Quantized Berry Phases as a Local Order Parameter of a Quantum Liquid

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, 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.