# Gapless Coulomb state emerging from a self-dual topological tensor-network state

## TL;DR

This paper introduces a tensor network state that maps to the Ashkin-Teller model, revealing a novel gapless Coulomb phase with quasi-long-range order arising from a self-dual topological state, and analyzes phase transitions and anyon interactions.

## Contribution

It presents a new tensor network representation of a topological state that exactly maps to the Ashkin-Teller model, uncovering a gapless Coulomb phase from self-duality.

## Key findings

- Exact mapping to the Ashkin-Teller model
- Identification of a gapless Coulomb phase with quasi-long-range order
- Derivation of anyon correlation scaling behaviors

## Abstract

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 (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT 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 confining phase.