular magnetic field. Landau levels form a key element of theoretical models of the quantum Hall effect, which inspired the study of topological insulators. Each Landau level is highly degenerate or flat, and is topologically non-trivial. Motivated by Landau levels, we study the topological properties of tight-binding Hamiltonians which only have flat energy levels. We find that the spectral projectors of such Hamiltonians are strictly local. In chapters 2 and 3, we show that in one dimension, compact Wannier functions (and their analogs in the absence of lattice translational invariance) can be constructed if and only if the subspace they span is described by a strictly local projector. Using this insight, in Chapter 4, we present and prove a no-go theorem which says that if a strictly local tight-binding Hamiltonian in two dimensions only has flat bands, then each of the bands must have a Chern number of zero. All results are proven without the requirement of lattice translational invariance. The role of an inequality relating the number of energies of the Hamiltonian and the system size is also clarified.In the second part of this dissertation (Chapter 5), we present some results concerning a delocalization transition that arises in a certain class of Floquet topological insulators.Specifically, we study chiral Floquet topological insulators in one dimension, and show that the localization lengths of eigenstates of the time evolution operator diverge with a universal exponent of two as the time approachs a special point in drive.

High energy physics, like many other scientific disciplines, has entered an exciting new era of big data, where both particle accelerators at the \emph{energy frontier} and astrophysical surveys at the \emph{cosmic frontier} are producing an enormous amount of data which may hold the very key to the most fundamental questions about nature. Mining such gold inevitably calls for revolutionary designs of ever more powerful and efficient statistical analysis frameworks, while at the same time scientific rigorousness places an additional requirement on the interpretability of any novel model proposed. Among a plethora of available modern machine learning techniques, the theory of \emph{optimal transport} stands out as a distinct approach that is both high performing and mathematically well grounded. By equipping the space of data represented as distributions with a suitable metric, optimal transport replaces \emph{ad hoc} notions of similarity with a well-defined distance, opening up a range of new applications with profound theoretical implications. This thesis introduces the theory of optimal transport with an eye towards its usage in physics. Special emphasis is put on two particular optimal transport distances which enjoy unique geometric properties. Utilizing their geometric structure, we develop a computationally efficient linearization framework for the two distances and highlight their approximations for discrete distributions encountered in practice. We then showcase the power of this linearized optimal transport framework by applying it to two use cases---one in collider physics at the energy frontier and the other in dark matter astrophysics at the cosmic frontier. As the adoption of optimal transport in high energy physics is still in its early stage, the present thesis invites the readers to think of other potential applications for their own research.

In this dissertation we describe several recent advancements in the study of Coon amplitudes.In the first chapter, we detail the properties of the Veneziano, Virasoro, and Coon amplitudes. These tree-level four-point scattering amplitudes may be written as infinite products with an infinite sequence of simple poles. Our approach for the Coon amplitude uses the mathematical theory of q-analysis. We interpret the Coon amplitude as a q-deformation of the Veneziano amplitude for all q ≥ 0 and discover a new transcendental structure in its low-energy expansion. We show that there is no analogous q-deformation of the Virasoro amplitude.In the second chapter, we analyze so-called generalized Veneziano and generalized Virasoro amplitudes. Under some physical assumptions, we find that their spectra must satisfy an over-determined set of non-linear recursion relations. The recursion relation for the generalized Veneziano amplitudes can be solved analytically and yields a two-parameter family which includes the Veneziano amplitude, the one-parameter family of Coon amplitudes, and a larger two-parameter family of amplitudes with an infinite tower of spins at each mass level. In the generalized Virasoro case, the only consistent solution is the string spectrum.

We study the obstacles to constructing metastable de Sitter space in string theory. We explain that it is very difficult to find stationary points for which both the string coupling is small and compactification radii are large even allowing the possibility of arbitrarily large fluxes, and a set of small perturbations of any would-be metastable de Sitter state, classically, will evolve to uncontrollable singularities. We study the Transplanckian Censorship Conjecture and show a conflict between the TCCand conventional conjectures about the string landscape. We calculate, as a function of the primordial black holes mass and initial abundance, the combination of dark matter particle masses and number of effective dark degrees of freedom leading to the right abundance of dark matter today, whether or not evaporation stops around the Planck scale.

