Correlated degenerate systems represent a fascinating subset of quantum manybody systems where the interaction effects among particles lead to emergent phenomena beyond conventional perturbation theory. In this dissertation, we survey several such systems, including flat band systems, moat band systems, and interacting conformal field theories (CFTs), elucidating their chaotic, integrable, and topological properties. We begin by exploring the Sachdev-Ye-Kitaev (SYK) model, which captures chaotic non-Fermi liquid behavior and holographic duality properties. Leveraging a system of spinless fermionic atoms in an optical Kagome lattice with flat band spectra, we demonstrate the emergence of the SYK Hamiltonian, providing a platform for experimental exploration of its exotic behavior. Transitioning to magic-angle twisted bilayer graphene (TBG) under strong Coulomb disorder, we unveil an emergent quantum chaotic strange metal (SM) phase characterized by weakly coupled SYK bundles. We propose a finite-temperature phase diagram for TBG, discussing implications for experimental observations. In the realm of strongly interacting bosons with moat band dispersion in two dimensions, we investigate the propensity for stabilizing a chiral spin liquid (CSL) ground state. Through Monte Carlo simulations and variational analysis, we identify parametric windows favoring the uniform CSL state and provide density estimates for experimental relevance. Lastly, we delve into the low-energy properties of the one-dimensional spin-1/2 XXZ chain with time-reversal symmetry-breaking pseudo-scalar chiral interaction, unveiling a comprehensive phase diagram using thermodynamic Bethe ansatz. We identify emergent conformal field theories describing phases with distinct ground state symmetries and investigate finite-size effects around the critical transition point.