Phases and Phase Transitions of Spin Chains with Non-invertible Symmetries

Non-invertible symmetries are often emergent at low-energies in gapless states of quantum matter. It is useful to construct lattice models that have these as exact symmetries in order to provide a UV-complete setting in which they are well-controlled. To that end, we propose to study one-dimensional Hamiltonians defined on tensor product Hilbert spaces with finite on-site dimension — referred to as “spin chains” in short — with exact non-invertible symmetries. We focus on two concrete examples: a spin chain with (invertible) S_3 symmetry and one with (non-invertible) Rep(S_3) symmetry. These models are largely analytically tractable and demonstrate all spontaneous symmetry breaking (SSB) phases of these symmetries. With the aid of tensor network algorithms, we systematically study the phase transitions between these SSB phases. Both models possess (intrinsically) non-invertible self-duality symmetries, for which we provide sequential circuit implementations. On the self-dual manifold in parameter space, we discover an incommensurate gapless phase with an anomalous U(1) symmetry which emanates from lattice translation.

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