The investigation of microscopic nonequilibrium thermodynamic systems is a wide a varied field of study. While hard to pin down with one particular frame of reference, the technological importance of understanding what happens when small systems are driven out of equilibrium is undeniable. While it is theoretically possible to do most control processes without driving a system very far from an equilibrium state, it generally takes an inordinate amount of time to do so. Computing itself is the process of preserving, transforming and translating a physical system’s states though various nonequlibrium procedures in finite time; thus, the mechanical computers that we all rely on are, at a fundamental level, nanoscale nonequilibrium thermal systems. In the following, various properties of nonequilibrium systems are discussed with an eye towards useful operations in computing. In chapter 1 the issue is tackled from a historical perspective; we see that controlling information can have a cost even using only equilibrium considerations. Next, chapter 2 moves away from purely equilibrium considerations by considering the costs that come from operating in finite time. Chapter 3 reviews a suite of relatively recent equalities that extend arbitrarily far outside the regime of equilibrium, as well as introducing novel equalities and applications for these new results. Chapter 4 investigates the applicability and scope of the new results, and in doing so, reveals the connection between a class of highly nonequlibrium processes and the precision of currents within them. Finally, in chapter 5, this class of protocol is leveraged to design highly efficient devices that are capable of universal computation that operate on similar timescales of todays state of the art machines, but hold the promise of being 4 or 5 orders of magnitude more efficient energetically.

While work in the past decade has strongly cemented the connection between quantuminformation and quantum gravity, much of the research studies the connection from theperspective of applying quantum information tools to quantum gravity. In this dissertation,we aim to study the connection in both directions: not only quantum information forgravity, but how gravity may offer a window into otherwise intractable problems in quantuminformation theory. We begin by exploring the properties of thermal quantum error-correctioncodes, derived from the intersection of quantum information and many-body physics, inAdS/CFT, where the thermodynamic properties of black holes can be more tractable to studythan many-body systems. We then propose a multipartite generalization and prove severalproperties of reflected entropy, an entanglement measure derived from studying semiclassicalgeometries, where again, the holographic duality of AdS/CFT enables tractable calculations.Next, we further expand on this program by demonstrating that, under a reasonable set ofassumptions, holographic states saturate a large class of entanglement measures that are oftenintractable to compute, such as squashed entanglement. AdS/CFT may therefore provide auseful arena by which to study various quantum information-theoretic quantities that maybe too complicated to understand in generic quantum systems. We then turn our focusto generalizing graph-theoretic techniques for studying entanglement in quantum gravityto quantum information, more broadly. We do this in a systematic fashion, in increasingorder of generality (graphs to hypergraphs to topological knots), showing that they capturebroader swathes of quantum entropies. We complete this line of study with a mathematicalinvestigation of random tensor networks with non-trivial link states, in which we take acomputational tool common in condensed matter and quantum gravity, and generalize thetool to capture entanglement properties for a broader class of quantum states.

This work furthers the program of understanding the black hole information paradox, the firewall paradox, and the relevance of concepts from quantum information to quantum gravity. A gravitational calculation was recently shown to yield the Page curve for the entropy of Hawking radiation. In this work, we generalize the Ryu-Takayanagi (Quantum Extremal Surface) prescription for von Neumann entropies to various settings relevant for this calculation. We examine a puzzle in the calculation, which we call the state paradox: that the calculation makes use of Hawking's result that the radiation entropy becomes large at late times. The paradox is resolved if the gravitational path integral computes averaged quantities in a suitable ensemble of unitary theories. We apply the insights from these calculations to cosmology by searching for entanglement islands in cosmologies with spatial curvature containing thermal radiation purified by a reference spacetime. We show arbitrarily small positive curvature guarantees that the entire universe is an island. Proper subsets of the time-symmetric slice of a closed or open universe can be islands, but only if the cosmological constant is negative and sufficiently large in magnitude. We extend this analysis to cosmologies containing a general fluid, finding that flat universes with time-symmetric slices always have islands on that slice. We find islands in the Simple Harmonic Universe model, which has no classical singularity at the background level. We then argue that the Page curve calculations do not resolve the firewall paradox. We exhibit a sharpened version of the paradox which consists of the Hayden-Preskill thought experiment in which the message thrown into the black hole is itself a smaller black hole. We argue that accounting for decoherence resolves the paradox. Finally, we conclude with an argument that quantum complexity theory can quantify the difficulty of distinguishing eigenstates obeying the Eigenstate Thermalization Hypothesis (ETH).

