We study the classification of symmetry protected topological phases on quantum spin lattices in one and two dimensions. In one dimensional systems, we present how to define a topological in- dex with on-site symmetry using an operator algebraic approach. In two dimensions, we provide the proof of the 3-cocycle relation for the topological index. The abstract theory of Finitely correlated states and Valence bond states is thoroughly analyzed and further developed to derive covariance relations for on-site symmetries. The formalism and GNS construction of a newly-developed class of states is introduced. Finitely correlated states will arise as a variant thereof. We conclude by establishing the AKLT ground state as a Finitely correlated state. By explicit computation, we will demonstrate that the S = 1 AKLT and trivial model belong to distinguishable symmetry protected phases with respect to the Z2 × Z2 -symmetry.
Laboratory for Theoretical and Computational Physics