Wave Function and Strange Correlator of Short Range Entangled states

We demonstrate the following conclusion: If $|\Psi\rangle$ is a $1d$ or $2d$ nontrivial short range entangled state, and $|\Omega \rangle$ is a trivial disordered state defined on the same Hilbert space, then the following quantity (so called strange correlator) $C(r, r^\prime) = \frac{\langle \Omega|\phi(r) \phi(r^\prime) | \Psi\rangle}{\langle \Omega| \Psi\rangle}$ either saturates to a constant or decays as a power-law in the limit $|r - r^\prime| \rightarrow +\infty$, even though both $| \Omega\rangle$ and $| \Psi\rangle$ are quantum disordered states with short-range correlation. $\phi(r)$ is some local operator in the Hilbert space. This result is obtained based on both field theory analysis, and also an explicit computation of $C(r, r^\prime)$ for four different examples: $1d$ Haldane phase of spin-1 chain, $2d$ quantum spin Hall insulator with a strong Rashba spin-orbit coupling, $2d$ spin-2 AKLT state on the square lattice, and the $2d$ bosonic symmetry protected topological phase with $Z_2$ symmetry. This result can be used as a diagnosis for short range entangled states in $1d$ and $2d$. A possible diagnosis for $3d$ short range entangled states is also proposed.