Density functional theory (DFT), combined with standard exchange-correlation approximations, is a usefully accurate and efficient tool to generate computational predictions in chemistry and material sciences. In the past decade, machine learning has been used extensively to build density functional approximations that concur with human-defined standards. This thesis details the effort to construct and characterize exchange-correlation approximations in DFT with physics-informed machine learning. The Kohn-Sham regularizer (KSR) is a differentiable approach for making machine-learned density functionals. It allows approximating the exchange-correlation functional while self-consistently solving the Kohn-Sham equations. It was initially formulated to generate accurate predictions for strongly-correlated molecules. Here I discuss a spin-adapted extension of the KSR that machine-learns the exchange-correlation energy densities as a functional of the spin densities and substantially improves generalizability for weakly correlated molecules. With a neural network approximation that accounts for nonlocal interactions, a training set of just five atoms and ions in 1D can predict the ground-state properties of several molecules with near chemical accuracy. The differentiability of spin-adapted KSR ensures a fast convergence during training and yields accurate predictions of the exchange-correlation potentials and other properties, often complying with known exact behaviors. While this serves as a proof of concept for what machine learning can achieve, such methods, in principle, can add to the complexity of the existing diverse approaches for designing exchange-correlation approximations, further deluding the existence of a unified scheme for systematic improvement of density functionals. On the other hand, machine learning, especially unsupervised learning algorithms, can help categorize different exchange-correlation approximations without introducing human bias or considering any absolute errors. To answer the question of how several exchange-correlation functionals are similar or different from each other, we propose a novel approach to group these functionals based on statistical learning tools. This approach does not use any exact information, accounts for density-driven differences in approximations based on the theory of density-corrected DFT, and avoids any form of bias between empirical, partially-empirical, and non-empirical approximations. It sorts exchange-correlation functionals based on similarities predicted using a novel, parameter-free unsupervised clustering algorithm. For 33 popular exchange-correlation functionals and the MGAE109 dataset, this scheme generates categories of functionals that somewhat mimic the popular Jacob’s ladder categorization while depicting that Minnesota functionals of recent vintage might have strayed far from the path of typical functional development.

This thesis concerns the construction and analysis of a new family of exact general relativistic shock waves. The construction resolves the open problem of determining the expanding waves created behind a shock-wave explosion into a static isothermal spacetime with an inverse square density and pressure profile. The construction involves matching two self-similar families of solutions to the perfect fluid Einstein field equations across a spherical shock surface. The matching is accomplished in Schwarzschild coordinates where the shock waves appear one derivative less regular than they actually are. Separately, both families contain singularities, but as matched shock-wave solutions, they are singularity free. There was no guarantee ahead of time that the matching of the two families could be achieved within the regions where both families are nonsingular. Indeed, for pure radiation equations of state, the matching occurs very near the singular point of the interior expanding wave, and this makes the analysis quite delicate, both numerically and formally. It is for this reason the construction is accompanied by a novel existence proof in the pure radiation case. The analysis is extended to demonstrate Lax stability in the pure radiation case and provide a criterion for stability in all other cases. These shock-wave solutions represent an intriguing new mechanism in General Relativity for exhibiting accelerations in perturbed Friedmann spacetimes, analogous to the accelerations modelled by the cosmological constant in the Standard Model of Cosmology. However, unlike in the Standard Model of Cosmology, these shock-wave solutions solve the Einstein field equations in the absence of a cosmological constant, opening up the question of whether a purely mathematical mechanism could account for the cosmic acceleration observed today, rather than dark energy.

