Bulk Detection of Average Symmetry-Protected Topological Phases Through Strange Density Matrix

We demonstrate that the recently established average symmetry-protected topological (ASPT) phases can be directly detected in the bulk by the strange density matrix and the corresponding strange correlator. We obtain that even though the ASPT state as a density matrix $ ho$ we detect is short-range entangled (SRE), its strange correlator has long-range behavior if the density matrix $ ho$ is a nontrivial ASPT phase. By explicit calculations and Monte-Carlo simulations, we confirm the strange correlator detection of ASPT phases by two typical examples: (1+1)D cluster state with random disorder and (2+1)D ASPT protected by $SO(3) imesmathbb{Z}_2^{mathrm{ave}}$ and prove that the SPT phases without ``domain wall decoration'' configuration in the clean system have no ASPT counterpart by vanishing strange density matrix. We further make some connections with the correlation function of statistical models.