Multicriticality between Purely Gapless SPT Phases with Unitary Symmetry

Symmetry-protected topological (SPT) phases are commonly required to have an energy gap, but recent work has extended the concept to gapless settings. This raises a natural question: what happens at transitions between inequivalent gapless SPTs? We address this for the simplest known case among gapless SPTs protected by a unitary symmetry group acting faithfully on the low-energy theory. To this end, we consider a qutrit version of the nearest-neighbor XX chain. Trimerizing the chain explicitly breaks an anomalous symmetry and produces three distinct gapped SPT phases protected by a unitary ${\mathbb{Z}}_3\times {\mathbb{Z}}_3$ symmetry. Their phase boundaries are given by three inequivalent gapless SPTs without any gapped symmetry sectors, each described by a symmetry-enriched version of an orbifolded Potts${}^2$ conformal field theory with central charge $c=\frac{8}{5}$. We provide an analytic derivation of this critical theory in a particular regime and confirm its stability using tensor network simulations. Remarkably, the three gapless SPTs meet at a $c=2$ multicritical point, where the protecting ${\mathbb{Z}}_3\times {\mathbb{Z}}_3$ symmetry exhibits a mixed anomaly with the ${\mathbb{Z}}_3$ entangler symmetry that permutes the SPT classes. We further explore how discrete gauging gives dipole-symmetric models, offering insights into dipole symmetry-breaking and SPTs, as well as symmetry-enriched multiversality. Altogether, this work uncovers a rich phase diagram of a minimal qutrit chain, whose purely nearest-neighbor interactions make it a promising candidate for experimental realization, including the prospect of critical phases with stable edge modes.