We use exact diagonalization to study the quantum phases and phase transitions when a single species of fermionic atoms at a Landau level filling factor ${\ensuremath{\nu}}_{f}=1$ in a rotating trap interact through a
p-wave Feshbach resonance. We show that under a weak pairing interaction, the system undergoes a second-order quantum phase transition from a ${\ensuremath{\nu}}_{f}=1$ fermionic integer quantum Hall (FIQH) state at positive detuning, to a ${\ensuremath{\nu}}_{b}=\frac{1}{4}$ bosonic fractional quantum Hall (BFQH) state at negative detuning. However, when the pairing interaction increases, a new phase between them emerges, corresponding to a fraction of fermionic atoms staying in a coherent superposition of a bosonic molecule state and an unbound pair. The phase transition from the FIQH phase to the new phase is of second order and that from the new phase to BFQH phase is of first order.
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer
n. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian
Z4 parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the
Z4 state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the
Z4 parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.
Interactions are crucial in driving topological phenomena in bosonic systems. Here, the authors construct exactly solvable models of topological phases of interacting bosons that preserve electric charge. They focus on the
E8 bosonic integer quantum Hall state and show how its intrinsic
E8 symmetry allows for the emergence of fractional topological phases of bosons. They detail the charge and statistics of the quasiparticle excitations of the resulting fractional quantum Hall states.