Strong-to-weak spontaneous symmetry breaking meets average symmetry-protected topological order

The authors uncover here a new quantum phase --- the double average symmetry-protected topological (ASPT) phase --- that arises uniquely in open quantum systems, where symmetry breaking and topological order coexist. Using the imaginary-time Lindbladian formalism, they establish a rich phase diagram featuring distinct gapless phases and topological triple points. This work highlights how environmental interactions can lead to novel quantum phenomena with no analog in closed systems.