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

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. 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,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 H d+1 (G,R/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.