Quantized Berry Phases as Local Order Parameters of Quantum Liquids

We propose to use quantized Berry phases as local order parameters of gapped quantum liquids, which are invariant under some anti-unitary operation. After presenting a general prescription, the scheme is applied for Heisenberg models with frustrations and two dimensional extended Su-Schrieffer-Heeger models associated with a random dimer covering. In each phases, the quantized Berry phases, as 0 or $\pi$, describe the ground state texture pattern of local singlet bonds and dimer bonds. Also possible applications to large classes of correlated electron systems are discussed.