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 $H^{d+1}(G,\mathbb{R}\mathrm{/}\mathbb{Z})$ 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 (ie allow us to measure the cocycles in $H^{d+1}(G,\mathbb{R}\mathrm{/}\mathbb{Z})$ that characterize the SPT states). We give an explicit and detailed construction of symmetry-protected topological invariants for 2+1D SPT states. Such a construction can be directly generalized to other dimensions.