Fractionalization via $Z_{2}$ Gauge Fields at a Cold Atom Quantum Hall Transition

We study a single species of fermionic atoms in an ``effective'' magnetic field at total filling factor $\nu_{f}=1$, interacting through a p-wave Feshbach resonance, and show that the system undergoes a quantum phase transition from a $\nu_{f} =1 $ fermionic integer Quantum Hall state to $\nu_{b} =1/4 $ bosonic fractional Hall state as a function of detuning. The transition is in the $(2+1)$D-Ising universality class. We formulate a dual theory in terms of quasiparticles interacting with a $Z_{2}$ gauge field, and show that charge fractionalization follows from this topological quantum phase transition. The resultant effective theory contains the lattice $Z_{2}$ gauge theory action along with a ``Hopf'' term which encodes the quasiparticle statistics. The transition occurs in the $Z_{2}$ sector and is a confinement-deconfinement transition for the quasiparticles.