Symmetry-protected topological phases in the $\mathrm{SU}\left(N\right)$ Heisenberg spin chain: A Majorana fermion approach

The nature of symmetry-protected topological phases of Heisenberg spin chains in totally symmetric representations of rank $N$ of the $\mathrm{SU}(N$) group is investigated through a Majorana fermion study starting from an integrable point. The latter approach generalizes the one pioneered by Tsvelik [Phys. Rev. B 42, 10499 (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 $N$'s, the emergence of a nondegenerate gapped phase with edge states whose topological protection depends on the parity of $N$. Whereas 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 $\mathrm{SU}\left(N\right)$ group.