From Topological Order to Mixed-State Phases: A Ground-State Probe of Fractionalized Excitations

How do we detect topological phases from a single ground state? Entanglement entropy and spectrum have long been the standard tools – but the reduced density matrix (RDM) itself contains far more information. We show that the RDM of a 2D topologically ordered system, expressed at the entanglement cut, realizes a 1D mixed-state phase. For the ${\mathbb{Z}}_2$ toric code phase, it is a 1D ${\mathbb{Z}}_2$ strong-to-weak spontaneous symmetry breaking (SW-SSB) phase, where deconfinement of anyons manifests as the short-range correlation of both ${\mathbb{Z}}_2$ charge and ${\mathbb{Z}}_2$ domain-wall in the RDM. The bulk $e$-$m$ duality translates into a Kramers–Wannier self-duality of the SW-SSB phase. Extending the framework to gapped ${\mathbb{Z}}_2$ spin liquids, the global spin-rotation symmetry manifests as an additional weak symmetry for the 1D RDM. Spin-$\frac{1}{2}$ spinons result in a cusp on the disorder parameter of spin-rotation at $\theta =\pi$, providing a direct, ground-state signature of symmetry fractionalization. We verify this prediction analytically using the matrix product density operator formalism and numerically for the kagome-lattice resonating valence bond state. The proposed observable requires only a single ground-state wavefunction, making it amenable to quantum simulation platforms.