We explore the states of matter arising from the spontaneous symmetry breaking (SSB) of ${\mathbb{Z}}_2$ non-onsite symmetries. In one spatial dimension, we construct a frustration-free lattice model exhibiting SSB of a non-onsite symmetry, which features the coexistence of two ground states with distinct symmetry-protected topological (SPT) orders. We analytically prove the twofold ground-state degeneracy and the existence of a finite-energy gap. Fixing the symmetry sector yields a long-range entangled ground state that features long-range correlations among noninvertible charged operators. We also present a constant-depth measurement-feedback protocol to prepare such a state with a constant success probability in the thermodynamic limit, which may be of independent interest. Under a symmetric deformation, the SSB persists up to a critical point, beyond which a gapless phase characterized by a conformal field theory emerges. In two spatial dimensions, the SSB of 1-form non-onsite symmetries leads to a long-range entangled state (SPT soup)—a condensate of one-dimensional SPT along any closed loops. On a torus, there are four such locally indistinguishable states that exhibit algebraic correlations between local operators, which we derived via a mapping to the critical $O\left(2\right)$ loop model. This constitutes an intriguing example of ‘‘topological quantum criticality.’’

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• Quantum circuits
• Quantum phase transitions
• Spontaneous symmetry breaking
• Symmetry protected topological states