Gapped quantum phases for the S=1 spin chain with D2h symmetry

We study different quantum phases in integer spin systems with on-site D-2h = D-2 circle times Z(2) and translation symmetry. We find four distinct nontrivial phases in S = 1 spin chains despite the fact that they all have the same symmetry. All four phases have gapped bulk excitations, doubly degenerate end states, and the doubly degenerate entanglement spectrum. These nontrivial phases are examples of symmetry-protected topological (SPT) phases introduced by Gu andWen [Phys. Rev. B 80, 155131 (2009)]. One of the SPT phases corresponds to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different responses of the end states to weak external magnetic fields. According to the Chen-Gu-Wen classification, the D-2h symmetric spin chain can have a total of 64 SPT phases that do not break the symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of the D-2h group. Four of these are found in S = 1 spin chains and studied in this paper.