Wave function and strange correlator of short-range entangled states

We demonstrate the following conclusion: If |Ψ" is a one-dimensional (1D) or two-dimensional (2D) nontrivial short-range entangled state and |Ω" is a trivial disordered state defined on the same Hilbert space, then the following quantity (so-called "strange correlator") C(r,r′)="Ω|φ(r)φ(r′) |Ψ"/"Ω|Ψ" either saturates to a constant or decays as a power law in the limit |r-r′|→, even though both |Ω" and |Ψ" are quantum disordered states with short-range correlation; φ(r) is some local operator in the Hilbert space. This result is obtained based on both field theory analysis and an explicit computation of C(r,r′) 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 Affleck-Kennedy-Lieb-Tasaki state on the square lattice, and the 2D bosonic symmetry-protected topological phase with Z2 symmetry. This result can be used as a diagnosis for short-range entangled states in 1D and 2D.