Analytic Theory of Phase Transitions in Optical Metamaterials

Optical metamaterials provide a versatile platform for engineering homogeneous electromagnetic media whose distinct phases are characterized by phase diagrams in constitutive-parameter space. However, existing studies of hyperbolicity, topological properties, and exceptional-point formation often rely on highly symmetric models or case-by-case numerical parameter scans, leaving a unified analytic framework that identifies phases and phase transitions directly from the constitutive tensors lacking. Here, we develop a general theory that yields exact analytic criteria for topological transitions, exceptional-point transitions, pinch-off Lifshitz transitions, and optical Lifshitz transitions in homogeneous media. Applying this framework to a tractable example of a gyroelectric medium with anisotropic chirality, we uncover exceptional rings and negative refraction induced by gyroelectric-chiral coupling. By enabling the exact determination of phase boundaries, our theory provides a predictive framework for discovering previously unexplored electromagnetic phases and offers new principles for the systematic design of optical metamaterials.