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

Recent studies have unveiled possibilities for discovering intrinsic quantum phases that are unique to open systems, including phases with average symmetry-protected topological (ASPT) order and strong-to-weak spontaneous symmetry-breaking order in systems with global symmetry. In this work, we propose a class of phases, termed the double ASPT phase, which emerges from a nontrivial extension of these two orders. This phase is absent from prior studies and cannot exist in conventional closed systems. Using the recently developed imaginary-Lindbladian formalism, we explore the phase diagram of a one-dimensional open system with symmetry. We identify universal critical behaviors along each critical line and observe the emergence of an intermediate phase that completely breaks the symmetry, leading to the formation of two triple points in the phase diagram. These two triple points are topologically distinct and connected by a domain-wall decoration duality map. Our results promote the establishment of a complete classification for quantum phases in open systems with various symmetry conditions.