Gapless Coulomb State Emerging from a Self-Dual Topological Tensor-Network State

In the tensor network representation, a deformed Z_{2} topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the Ashkin-Teller model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or the confining phase.