A way to detect symmetry protected topological (SPT) phases from a given ground state wave function is discussed. Many body topological invariants are defined as partition functions of topological quantum field theory (TQFT) on space-time manifolds, for example, real projective spaces. It is expected that by translating those to the operator formalism, one can get a definition of many body topological invariant made from ground state wave functions and symmetry operations. We propose that a kind of non-local operation, the “partial point group transformation”, on a short-range entangled quantum state is a unified measure to detect topological nontrivial phases with point group symmetry.

In this talk, I introduce the many body Z2 invariant (Arf invariant) of the (1+1)d Kitaev chain to explain the relation between ground state wave functions and TQFT. Next, I give our proposals: the Z8 invariant from the partial reflection on (1+1)d superconductor, the partial rotations on the (2+1)d superconductors, and the Z16 invariant from the partial inversion on (3+1)d superconductors. All the invariants can be analytically calculated from the boundary theory and we confirmed these results numerically.

KS, Hassan Shapourian, and Shinsei Ryu, arXiv:1609.05970