# Symmetry Protected Topological (SPT) phases of spin-1 chains

Let's consider this family of 1D spin-1 of hamiltonians:

$$\sum_{i}[S^x_{i}S^{x}_{i+1}+S^y_{i}S^{y}_{i+1}+\lambda S^z_{i}S^{z}_{i+1} + D(S^{z}_{i})^2].$$

If I understand it right, these models have:

1. Translational symmetry

2. Inversion symmetry ($$S^{x,y,z}_{i}\to S^{x,y,z}_{-i+1}$$)

3. Time reversal symmetry ($$S^{x,y,z}_{i}\to - S^{x,y,z}_{i}$$)

4. $$Z_2\times Z_2$$ symmetry ($$\pi$$-rotation about $$x, y$$ anz $$z$$ axes $$S^{\alpha,\beta}_{i}\to - S^{\alpha,\beta}_{i}, S^{\gamma}_{i}\to S^{\gamma}_{i}$$).

Then according to [1] there should be 1024 different SPT phases. But other than the Haldane phase and the trivial topological phase I don't know any other topological phase for this class of hamiltonians [2]. Where are the other phases?

References:

[2] Kennedy, T. & Tasaki, H. Commun.Math. Phys. (1992) 147: 431. https://doi.org/10.1007/BF02097239

• 1024, lovely. This model only realizes two SPT phases: the archetypal Haldane phase ($\lambda =0$; $D=0$) and the trivial phase ($D \to \infty$) (as well as a symmetry-breaking phase for $\lambda \to \infty$). For the other 1022 phases, you need other lattice models. There are papers on this as well, in particular by Wen. However, I haven't really seen physically plausible models that realize those other phases. – Ruben Verresen Nov 11 '18 at 23:26
• In fact, scratch that: this nice paper shows that a physically reasonable spin-$2$ model realizes a variety of SPT phases with your above symmetry groups: arxiv.org/abs/1412.3370 – Ruben Verresen Nov 12 '18 at 23:56