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

Abstract

Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point group transformations restricted to a spatial subregion on a given ground-state wave function. Through analytical and numerical calculations, the authors show that the complex $U(1)$ phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the ${Z}_{8}$ and ${Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, as well as the lens space topological invariants in $(2+1)$-dimensional fermionic topological phases.