Quantum many-body scars (QMBS) provide a route to weak ergodicity breaking in isolated many-body systems. Unlike conventional mechanisms for the failure of thermalization, such as integrability or many-body localization, where atypical behavior pervades the entire spectrum, QMBS systems harbor only a small, measure-zero set of nonthermal eigenstates embedded within an otherwise thermalizing bulk. Conventionally, these atypical states ($scar$ states) are distinguished by their anomalously low entanglement entropy relative to nearby thermal eigenstates. When a system is initialized in a state with significant overlap onto this scarred subspace, its subsequent dynamics evade rapid thermalization, instead producing long-lived coherent oscillations in local observables. QMBS have now been identified across a diverse range of platforms, including lattice models, Floquet systems, and various quantum simulators. Nevertheless, a systematic microscopic understanding of how scar eigenstates arise and how they can be analytically constructed remains incomplete. The central aim of this dissertation is to use interacting spin models as a concrete setting to investigate the explicit construction, algebraic organization, and entanglement structure of a hierarchy of QMBS, including states that lie well beyond the low-entanglement paradigm. The first part of this thesis studies the spin-$1$ Kitaev chain, where conserved bond parity operators fragment the Hilbert space into exponentially many dynamically disconnected sectors. Beyond the previously identified sector that maps exactly onto the paradigmatic PXP model, we show that pronounced scarred dynamics also occur in other sectors. Suitably chosen product states in these sectors display strong and long-lived fidelity revivals, which we analyze using the forward-scattering approximation, revealing how constrained Hilbert spaces give rise to robust non-ergodic dynamics. We then develop a general algebraic construction of exact zero-energy eigenstates with volume-law entanglement in a large class of spin chains with periodic boundary conditions. These states are formed by entangled dimers on antipodal sites and arise in the middle of the spectrum for models including the transverse-field Ising model, the PXP model, and spin-$S$ $XY$ and Kitaev chains. Although they exhibit thermal expectation values for all local observables, they remain atypical under suitable nonlocal few-body probes, thereby extending the notion of scar-like eigenstates decisively beyond the low-entanglement paradigm. The construction generalizes to spin models on arbitrary graphs in higher dimensions. Finally, we revisit the spin-$1$ $XY$ chain and construct new families of exact scars within its extensively degenerate zero-energy manifolds, arising from the interplay between $U(1)$ magnetization conservation and chiral symmetries. These include interference-protected Fock-space-cage-like states, volume-entangled towers, and mirror-dimer states. Using the commutant algebra framework, we show that these states are simultaneous eigenstates of non-commuting local operators, providing a unifying algebraic organizing principle and systematic routes to identifying and classifying QMBS in generic many-body systems.

We introduce a matrix model for SU(2)-gauge theory, and introduce its symmetries. We construct its gauge invariant partition function by projecting out colored states from the path integral, and simulate the theory on a lattice. We observe that the theory has two minima, one of which is metastable. We find phase transitions, and draw the phase diagram.

Amorphous solids, unlike crystals, lack long-range order, and their mechanical stability arises from disordered networks of force-balanced constituents and quenched internal stresses. In this talk, I will discuss how these internal stresses influence the mechanical response of athermal jammed solids. I will introduce a stress-based theoretical framework for describing elasticity in systems without a well-defined stress-free reference state, in which mechanical equilibrium takes the form of Gauss's law in a rank-2 tensor theory with vector charges, with internal stresses playing a role analogous to polarization in a dielectric medium. I will then discuss how incorporating a characteristic microscopic length scale leads to a scale-dependent response, capturing the crossover from continuum behavior at long wavelengths to the suppression of stress fluctuations at short distances. Finally, I will discuss the response to external forces and the behavior of the emergent elasticity as the system approaches the unjamming transition.

A common envelope (CE) event occurs when a giant star, consisting of a dense core and bloated envelope, engulfs a much smaller binary companion. This leads to the core and companion inspiralling inside a shared envelope, to which they transfer orbital energy and angular momentum via gravitational drag (gas dynamical friction). This can result in a merger between the core and companion, which may produce a luminous red nova, or the ejection of the envelope, resulting in a stable short-period `post-CE' binary. Global three-dimensional hydrodynamical simulations can be used to better understand the key physical processes that govern CE evolution. We are currently focusing on evaluating the applicability of various drag force models from the literature by comparing their predictions with the drag force measured from the simulations. We find that certain analytic drag models can reproduce the simulation results with remarkable accuracy. Due to their much lower computational cost, one-dimensional spherically symmetric simulations may be better suited for certain science goals, but should reliably reproduce key outcomes of the 3D simulations. I will discuss our ongoing effort to develop global 1D hydrodynamical CE simulations, which involves deploying the analytic models mentioned above in the 1D simulations and comparing 1D and 3D simulations at each stage of model development.