Symmetry protection of topological phases in one-dimensional quantum spin systems

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 nontrivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\ensuremath{\pi}$ rotations about the $x$, $y$, and $z$ axes, (ii) time-reversal symmetry ${S}^{x,y,z}\ensuremath{\rightarrow}\ensuremath{-}{S}^{x,y,z}$, and (iii) link inversion symmetry (reflection about a bond center), consistent with previous results [Phys. Rev. B 81, 064439 (2010)]. On the other hand, an even-$S$ Haldane phase is not topologically protected (i.e., it is indistinct from a trivial,…

• NS
  National Science Foundation

  Awards: 0705472, 0757145
• MO
  Ministry of Education, Culture, Sports, Science and Technology

  Award: 20102008
• JS
  Japan Society for the Promotion of Science

  Award: 20654030
• DO
  Division of Materials Research
• AR
  Army Research Office

  Award: W911NF