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

Recent studies have unveiled new 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 (SWSSB) order in systems with global symmetry. In this work, we propose a new class of phases, termed the double ASPT phase, which emerges from a nontrivial extension of these two orders. This new 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 $\mathbb{Z}_2^{\sigma}\times \mathbb{Z}_2^{\tau}$ symmetry. We identify universal critical behaviors along each critical line and observe the emergence of an intermediate phase that completely breaks the $\mathbb{Z}_2^{\sigma}$ 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.