Wave Function and Strange Correlator of Short-Range Entangled States

We demonstrate the following conclusion: If $|\mathrm{\Psi }⟩$ is a one-dimensional (1D) or two-dimensional (2D) nontrivial short-range entangled state and $|\mathrm{\Omega }⟩$ is a trivial disordered state defined on the same Hilbert space, then the following quantity (so-called “strange correlator”) $C(r,r^{\prime })=⟨\mathrm{\Omega }|\varphi (r)\varphi (r^{\prime })|\mathrm{\Psi }⟩/⟨\mathrm{\Omega }|\mathrm{\Psi }⟩$ either saturates to a constant or decays as a power law in the limit $|r-r^{\prime }|\to +\infty$, even though both $|\mathrm{\Omega }⟩$ and $|\mathrm{\Psi }⟩$ 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 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 Affleck-Kennedy-Lieb-Tasaki 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.

Figure 1

(a) $|\mathrm{\Psi }⟩$ and $⟨\mathrm{\Omega }|$ are given by infinite time evolution of their QFTs from below and above, respectively. The strange correlator can be viewed as the correlator at the $\tau =0$ interface. (b) Under the Lorentz rotation, the two QFTs are separated by the $x=0$ interface, and the strange correlator can be interpreted as the correlation function on this spatial interface.

Figure 2

(a) The amplitude of strange correlator in the momentum space. The inset shows the Brillouin zone and the high symmetry points. (b) $|C_k{|}^{-1}$ exhibits nice linearity around the $K$ point, establishing the $1/|k|$ divergence in $|C_k|$.

Figure 3

(a) Tensor network representation of the 2D AKLT state. The red (blue) legs represent the physical (internal) degrees of freedom. (b) Strange correlator of the 2D AKLT state measured along the horizontal direction. (c) The amplitude follows a power-law behavior in the log-log plot. The final deviation is due to the finite-size effect.

Figure 4

(a) The strange correlator of the SPT state (in blue) at infinite distance $|r-r^{\prime }|\to \infty$, in comparison with that of the trivial state (in red). The SPT strange correlator follows the power-law behavior (b) at the critical point and (c) in the dense loop phase.