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Recent developments have revealed that symmetries need not form a group, but instead can be non-invertible. I will use analytical arguments and numerical evidence to illuminate how spontaneous symmetry breaking of a non-invertible symmetry is similar yet distinct from ordinary symmetry breaking. I will consider one-dimensional chains of group-valued qudits, whose local Hilbert space is spanned by elements of a finite group $G$ (reducing to ordinary qubits when $G=\mathbbm{Z}_2$). I will construct Ising-type transverse-field Hamiltonians with Rep($G$) symmetry whose generators multiply according to the tensor product of irreducible representations (irreps) of the group $G$. For non-Abelian $G$, the symmetry is non-invertible. In the symmetry broken phase there is one ground state per irrep on a closed chain. The symmetry breaking can be detected by local order parameters but, unlike the invertible case, different ground states have distinct entanglement patterns. I will show that for each irrep of dimension greater than one the corresponding ground state exhibits string order, entanglement spectrum degeneracies, and has gapless edge modes on an open chain -- features usually associated with symmetry-protected topological order. Consequently, domain wall excitations behave as one-dimensional non-Abelian anyons with non-trivial internal Hilbert spaces and fusion rules. This work identifies properties of non-invertible symmetry breaking that existing quantum hardware can probe and raises new questions in quantum computing.
