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