Quantized Berry phases as a local order parameter of a quantum liquid

We propose the use of quantized Berry phases as a local topological order parameter of a gapped quantum liquid in any dimension that is invariant under some antiunitary operation. The Berry connection is constructed by the response of the quantum manybody state to a local perturbation. Due to the anti-unitary invariance, the Berry phases are quantized as 0 or pi unless the energy gap closes by the local perturbation. Nontrivial characterizations are demonstrated for ground states of frustrated Heisenberg models and manybody ground states of half-filled random-hopping models. The local topological-order parameters in the models provide quantized texture patterns of local singlet pairs and fermionic local covalent bonds. The Haldane phase of the spin I chain is also characterized by the uniform pi Berry phases.