Mapping from quantum antiferromagnets to a semiclassical field theory such as nonlinear sigma models with topological terms has contributed insights into the nontrivial behavior of quantum fluctuations since the work by Haldane in 1980’s. Recently, another aspect of AKLT states is focused, i.e., short-range entangled state protected by some symmetry. We explain how such symmetry protected topological (SPT) properties in AKLT-type VBS states are described using an effective field theory in spatial dimensions one through three.
Starting from the well-understood case of one-dimensional antiferromagnets and thereby establishing its validity, we proceed to the two-dimensional square lattice. Through a path integral representation of the ground state wave functional,we conclude that the ground state is an SPT state for the spin quantum number equal to two times odd integer while it is topologically trivial for two times even integer.
We also show that this representation of the ground state wave functional is closely related with the strange correlator,which is proposed as an indicator of SPT phase. Finally, we discuss generalization of the preceding approach to the case of a cubic lattice.