In this work we propose a parton state as a candidate state to describe the fractional quantum Hall effect in the half-filled second Landau level. The wave function for this parton state is PLLLΦ 3 1 [Φ * 2 ] 2 ∼ Ψ 2 2/3 /Φ1 and in the spherical geometry it occurs at the same flux as the anti-Pfaffian state. This state has a good overlap with the anti-Pfaffian state and with the ground state obtained by exact diagonalization, using the SLL Coulomb interaction pseudopotentials for an ordinary semiconductor such as GaAs. By calculating the entanglement spectrum we show that this state lies in the same phase as the anti-Pfaffian state. A major advantage of this parton state is that its wave function can be evaluated for large systems, which makes it amenable to variational calculations. In the appendix of this work we have numerically assessed the validity of another candidate state at 5/2, namely the particle-hole-symmetric Pfaffian (PH-Pfaffian) state. We find that the proposed candidate wave function for the PH-Pfaffian state is particle-hole symmetric to a high degree but it does not appear to arise as the ground state of any simple two-body Hamiltonian.
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X 5, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on non-relativistic composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particlehole transformation of electrons at filling factor ν = 1/2 corresponds to an effective time reversal transformation (i.e. {k j }→{−k j }) for composite fermions, and discuss how this connects to the absence of 2kF backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ levels (analogs of electronic Landau levels) we determine the Λ level spacing and find it to be very nearly independent of the Λ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a "normal form" for the unprojected wave functions of the n/(2pn − 1) states that correctly capture the topological properties even without lowest Landau level projection.
Motivated by recent experiments that reveal expansive fractional quantum Hall states in the n = 1 graphene Landau level and suggest a nontrivial role of the spin degree of freedom [Amet et al., Nat. Commun. 6, 5838 (2014)], we perform accurate quantitative study of the the competition between fractional quantum Hall states with different spin polarizations in the n = 1 graphene Landau level. We find that the fractional quantum Hall effect is well described in terms of composite fermions, but the spin physics is qualitatively different from that in the n = 0 Landau level. In particular, for the states at filling factors ν = s/(2s ± 1), s integer, a combination of exact diagonalization and the composite fermion theory shows that the ground state is fully spin polarized and supports a robust spin wave mode even in the limit of vanishing Zeeman coupling. Thus, even though composite fermions are formed, a mean field description that treats them as weakly interacting particles breaks down, and the exchange interaction between them is strong enough to cause a qualitative change in the behavior by inducing full spin polarization. We also verify that the fully spin polarized composite fermion Fermi sea has lower energy than the paired Pfaffian state at the relevant half fillings in the n = 1 graphene Landau level, indicating an absence of composite-fermion pairing at half filling in the n = 1 graphene Landau level.
The observation of extensive fractional quantum Hall states in graphene brings out the possibility of more accurate quantitative comparisons between theory and experiment than previously possible, because of the negligibility of finite width corrections. We obtain accurate phase diagram for differently spin-polarized fractional quantum Hall states, and also estimate the effect of Landau level mixing using the modified interaction pseudopotentials given in the literature. We find that the observed phase diagram is in good quantitative agreement with theory that neglects Landau level mixing, but the agreement becomes significantly worse when Landau level mixing is incorporated assuming that the corrections to the energies are linear in the Landau level mixing parameter λ. This implies that a first order perturbation theory in λ is inadequate for the current experimental systems, for which λ is typically on the order of or greater than one. We also test the accuracy of the composite-fermion theory and find that it is very accurate for the states of the form n/(2n + 1) but for the states at n/(2n − 1) the results are sensitive to the lowest Landau level projection method. An earlier prediction of an absence of spin transitions for the n/(4n + 1) states is confirmed by more rigorous calculations, and new predictions are made regarding spin physics for the n/(4n − 1) states.
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