Many-body topological invariants in fermionic symmetry-protected topological phases: Cases of point group symmetries

We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Here, partial point group transformations gD are defined by point group transformations restricted to a spatial subregion D, which is closed under the point group transformations and sufficiently larger than the bulk correlation length ξ. By analytical and numerical calculations, we find that the ground state (GS) expectation value of the partial point group transformations behaves generically as (GS|gD|GS)∼exp[iθ+γ-αArea(∂D)ξd-1]. Here, Area(∂D) is the area of the boundary of the subregion D, and α is a dimensionless constant. The complex phase of the expectation value θ is quantized and serves as the topological invariant, and γ is a scale-independent topological contribution to the amplitude. The examples we consider include the Z8 and Z16 invariants of topological superconductors protected by inversion symmetry in (1+1) and (3+1) dimensions, respectively, and the lens space topological invariants in (2+1)-dimensional fermionic topological phases. Connections to topological quantum field theories and cobordism classification of symmetry-protected topological phases are discussed.