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Using the Matrix Product State framework, we generalize theAffleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquidswith global color ${\rm SU}(N)$ symmetry, finite correlation lengths, and edgestates that can belong to any self-conjugate irreducible representation (irrep)of ${\rm SU}(N)$. In particular, ${\rm SU}(2)$ spin-$1$ AKLT states with edgestates of arbitrary spin $s=1/2,1,3/2,\cdots$ are constructed, and a generalformula for their correlation length is given. Furthermore, we show how toconstruct local parent Hamiltonians for which these AKLT states are uniqueground states. This enables us to study the stability of the edge states byinterpolating between exact AKLT Hamiltonians. As an example, in the case ofspin-$1$ physical degrees of freedom, it is shown that a quantum phasetransition of central charge $c = 1$ separates the Symmetry ProtectedTopological (SPT) phase with spin-$1/2$ edge states from a topologicallytrivial phase with spin-$1$ edge states. We also address some specificities ofthe generalization to ${\rm SU}(N)$ with $N>2$, in particular regarding theconstruction of parent Hamiltonians. For the AKLT state of the ${\rm SU}(3)$model with the $3$-box symmetric representation, we prove that the edge statesare in the $8$-dimensional adjoint irrep, and for the ${\rm SU}(3)$ model withadjoint irrep at each site, we are able to construct two differentreflection-symmetric AKLT Hamiltonians, each with a unique ground state whichis either even or odd under reflection symmetry and with edge states in theadjoint irrep. Finally, examples of two-column and adjoint physical irreps for${\rm SU}(N)$ with $N$ even and with edge states living in the antisymmetricirrep with $N/2$ boxes are given, with a conjecture about the general formulafor their correlation lengths.

Novel families of ${\rm SU}(N)$ AKLT states with arbitrary self-conjugate edge states