Natural systems are regularly "far from equilibrium:" transient, dynamical, and in constant interaction with their environments. These systems are often complex, comprising many mutually-interacting parts and giving rise to emergent behavior at observable scales—the human body, forest ecosystems, and the planet's climate are but a few pertinent examples. In cases like these, traditional techniques for analysis fall short: we cannot feasibly deduce the observed behavior either from individual, microscopic degrees of freedom (writing a system Hamiltonian, for example) nor from macroscopic statistical treatments as in equilibrium thermodynamics. How can we understand this class of systems? How do they absorb and dissipate energy; how do they store, process, and produce information; broadly, how do they function?This dissertation provides a suite of mathematical and computational tools for probing these and related questions. Specifically: it extends ideas from stochastic thermodynamics to develop a so-called trajectory class fluctuation theorem for nonequilibrium complex systems, freed from many previously restrictive assumptions; derives a stochastic dynamical "first law" linking nonequilibrium behavior to information transduction; and applies these theoretical results to a collection of example systems: the ion channels of mammalian neural membranes and autonomous Maxwellian ratchets.The contributions are organized as follows: Ch. 2 introduces key ideas from the languages of information theory, stochastic processes, and computational mechanics. Ch. 3 then continues by introducing fluctuation theorems and the core language of stochastic thermodynamics, before extending it with a nonequilibrium steady-state trajectory class fluctuation theorem and applying the theory to uncover surprising implications for the ion channels of mammalian neural membranes. Ch. 4 then combines thermodynamic and information-processing paradigms to derive generalized information processing first and second laws, applying them to the limits on how autonomous Maxwellian ratchets can leverage information to perform useful work. Finally, Ch. 5 closes and suggests rich ongoing and future lines of inquiry.

In this dissertation, we discuss half-BPS solutions of gauged supergravity that are holographic realizations of conformal line defects and interfaces. These solutions are constructed by taking a suitable ansatz for the geometry, consisting of a warped product of an AdS spacetime and a sphere, if necessary, over a line, and solving the supersymmetry variations. Quantities such as one-point functions in the presence of the defect and the entanglement entropy are calculated holographically.In chapter 1, we review the AdS/CFT correspondence and holography. In chapter 2, we relate two different formulations of AdS6 solutions in type IIB supergravity. In chapter 3, we construct solutions in six-dimensional F(4) gauged supergravity that are dual to line defects. In chapter 4, we construct solutions in four-dimensional N = 2 gauged supergravity that are dual to line defects, obtained by a double-analytic continuation of BPS black hole solutions. In chapter 5, we construct Janus solutions in three-dimensional N = 8 gauged supergravity that are dual to interface CFTs.

