Crossing exceptional points without phase transition

We show that the theoretical framework linking exceptional points (EPs) to phase transitions in parity-time (PT) symmetric Hamiltonians is incomplete. Particularly, we demonstrate that the application of the squaring operator to a Jx PT lattice dramatically alter the topology of its Riemann surface, eventually resulting in a system that can cross an EP without undergoing a symmetry breaking. We elucidate on these rather surprising results by invoking the notion of phase diagrams in higher dimensional parameter space. Within this perspective, the canonical PT symmetry breaking paradigm arises only along certainprojections of the Riemann surface in the parameter space.