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arXiv:1012.0046 · Physical Review Letters

Fractionalization via Z2 Gauge Fields at a Cold-Atom Quantum Hall Transition

Yafis Barlas, Kun Yang

Published Apr 29, 2011• 3 citations
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Abstract

We study a single species of fermionic atoms in an "effective" magnetic field at total filling factor ν(f)=1, interacting through a p-wave Feshbach resonance, and show that the system undergoes a quantum phase transition from a ν(f)=1 fermionic integer quantum Hall state to ν(b)=1/4 bosonic fractional quantum 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. Experimental consequences and possible tests of our theoretical predictions are discussed.

DOI
10.1103/physrevlett.106.170403

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We study a single species of fermionic atoms in an "effective" magnetic field at total filling factor ν(f)=1, interacting through a p-wave Feshbach resonance, and show that the system undergoes a quantum phase transition from a ν(f)=1 fermionic integer quantum Hall state to ν(b)=1/4 bosonic fractional quantum 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. Experimental consequences and possible tests of our theoretical predictions are discussed.
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 ppp-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.
We numerically investigate the quantum phases and phase transition in a system made of two species of fermionic atoms that interact with each other via sss-wave Feshbach resonance and are subject to rotation or a synthetic gauge field that puts the fermions at Landau level filling factor ${\ensuremath{\nu}}_{f}=2$. We show that the system undergoes a continuous quantum phase transition from a ${\ensuremath{\nu}}_{f}=2$ fermionic integer quantum Hall state formed by atoms to a ${\ensuremath{\nu}}_{b}=1/2$ bosonic fractional quantum Hall state formed by bosonic diatomic molecules. In the disk geometry we use, these two different topological phases are distinguished by their different gapless edge excitation spectra, and the quantum phase transition between them is signaled by the closing of the energy gap in the bulk. Comparisons will be made with field-theoretical predictions and the case of ppp-wave pairing.
We consider two-species of fermions in a rotating trap that interact via an s-wave Feshbach resonance, at total Landau level filling factor two (or one for each species). We show that the system undergoes a quantum phase transition from a fermion integer quantum Hall state to a boson fractional quantum Hall state as the pairing interaction strength increases, with the transition occurring near the resonance. The effective field theory for the transition is shown to be that of a (emergent) massless relativistic bosonic field coupled to a Chern-Simons gauge field, with the coupling giving rise to semionic statistics to the emergent particles.
We find a series of possible continuous quantum phase transitions between fractional quantum Hall states at the same filling fraction in two-component quantum Hall systems. These can be driven by tuning the interlayer tunneling and/or interlayer repulsion. One side of the transition is the Halperin (p,p,p-3) Abelian two-component state, while the other side is the non-Abelian Z4 parafermion (Read-Rezayi) state. We predict that the transition is a continuous transition in the 3D Ising class. The critical point is described by a Z2 gauged Ginzburg-Landau theory. These results have implications for experiments on two-component systems at ν=2/3 and single-component systems at ν=8/3.
We study U(1)×U(1)⋊Z2 Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the U(1)×U(1)⋊Z2 Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z2 vortices in the U(1)×U(1)⋊Z2 Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z2 vortices on a sphere. These results allow us to show that l=3 U(1)×U(1)⋊Z2 Chern-Simons theory is the low-energy effective theory for the Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction ν=2/2k−3. The U(1)×U(1)⋊Z2 theory is more useful than an alternative SU(2)4×U(1)∕U(1) Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the ν=2∕3 phase diagram.
We consider the bosonic fractional quantum Hall (FQH) effect in the presence of a non-Abelian gauge field in addition to the usual Abelian magnetic field. The non-Abelian field breaks the twofold internal state degeneracy, but preserves the Landau level degeneracy. Using exact diagonalization, we find that for moderate non-Abelian field strengths the system's behaviour resembles a single internal state quantum Hall system, while for stronger fields there is a phase transition to either two internal state behaviour or the complete absence of FQH plateaus. Usually the energy gap is reduced by the presence of a non-Abelian field, but some non-Abelian fields appear to slightly increase the gap of the ν=1 and ν=3/2 Read–Rezayi states.