Wave Function and Strange Correlator of Short Range Entangled states

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We demonstrate the following conclusion: If |Ψ is a 1d or 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 also 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 AKLT 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. A possible diagnosis for 3d short range entangled states is also proposed.