I present a way to define the manybody topological invariant (“order parameter”) for some classes of fermionic topological phases protected by point group symmetries. We introduced the “partial point group transformation”, which is defined by the point group transformation on a subregion that is closed under the point group transformation. The topological invariant appears as the complex U(1) phase of the ground state expectation value of the partial point group transformation. Since such a topological invariant is made only from the ground state and the symmetry operator, it is applicable to systems with interaction and disorder. I explain why such the quantity gives the topological invariant from the viewpoints of (i) the topological quantum  field theory and (ii) the surface gapless theory. I also show the numerical calculations of our topological invariants for some simple free fermionic ground states.

Refs:
Hassan Shapourian, KS, Shinsei Ryu, arXiv:1607.03896
KS, Hassan Shapourian, Shinsei Ryu, arXiv:1609.05970

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