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Since the discovery of topological insulators, topological aspects of quantum matters have attracted growing interests. In recent years, topological insulators are regarded as members of a larger class of topological phases called symmetry protected topological (SPT) phases. SPT phases are gapped phases that cannot be adiabatically connected to trivial insulators in the presence of certain symmetry and accompany gapless excitations at the boundary. The notion of SPT phases is not restricted to systems of non-interacting fermions, but can also be applied to systems of bosons and interacting fermions. From this viewpoint, the Haldane phase of an S=1 spin chain can be understood as an SPT phase protected by Z₂×Z₂ symmetry of spin π rotations around x, y, and z axes.
In this talk, I first review the classification theory of one dimensional bosonic SPT phases, and then show our recent attempt to generalize the Haldane phase into a Z3 SPT phase realized in SU(3) spin chains protected by Z₃×Z₃ symmetry [1]. The parent Hamiltonian of the Z3 SPT phase is constructed and turns out to be an SU(3) version of the AKLT bilinear-biquadratic model. We have studied general SU(3) bilinear-biquadratic models with iDMRG and obtained a phase diagram.
I would also like to briefly report on the classification theory of SPT phases of interacting fermions in arbitrary dimensions by using the nonlinear sigma model [2].

[1] Takahiro Morimoto, Hiroshi Ueda, Tsutomu Momoi, and Akira Furusaki, Phys. Rev. B 90, 235111 (2014).
[2] Takahiro Morimoto, Akira Furusaki, and Christopher Mudry, Phys. Rev. B 92, 125104 (2015).

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