A geometric formulation of scattering amplitudes in perturbative quantum field theory has surfaced in recent years where, in place of the summations of all off-shell Feynman diagrams for an amplitude, a single geometric object in the kinematic space of on-shell scattering data completely dictates the final answer. By far the most sophisticated example of this new formulation is the \emph{amplituhedron} geometry for the planar $\mathcal N=4$ super Yang-Mills theory, whose boundary structure encodes all singularities of any amplitude at any loop-order. While an amplituhedron describes an amplitude (prior to loop integration) as a differential form with logarithmic singularities on its boundary, a \emph{dual amplituhedron} whose \emph{bulk volume} directly reproduces the same amplitude is much desired. In chapter two of this thesis, we provide nontrivial evidence for the dual amplituhedron from the perspective of \emph{local triangulation}. We classify all the elementary geometric spaces relevant for one-loop Feynman integrands which allow for further identification of geometrical spaces associated with the chiral pentagon-integrands in the local expansion of one-loop MHV amplitudes. Using projective duality maps on certain two-dimensional boundaries where the geometry reduce to polygons, we show that the real purpose of life of chiral pentagons is to triangulate the putative dual amplituhedron.The discovery of amplituhedron originated from a reformulation of amplitudes at the level of the \emph{integrand} --the unintegrated sum of Feynman diagrams, in terms of \emph{positive Grassmannians} and \emph{on-shell diagrams}. Extending the geometric formulation to more general theories after the example of $\mathcal N=4$ sYM requires both a wealth of theoretical data from explicit computations of loop integrands and further understanding of the connection between on-shell diagrams and Grassmannian geometry without planarity.In chapter three, we discuss one-loop \emph{integrand} construction for planar Yang-Mills amplitudes with $1\leq \mathcal N < 4$ supersymmetry based on a \emph{generalized unitarity} method. We trace the source of ambiguity in defining ``the loop integrand'' for non-maximally supersymmetric amplitudes to the ill-defined non-singlet massless “bubble-cuts”, and pinpoint two prescriptions for \emph{defining} the loop integrand, paving the way for future exploration of amplituhedron-like geometric objects for less supersymmetric Yang-Mills theories.In chapter four, we go beyond the planar sector of $\mathcal N=4$ sYM, exploring the geometry of non-planar on-shell diagrams --- essential building blocks of any non-planar amplitudes --- that arise in the context of BCFW recursion relations of tree-amplitudes using non-adjacent shifts of momenta. We show that individual terms in the non-adjacent BCFW expansions are associated with a particular class of non-planar positive geometries which generalize the positroid cells of the positive Grassmannian into the Grassmannian.

Amplitudes methods have been successfully applied to various physical situations. In this manuscript we present their application to two quite distinct problems: classical gravitational-wave physics and constraining gravitational theories beyond general relativity. In chapter 1 we develop two methods for simplifying a computationally intense step in amplitudes calculations, specifically the sum over the physical states of photons and gravitons. These methods, even though generic, are developed with the application of gravitational-wave physics in mind. In chapters 2 and 3 we compute novel corrections to the Newtonian gravitational potential experienced by two spinning objects. These corrections originate in general-relativity effects. In chapter 4 we obtain bounds on Wilson coefficients of four-dimensional gravitational effective field theories. By calculating explicit examples of consistent gravitational theories and extracting the corresponding Wilson coefficients, we find that they occupy tiny islands in the Wilson-coefficient space derived. In chapter 5 we extend this analysis to dimensions higher than four. We again construct explicit gravitational theories and observe that the corresponding Wilson coefficients land in similar regions.

In this dissertation, we study nonlocal quantum field theories obtained by delocalizing the fields in the interaction terms of some local Lagrangian. The delocalization kernels in momentum space are exponentials of entire functions, thus no additional mass poles are introduced, but they ensure UV finiteness of loop integrals of Feynman diagrams. The transition matrix is defined in the Euclidean regime, followed by analytic continuation of the external momenta. Because of the delocalization kernels, manipulations valid in local theories are prohibited in the nonlocal theories, including Wick rotation and dispersion relation. We study the analyticity structure of the transition matrix of such theories at the perturbative level, using the Landau equations. The result shows that singularities that occur in local theories are also present in the nonlocal theories, except for singularities of the second type. The Cutkosky rule is then derived using Cutkosky's original argument, and the unitarity condition follows as an application of the rule. We then quantify the acausality present in such nonlocal theories using the Bogoliubov causality condition (BCC). First we study the acausality of the effective propagator, which is shown to be exponentially suppressed for large space-time separation. Mellin transform is introduced to help find the asymptotic behavior of acausality. Next we prove that in local theories, for each diagram, the BCC equation amounts to a chain of retarded propagator between two vertices; while in nonlocal theories, the chain consists of $\tilde\Delta_R$, which is the retarded propagator delocalized by the nonlocal kernel and is therefore no longer causal. The acausal amplitudes are then integrated over wave packets to make experimental predictions. We show that for a nonlocality scale of order Plank's length, the acausality will be experimentally immeasurable. Along the way, by extending the BCC to unitarity equations, we show that the theory defined directly in Minkowksi space is not unitary. This reaffirms that the theory has to be defined in the Euclidean regime and then analytically continued.

