Caroline de Groot

OS Mathematical Physics December 2 16:00

(Symmetry protected) topological phases in open systems

We address the question of whether symmetry protected topological order (SPTO) exists in open systems by preserving both equivalence classes and order parameters. To this end we consider which symmetric channels map within a manifold of SPT mixed states. The first important result is proving that a strong symmetry (SS) condition on channels maps a one-dimensional SPT phase to itself and equivalently maps a string order parameter's pattern of zeros back to the same pattern of zeros at finite time. This is a sufficient condition for preserving the notion of a cocycle. This allows us to define an equivalence class of SPT mixed states through fast dissipative Lindbladian evolution (Coser and Perez-Garcia, 2019).

We crucially show that a notion of weak symmetry (WS) is too coarse-grained to preserve SPTO, which is evident in the string order parameter that is destroyed instantly with any finite decay. Asking which conditions transform between *different* SPTO leads to our second main result. Now requiring also *necessity of preservation of the cocycle* leads us to introduce twisted SS conditions which induce endomorphisms on the SPT phase and map SPT mixed states to SPT mixed states in a different phase.