Long-range mutual information and topological uncertainty principle
Ordered phases in Landau paradigm can be diagnosed by a local order parameter, whereas topologically ordered phases cannot be detected in such a way. In this work, we propose long-range mutual information(LRMI) as a unified diagnostic for both conventional long-range order and topological order. Physically, LRMI describes condensation of extended objects such as strings and membranes. As a key feature of topological order to be distinguished from conventional order, we propose and study the topological uncertainty principle, which describes the non-commuting nature of non-local order parameters in topological orders. The topological uncertainty principle guarantees that nontrivial entanglement always exists in ground states of topologically ordered states, and proposes a new framework for understanding higher dimensional topologically ordered states.
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