Quantum criticality in open quantum systems from the purification perspective

Open quantum systems host mixed-state phases that go beyond the symmetry-protected topological and spontaneous symmetry-breaking paradigms established for closed, pure-state systems. Developing a unified and physically transparent classification of such phases remains a central challenge. In this work, we introduce a purification-based framework that systematically characterizes all mixed-state phases in one-dimensional systems with $\mathbb{Z}_2^{\sigma} \times \mathbb{Z}_2^{\tau}$ symmetry. By introducing an ancillary $\kappa$ chain and employing decorated domain-wall constructions, we derive eight purified fixed-point Hamiltonians labeled by topological indices $(\mu_{\sigma\tau},\mu_{\tau\kappa},\mu_{\kappa\sigma}) \in \{\pm1\}^3$. Tracing out the ancilla recovers the full structure of mixed-state phases, including symmetric, strong-to-weak spontaneous symmetry breaking, average symmetry-protected topological phases, and their nontrivial combinations. Interpolations between the eight fixed points naturally define a three-dimensional phase diagram with a cube geometry. The edges correspond to elementary transitions associated with single topological indices, while the faces host intermediate phases arising from competing domain-wall decorations. Along the edges, we identify a class of critical behavior that connects distinct strong-to-weak symmetry-breaking patterns associated with distinct strong subgroups, highlighting a mechanism unique to mixed-state settings. Large-scale tensor-network simulations reveal a rich phase structure, including pyramid-shaped symmetry-breaking regions and a fully symmetry-broken phase at the cube center. Overall, our purification approach provides a geometrically transparent and physically complete classification of mixed-state phases, unified with a single $\mathbb{Z}_2^{\sigma} \times \mathbb{Z}_2^{\tau} \times \mathbb{Z}_2^{\kappa}$ model.