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The nature of symmetry-protected topological phases of Heisenberg spin chainsin totally symmetric representations of rank N of the SU(N) group isinvestigated through a Majorana fermion study starting from an integrablepoint. The latter approach generalizes the one pioneered by Tsvelik [A. M.Tsvelik, Phys. Rev. B 42, 10 499 (1990)] to describe the low-energy propertiesof the Haldane phase of the spin-1 Heisenberg chain from three massive Majoranafermions. We find for all N the emergence of a non-degenerate gapped phase withedge states whose topological protection depends on the parity of N. While forN odd there is no such protection, the phase with even N is shown to betopologically protected. We find that the phase belongs to the same topologicalclass as the phase with edge states living in self-conjugate fullyantisymmetric representation of the SU(N) group.

Symmetry-protected topological phases in the SU(N) Heisenberg spin chain: a Majorana-fermion approach