  Gapped quantum spin chains with symmetry PSU(N)=SU(N)/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. Motivated by the potential realization in
alkaline-earth atomic Fermi gases, we explicitly construct previously unknown
Hamiltonians for two classes of chiral AKLT states and we discuss their
physical properties. We also point out a deep connection between symmetry
protection in gapped and gapless 1D quantum spin systems and its implications
for a potential multicritical nature of topological phase transitions.
