Quantized Berry Phases as a Local Order Parameter of a Quantum Liquid
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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 π 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 1 chain is also characterized by the uniform π Berry phases.
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