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Emergence of half-integer filling factor states, such as nu=5/2 and 7/2, is found in quantum dots by using numerical many-electron methods. These states have interesting similarities and differences with their counterstates found in the two-dimensional electron gas. The nu=1/2 states in quantum dots are shown to have high overlaps with the composite fermion states. The lower overlap of the Pfaffian state indicates that electrons might not be paired in quantum dot geometry. The predicted nu=5/2 state has high spin polarization which may have impact on the spin transport through quantum dot devices.
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Fractional quantum Hall states at half-integer filling factors have been observed in many systems beyond the $5/2$ and $7/2$ plateaus in GaAs quantum wells. This includes bilayer states in GaAs, several half-integer plateaus in ZnO-based heterostructures, and quantum Hall liquids in graphene. In all cases, Cooper pairing of composite fermions is believed to explain the plateaus. The nature of Cooper pairing and the topological order on those plateaus are hotly debated. Different orders are believed to be present in different systems. This makes it important to understand experimental signatures of all proposed orders. We review the expected experimental signatures for all possible composite-fermion states at half-integer filling. We address Mach-Zehnder interferometry, thermal transport, tunneling experiments, and Fabry-P{e}rot interferometry. For this end, we introduce a uniform description of the topological orders of Kitaevs sixteenfold way in terms of their wave-functions, effective Hamiltonians, and edge theories.
In this review the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ($p$-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the PH Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern-Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.
We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by $lambda$ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglement (and its fluctuations) are analyzed for a set of $U(4)$ coherent (emph{quasiclassical}) states generalizing the standard pseudospin $U(2)$ coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta $lambda$. Interlayer entanglement of an equilibrium thermal state and its dependence with temperature and bias voltage is also studied for a pseudo-Zeeman interaction.
We theoretically study the quantum Hall effect (QHE) in graphene with an ac electric field. Based on the tight-binding model, the structure of the half-integer Hall plateaus at $sigma_{xy} = pm(n + 1/2)4e^2/h$ ($n$ is an integer) gets qualitatively changed with the addition of new integer Hall plateaus at $sigma_{xy} = pm n(4e^2/h)$ starting from the edges of the band center regime towards the band center with an increasing ac field. Beyond a critical field strength, a Hall plateau with $sigma_{xy} = 0$ can be realized at the band center, hence restoring fully a conventional integer QHE with particle-hole symmetry. Within a low-energy Hamiltonian for Dirac cones merging, we show a very good agreement with the tight-binding calculations for the Hall plateau transitions. We also obtain the band structure for driven graphene ribbons to provide a further understanding on the appearance of the new Hall plateaus, showing a trivial insulator behavior for the $sigma_{xy} = 0$ state. In the presence of disorder, we numerically study the disorder-induced destruction of the quantum Hall states in a finite driven sample and find that qualitative features known in the undriven disordered case are maintained.
We observe two-fold shell filling in the spectra of closed one-dimensional quantum dots formed in single-wall carbon nanotubes. Its signatures include a bimodal distribution of addition energies, correlations in the excitation spectra for different electron number, and alternation of the spins of the added electrons. This provides a contrast with quantum dots in higher dimensions, where such spin pairing is absent. We also see indications of an additional fourfold periodicity indicative of K-K subband shells. Our results suggest that the absence of shell filling in most isolated nanotube dots results from disorder or nonuniformity.
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