We investigate the nature of the quantum phase transition in modulated $\mathrm{SU}\left(N\right)$ Heisenberg spin chains. In the odd-$N$ case, the transition separates a trivial nondegenerate phase to a doubly degenerate gapped chiral $\mathrm{PSU}\left(N\right)$ symmetry-protected topological (SPT) phase which breaks spontaneously the inversion symmetry. The transition is not an Ising transition associated to the breaking of the ${\mathbb{Z}}_2$ inversion symmetry, but is governed by the delocalization of the edge states of the SPT phase. In this respect, a modulated $\mathrm{SU}\left(N\right)$ Heisenberg spin chain provides a simple example in one dimension of a non-Landau phase transition which is described by the $\mathrm{SU}{\left(N\right)}_1$ conformal field theory. We show that the chiral SPT phase exhibits fractionalized spinon excitations, which can be confined by slightly changing the model parameters.

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• Fractionalization
• Quantum phase transitions
• Symmetry protected topological states
• 1-dimensional spin chains
• Heisenberg model
• SU(N) symmetries
• Tensor network methods