We propose the definitions of many-body topological invariants to detectsymmetry-protected topological phases protected by point group symmetry, usingpartial point group transformations on a given short-range entangled quantumground state. Partial point group transformations $g_D$ are defined by pointgroup transformations restricted to a spatial subregion $D$, which is closedunder the point group transformations and sufficiently larger than the bulkcorrelation length $\xi$. By analytical and numerical calculations,we find thatthe ground state expectation value of the partial point group transformationsbehaves generically as $\langle GS | g_D | GS \rangle \sim \exp \Big[ i \theta+\gamma - \alpha \frac{{\rm Area}(\partial D)}{\xi^{d-1}} \Big]$. Here, ${\rmArea}(\partial D)$ is the area of the boundary of the subregion $D$, and$\alpha$ is a dimensionless constant. The complex phase of the expectationvalue $\theta$ is quantized and serves as the topological invariant, and$\gamma$ is a scale-independent topological contribution to the amplitude. Theexamples we consider include the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$invariants of topological superconductors protected by inversion symmetry in$(1+1)$ and $(3+1)$ dimensions, respectively, and the lens space topologicalinvariants in $(2+1)$-dimensional fermionic topological phases. Connections totopological quantum field theories and cobordism classification ofsymmetry-protected topological phases are discussed.
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