A standard insight of the AdS/CFT correspondence is that some aspects of the geometry of a bulk state are encoded in the entanglement structure of its dual boundary state. As entanglement is not a linear quantum observable, this means that geometry in a quantum theory of gravity should likewise not be a linear observable.This is seemingly in tension with an understanding of quantum gravity as a gravity path integral summing over geometries, which inherently treats geometry as a linear observable. In a recent work the authors Marolf and Maxfield explore this tension in the context of a simple model of a gravity path integral where it is resolved by the appearance of null states, or in other words geometries that are equal to superpositions of other geometries. In Chapter 1 of this dissertation, we extend their 2d topological gravity model to have as its bulk action any open/closed TQFT obeying Atiyah's axioms. We describe the Hilbert spaces of these more general theories that remain after null states have been accounted for. The holographic duals of these topological gravity models are ensembles of 1d topological theories with random dimension. These holographic interpretations of our gravity models require projecting out negative-norm states from the baby universe Hilbert space, which Marolf and Maxfield achieved by adding a non-local boundary term to the bulk action. We describe the analogous solution in the framework of a TQFT with defects by coupling the boundaries of the gravity models to auxiliary 2d TQFTs in a non-gravitational (i.e. fixed topology) region. The gravity model is then holographically dual to an ensemble of boundary conditions in an open/closed TQFT without gravity.In Chapter 2 we explore linear dependencies between certain states with simple geometric duals: states made up of n copies of a thermofield double state and the states obtained from this one by permuting the n right hand sides. We derive expressions for the maximum fidelity between one such state and a linear combination of the others, and see that this fidelity approaches 1 as the number n of black holes increases. We also consider the possibility of obtaining a single thermofield double state as the partial trace of linear combinations of such states with topologies with no connection between the two untraced sides. We derive lower bounds for the fidelity between the thermofield double state and such partial traces and comment on the conceptual implications of the existence of such states.

In this disseration we explore the structure and construction of certain quantum many-body states, as well as operator dynamics in quantum many-body systems. First, we detail the implementation of a new tensor network ansatz for many-body ground states based on adiabatically evolving a state to repeatedly double in size. Next, we extend the notion of `strong zero modes, ' edge-localized operators stable at infinite temperature, to operators localized to boundaries between systems. We then discuss efforts to numerically characterize scrambling, a strong form of thermalization, in the Sachdev--Ye--Kitaev model. Finally, we show that a states with large momentum splittings can be created via symmetric Bloch oscillations in two optical lattices being accelerated in opposite directions.

Optimization is a natural language in which to express a multitude of problems from all reaches of the mathematical sciences. This is a dissertation in three parts, detailing work on three different problems in optimization.We begin in stochastic thermodynamics, with the problem of computing optimal operating protocols for Brownian machines by minimizing their dissipation on average. We develop a perturbative solution to the Fokker-Planck equation for one-dimensional driven Brownian motion in the limit of slow driving. Calculating the average dissipation in this formalism, we demonstrate the existence of an emergent Riemannian geometric structure in the space of control parameters, whereby the $n^{th}$ order term in the average dissipation contains a tensor of order $n$. We rigorously derive an exact formula for the order-$2$ tensor (the metric) and show that up to second-order terms in the perturbation theory, optimal dissipation-minimizing driving protocols minimize the length defined by this metric.Next, we move to the world of machine learning, and study the generalization properties of models trained with first-order optimizers on whitened data, and with exact second-order optimizers. For a general class of models trained with first-order optimizers, we find that the mutual information between the trained model and the dataset is strictly smaller when the data is whitened than when it isn't. This reduction of access to information relevant for test performance negatively impacts generalization in a dimension-dependent manner, worsening for high dimensional problems. Exploiting analogies between training linear models and wide neural networks with a first-order optimizer on whitened data and with an exact second-order optimizer, we also establish the existence of a generalization gap between first- and exact second-order optimizers in these model classes. Experiments demonstrate that our conclusions hold in a larger class of models than is supported by the theoretical results, but that regularizing a second-order optimizer appropriately can compensate for the reduction in generalization performance observed in the exact setting.Finally, we study the problem of setting step sizes for discrete-time gradient-based optimization. Taking the view that a discrete-time optimizer is a simulation of an underlying ordinary differential equation (ODE), we set step sizes for the former by controlling the error with which it tracks the latter. The resulting adaptive step size mechanism is highly efficient, requiring just one extra gradient call over all iterations, and applicable to any optimizer with a well-defined continuous-time representation as an ODE. We find empirically that this step size routine can achieve oracle lower bounds on strongly convex functions for first- and second-order optimizers, and that it can also perform competitively with the state of the art on a nonconvex problem, principal components analysis.

