Strong-to-weak Spontaneous Symmetry Breaking from Decoherence in SPT states

Symmetry-protected topological (SPT) phases are known for their robust boundary states and hidden order, yet their stability under decoherence remains an open question. We introduce decoherence to an SPT state, which can spontaneously induce strong-to-weak spontaneous symmetry breaking (SWSSB). Single-site dephasing, which leaves trivial states featureless, drives SWSSB in topological ones, revealing how the intrinsic entanglement structure of SPT influences the response to measurement. Using density-matrix renormalization group (DMRG) and Monte Carlo simulations, we show that decoherence generically induces long-range correlations in both one- and two-dimensional SPTs, including the ZXZ cluster chain, the spin-1 Haldane chain, and the quantum spin Hall state. Our framework connects decoherence, topology, and measurement-based quantum computation by viewing quantum channels as symmetry-preserving measurements on a doubled Hilbert space. These results identify SWSSB as a universal signature of SPTs under decoherence and suggest new experimental routes to probe topological order through noise-driven correlations.