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 ${\mathbb{Z}}_2^{\sigma }\times {\mathbb{Z}}_2^{\tau }$ 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.

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• Entanglement entropy
• Open quantum systems & decoherence
• Quantum phase transitions
• Spontaneous symmetry breaking
• Symmetry protected topological states
• Matrix product states