Solitons are a class of solution to equations that appear in the study of physics, particularly the study of fields and field theory. In the study of quantum field theory, solitons often present themselves as heavy degrees of freedom, and are therefore approached from a semi-classical perspective. Traditionally, the soliton is considered as the background field with which perturbative quantum states are defined with respect to. In this dissertation, we present the position that this approach is flawed. In particular, we suggest that using classical solitons as a starting point of analysis only narrowly covers the scope of the possibilities for solitons to manifest in quantum field theories. We show that there are classical solitons of infinite mass which in the quantum theory have their mass rendered finite. We additionally describe circumstances where quantum effects are large enough to significantly alter the net forces which act upon solitons, compared to the classical case. The conclusion is simple: quantum field theory generically contains a much larger space of dynamically accessible field configurations than classical field theory, and this facilitates the existence of quantum solitons that would not exist in the classical case.

The Standard Model (SM) of particle physics is one of the most complete mathematical models of physical phenomena to date. Even so, it cannot explain experimental results like the existence of particle dark matter and the fact that neutrino masses are non-zero. Explaining such results will necessitate developing a beyond the SM (BSM) theoretical description of particle physics. What form this BSM physics will take has become increasingly unclear; many elegant theories which were expected to appear in recent experiments have not emerged. Thus, we find ourselves at a cross-roads, in need of new perspectives and new computational frameworks to push our theoretical description of physics forward.New perspectives will come from challenging previously-held assumptions in the pursuit of fundamentally new descriptions, but challenging such assumptions often presents practical computational challenges. Therefore, these new perspectives must also be accompanied by new computational frameworks. Computational frameworks can come in many forms, from the purely mathematical to the largely numerical. In particular, in recent years machine learning (ML) has become an increasingly accessible and powerful computational tool for scientific applications. Crafting novel BSM theories will require us to investigate and embrace the full spectrum of computational frameworks. Additionally, one of the best ways to spark new insights is to closely collaborate with and draw inspiration from other fields, such as mathematics and computer science.In this thesis, we present two examples of how advanced computational frameworks can be used to aid in investigating new physics perspectives. In one example, we see how the purely mathematical framework of optimal transport (OT) theory can be used in tandem with advanced ML methods to enable a new perspective on particle physics simulations. The result is a novel strategy which lays the foundations for a completely data-driven, end-to-end simulation of particle collisions at the Large Hadron Collider. In a second example, we see work which considers a new perspective on what the history of our universe might have looked like. In particular, we consider how the abundance of a WIMP dark matter candidate could be altered by considering a phase of electroweak force confinement early in the universe. Considering this model while making relatively few assumptions was aided by the application of advanced numerical computational tools.We begin with both high-level and technical background on the topics relevant to these works. We conclude by discussing future directions for these works, as well as briefly giving general thoughts on strategies for applying the computational framework of ML to problems in theoretical particle physics more broadly.

