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The Dirac-like electronic structure can host a large number of competingorders in the form of mass terms. In particular, two different order parameterscan be said to be dual to each other, when a static defect in one of them trapsa quantum number (or "charge") of the other. We discuss that such complementarynature of the pair of the order parameters shows up in their correlationfunctions and dynamical properties when a quantum phase transition is driven byfluctuations of the one of the order parameters. Approaching the transitionfrom the disordered (paramagnetic) side, the order parameter correlationfunction at the critical point is reduced, while such fluctuations enhance thecorrelation of the dual order parameter. Such complementary behaviors in thecorrelation function can be used to diagnose the nature of quantum fluctuationsthat is the driving force of the quantum phase transition.

Competing orders in the Dirac-like electronic structure and the non-linear sigma model with the topological terms