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+1D SPT states. Such a construction can be directly generalized to other dimensions.
We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry-protected topological states can be adiabatically deformed into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the spatial group and glued together such that there is no open edge or end. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for noninteracting time-reversal symmetric electronic insulators with spin-orbit coupling and any one of the 230 space groups. This enumeration gives topological crystalline insulators a full classification.
Topological crystalline phases (TCPs) are topological states protected by spatial symmetries. A broad range of TCPs have been conventionally studied by formulating topological invariants (symmetry indicators) at invariant momenta in the Brillouin zone, which leaves open the question of their stability in the absence of translational invariance. In this work, we show that robust basis-independent topological invariants can be generically constructed for TCPs using projected symmetry operators. Remarkably, we show that the real-space topological markers of these invariants are exponentially localized to the fixed points of the spatial symmetry. As a result, this real-space structure protects them against the presence of impurities that are located away from the fixed points and also sufficiently slowly varying disorder potentials. By considering all possible symmetry centers in a crystalline system we can generate a mesh of real-space topological markers that can provide a local topological distinction for TCPs. We illustrate the robustness of this mesh of invariants with one- and two-dimensional TCPs protected by inversion, rotational, and mirror symmetries. Finally, we find that the boundary modes of these TCPs can also exhibit robust topological invariants with localized markers on the edges. We illustrate this with the gapless Majorana boundary modes of mirror-symmetric topological superconductors, and relate their integer topological edge invariant with a quantized effective edge polarization.