Robust Oscillations and Edge Modes in Nonunitary Floquet Systems

We explore oscillatory behaviour in a family of periodically driven spin chains which are subject to a weak measurement followed by post-selection. We discover a transition to an oscillatory phase with half the periodicity of the driving as the strength of the measurement is increased. By mapping these spin chains to free fermion models, we find that this transition is reflected in the opening of a gap in the imaginary direction. Interestingly, we find a robust, purely real, edge π-mode in the oscillatory phase. We establish a correspondence between the complex bulk spectrum and these edge modes. This transition is further found to coincide with a transition in the entanglement scaling of the steady state. These oscillations are numerically found to be stable against interactions and disorder.