We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-$S$ Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\pi$-rotations about $x,y$ and $z$ axes; (ii) time-reversal symmetry $S^{ x,y,z} \rightarrow - S^{ x,y,z}$; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B \textbf{ 81}, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.
Categories: cond-mat.str-el cond-mat.stat-mech