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In this paper a duality between the d=2 Wen-plaquette model in a transversefield and the d=1 Ising model in a transverse field is used to learn the natureof the quantum phase transition (QPT) between a spin-polarized phase and atopological ordered state with the string-net condensation. The QPT is notinduced by spontaneous symmetry breaking and there are no conventionalLandau-type local order parameters. Instead the string-like non-local orderparameters are introduced to describe the QPT. In particular, the dualitycharacter between the open-string and closed string for the QPT are explored.

Exact results of the quantum phase transition for the topological order