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arXiv:0804.1635 · Physical Review B

States of interacting composite fermions at the Landau level filling ν=2+3/8

Csaba Tőke, Chuntai Shi, Jainendra K. Jain

Published Jun 5, 2008• 12 citations
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Abstract

There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\ensuremath{\nu}=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\ensuremath{\nu}=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.

DOI
10.1103/physrevb.77.245305

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There is increasing experimental evidence for fractional quantum Hall effect at filling factor \ensuremathν=2+3/8\ensuremath{\nu}=2+3/8\ensuremathν=2+3/8. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at \ensuremathν=2+3/8\ensuremath{\nu}=2+3/8\ensuremathν=2+3/8. The resulting state is seen to be distinct from the Pfaffian wave function.
By developing an algorithm for evaluating the basis states for the composite fermions with negative effective magnetic field, we perform the composite-fermion-diagonalization study for the fully spin-polarized fractional quantum Hall states at the filling factors $\ensuremath{\nu}=3/10$, 4/13, and 5/17 in the range $2/7<\ensuremath{\nu}<1/3$. These observed states correspond to a partially filled second effective Landau level, for the composite fermions carrying four vortices, with filling factor $\overline{\ensuremath{\nu}}=1/2$, 1/3, and 2/3, respectively, analogous to the previously studied states of composite fermions with two attached vortices in the range $1/3<\ensuremath{\nu}<2/5$. We show that the character of these states in the range $2/7<\ensuremath{\nu}<1/3$ replicates the same for the states in the range $1/3<\ensuremath{\nu}<2/5$ having identical $\overline{\ensuremath{\nu}}$: chiral ppp-wave pairing with an anti-Pfaffian correlation of composite fermions carrying six quantized vortices produces an incompressible state at $\ensuremath{\nu}=3/10$; an unconventional interaction between composite fermions, resulting from the suppression of fermion pairs with relative angular momentum three and producing the fractional quantum Hall effect of composite fermions in the second effective Landau level with $\overline{\ensuremath{\nu}}=1/3$ and its particle-hole conjugate filling factor 2/3, reproduces incompressible states at 4/134/134/13 and 5/175/175/17 filling factors. We further estimate the thermodynamic limit of the ground-state energies and calculate the lowest energy gap for the neutral collective excitations of these states.
Pairing of composite fermions provides a possible mechanism for fractional quantum Hall effect at even denominator fractions and is believed to serve as a platform for realizing quasiparticles with non-Abelian braiding statistics. We present results from fixed-phase diffusion Monte Carlo calculations which predict that substantial Landau level mixing can induce a pairing of composite fermions at filling factors ν=1/2 and ν=1/4 in the l=-3 relative angular momentum channel, thereby destabilizing the composite-fermion Fermi seas to produce non-Abelian fractional quantum Hall states.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.