This dissertation is devoted to the theoretical study of strongly correlated quantum many-body systems. The central theme is to understand the universal properties of quantum matter from the perspective of renormalization group (RG) fixed points. The guiding principles are provided by symmetries and ’t Hooft anomalies, which are both preserved under RG flow and serve to constrain physical properties. The main body of this dissertation can be divided into four main parts.The first part concerns boundary critical phenomena associated with symmetry-protected topological (SPT) phases and unconventional quantum phase transitions. Our results include new stable boundary phases and exotic boundary phase transitions in various dimensions. For example, a continuous Neel-VBS transition can be potentially realized at the 1+1D boundary of a 2+1D SPT state protected by SO(3) symmetry.The second part is about two strongly correlated Moire materials. (1) The band topology in twisted bilayer graphene (TBG) severely complicates the standard lattice model descriptions. Therefore, we seek an alternative and provide a coupled-wire framework describing the correlated physics in TBG. (2) As for the experimentally observed continuous metal-insulator transition in MoTe2/WSe2 heterobilayer, we provide a theoretical proposal involving charge fractionalization, which potentially explains the observed anomalously large critical resistivity.The third part concerns exotic metallic states beyond Landau Fermi liquid theory. We study the problem mainly from two approaches. (1) In the perturbative RG approach, we show analytically controlled examples of marginal Fermi liquids involving non-Landau quantum critical points. In addition, we show charge fractionalization naturally leads to the bad metal behavior at low temperatures. (2) The other approach is based on exactly solvable toy models for quantum matter without quasiparticles. We construct a square-lattice model for the strange metal phase and generalize it for non-Fermi liquids with tunable transport scaling.The fourth part is about generalized symmetries and their ’t Hooft anomalies. (1) We illustrate how to unambiguously characterize generalized symmetries (including higher-form symmetries, categorical symmetries, and subsystem symmetries) at quantum phase transitions. (2) We discuss physical constructions and classifications of SPT states involving higher-form symmetries. Special attention is paid to anomaly constraints for condensed matter systems such as quantum dimer models.

Building off senior and master thesis work, we present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, $n_t$ and $n_p$, respectively. The case of $n_t=1$ and $n_p=1$ corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked precisely once, and form the surfaces of nested tori. We show that for $n_t \in \mathbb{Z}^+$ and $n_p=1$, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity and estimate the soliton lifetime in such a medium as a function of the toroidal winding number. Torus knot topology is also inherent in electromagnetic and gravitational radiation. We show this by constructing spin-$N$ fields based on the elementary states of twistor theory. The twistor functions corresponding to the elementary states admit a parameterization in terms of the torus knots' poloidal and toroidal winding numbers, allowing one to choose the degree of linking or knotting of the associated field configuration. We describe the topology of the gravitational fields and their physical interpretation in terms of their tidal and frame drag phenomenology. Using the gravito-electromagnetic formalism, we show that the torus knot structure is exhibited in the tendex and vortex lines for the analogous linearized gravitational solutions. We extend the definition of hopfions to include a larger class of spin-$h$ fields and use this to classify the electromagnetic and gravitational hopfions of different algebraic types. The fields are constructed through the Penrose contour integral transform; thus, the singularities of the generating functions are directly related to the geometry of the resulting physical fields. We discuss this relationship and how the topological structure of the fields is related to the Robinson congruence. Since the topology appears in the lines of force for both electromagnetism and gravity, the gravito-electromagnetic formalism is used to analyze the gravitational hopfions and describe the time evolution of their tendex and vortex lines. We thus obtain similar configurations based on the same topological structure but varying spin-$h$ algebraic types. The null and type N fields propagate at the speed of light, while the non-null and type D fields radiate energy outward from the center. Finally, we discuss the type III gravitational hopfion, which has no direct electromagnetic analog, but find that it still exhibits some characteristic features common to the other hopfion fields.In the complete non-linear theory we present an exact solution of Einstein's equation that describes the gravitational shockwave of a massless particle on the horizon of a Kerr-Newman black hole—generalizing the Dray-'t~Hooft solution to the case of a rotating background. The back-reacted metric is of the generalized Kerr-Schild form and is Type II in the Petrov classification. We show that if the background tetrad is aligned with shear-free null geodesics, and the background Ricci tensor satisfies a simple condition, all nonlinearities in the perturbation will drop out of the curvature scalars. We reformulate Einstein-Hilbert gravity in the first-order formalism of Elie Cartan and find that the Riemann curvature tensor is the Yang-Mills gauge curvature of a local Lorentz group gauge connection. We derive and make heavy use of the method of spin coefficients (the Newman-Penrose formalism) in its compacted form (the Geroch-Held-Penrose formalism).