Constant of Motion Identifying Excited-State Quantum Phases

We propose that a broad class of excited-state quantum phase transitions (ESQPTs) gives rise to two different phases. These phases are identified by means an operator, $\hat{\mathcal{C}}$, which is constant motion only in one them. Hence, the ESQPT critical energy splits spectrum into where equilibirium expectation values physical observables crucially depend on this motion, and another relevant thermodynamic magnitude. The trademark feature operator it has eigenvalues, $\pm1$, therefore acts as discrete symmetry first these This scenario observed systems with without additional symmetry; case, $\hat{\mathcal{C}}$ explains change from degenerate doublets non-degenerate eigenlevels upon crossing line. present stringent numerical evidence Rabi Dicke models, suggesting result exact limit, finite-size corrections decrease power-law.