Gapped quantum spin chains with symmetry $\text{PSU}\left(N\right)=\text{SU}\left(N\right)/{\mathbb{Z}}_N$ are known to possess $N$ distinct symmetry protected topological phases. Besides the trivial phase, there are $N-1$ Haldane phases which are distinguished by the occurrence of massless boundary spins. We give an expression for a universal AKLT Hamiltonian for an arbitrary symmetry group, including the case where inversion symmetry is broken. Using a diagrammatic method, we construct Hamiltonians for two of the topological classes in $\text{SU}\left(N\right)$ that arise when the physical spin is in the adjoint representation. Since our results are valid for all $N$, we also discuss the large-$N$ limit.

• Received 19 February 2017
• Revised 6 March 2018

DOI:https://doi.org/10.1103/PhysRevB.97.155148