Wigner Cat Phases: A finely tunable system for exploring the transition to quantum chaos

A transition to quantum chaos is characterized by a finely tunable mixed random matrix ensemble. The {\it mixed Gaussian Orthogonal Ensemble (mGOE)} offers a pedagogically accessible framework for simulating {\it Many-Body Localization (MBL)} transitions. It can be tuned from chaotic to localized and heavy-tailed phases in a continuous fashion, providing an opportunity to explore new phases. We numerically study how the spectral properties of mGOE evolve across these transitions. Characterization of the transition to quantum chaos is computed and analyzed via empirical spectral density, nearest-neighbor spacing, and adjacent gap, with statistical uncertainty quantification that strengthens the robustness of the evidence for transitions. The transitions are identified as the {\it Wigner Cat Phases}, due to the shape of empirical spectral density, which depend on the tunable parameter. These simulated phases in mGOE provide a valuable framework to study the {\it Eigenstate Thermalization Hypothesis (ETH)} and its related transitions, representing a family of systems with tunable localization and disorder.