Wave Function and Strange Correlator of Short Range Entangled states

Abstract

We demonstrate the following conclusion: If $|\Psi ⟩$ is a $1d$ or $2d$ nontrivial short range entangled state, and $|\Omega ⟩$ is a trivial disordered state defined on the same Hilbert space, then the following quantity (so called strange correlator) $C(r,r^{\prime })=\frac{⟨\Omega |\varphi (r)\varphi (r^{\prime })|\Psi ⟩}{⟨\Omega |\Psi ⟩}$ either saturates to a constant or decays as a power-law in the limit $|r-r^{\prime }|\to +\infty$, even though both $|\Omega ⟩$ and $|\Psi ⟩$ are quantum disordered states with short-range correlation. $\varphi (r)$ 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.

5 pages, 5 figures

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