Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions.

The ground-state wavefunction of double semion model on a two-dimensional hexagonal lattice is given by the superposition of closed domain wall loops weighted by a $\pi $ phase multiplying the number of loops. The presence of these negative signs makes the nature of this state distinguish from the toric code state. To explore the quantum effects of these intrinsic signs, a generic double semion wavefunction with tuning parameters is proposed. In tensor network representation, these phase factors are expressed in terms of auxiliary spins on the dual triangular lattice. The norm of this wavefunction is mapped to the partition function of a two-dimensional triangular Ashkin-Teller model with imaginary magnetic fields and imaginary $% 3$-spin triangular face interactions. Applying the numerical tensor network methods to this parity-time (PT) symmetric statistical model with negative Boltzmann weights, we establish a global phase diagram, where the double semion phase corresponds to the partial order phase of this Ashkin-Teller model. Adjacent to the double semion phase, we find dilute loop phase, dense loop phase with PT symmetry, and dense loop phase with spontaneously broken PT-symmetry. The latter two gapless phases contain non-abelian anyon excitations, characterized by non-unitary conformal field theories.