Universal symmetry-protected topological invariants for symmetry-protected topological states

Symmetry-protected topological (SPT) states are short-range entangled states with a symmetry G. They belong to a new class of quantum states of matter which are classified by the group cohomology Hᵈ⁺¹(G,R/Z) in d-dimensional space. In this paper, we propose a class of symmetry-protected topological invariants that may allow us to fully characterize SPT states with a symmetry group G [i.e., allow us to measure the cocycles in Hᵈ⁺¹(G,R/Z) that characterize the SPT states]. We give an explicit and detailed construction of symmetry-protected topological invariants for $2+1$D SPT states. Such a construction can be directly generalized to other dimensions.