  Bosonic symmetry-protected topological (SPT) states are gapped disordered
phases of matter possessing symmetry-preserving boundary excitations. It has
been proposed that, at long wavelengths, the universal properties of an SPT
system are captured by an effective non-linear sigma model field theory in the
presence of a quantized topological theta-term. By studying lattice models of
bosonic SPT states, we are able to identify, in their Euclidean path integral
formulation, (discrete) Berry phases that hold relevant physical information on
the nature of the SPT ground states. These discrete Berry phases are given
intuitive physical interpretation in terms of instanton effects that capture
the presence of a theta-term on the microscopic scale.

Strongly Correlated Electrons