Evidence for deconfined $U(1)$ gauge theory at the transition between toric code and double semion

Understand

Building on quantum Monte Carlo simulations, we study the phase diagram of a one-parameter Hamiltonian interpolating between trivial and topological Ising paramagnets in two dimensions, which are dual to the toric code and the double semion.

• We discover an intermediate phase with stripe order which spontaneously breaks the protecting Ising symmetry.
• Remarkably, we find evidence that this intervening phase is gapless due to the incommensurability of the stripe pattern and that it is dual to a $U(1)$ gauge theory exhibiting Cantor deconfinement.