### Abstract

We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the "unprojected" wave functions satisfying the correct (quasi)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called "proper states," which includes the incompressible ground states at electron filling factors ν=n2pn+1, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form ν=ν∗2pν∗+1, where n and p are integers and ν∗ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors ν=1/3, 2/5, and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.

Original language | English (US) |
---|---|

Article number | 195302 |

Journal | Physical Review B |

Volume | 96 |

Issue number | 19 |

DOIs | |

State | Published - Nov 7 2017 |

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### All Science Journal Classification (ASJC) codes

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics

### Cite this

*Physical Review B*,

*96*(19), [195302]. https://doi.org/10.1103/PhysRevB.96.195302

}

*Physical Review B*, vol. 96, no. 19, 195302. https://doi.org/10.1103/PhysRevB.96.195302

**Composite fermions on a torus.** / Pu, Songyang; Wu, Ying Hai; Jain, J. K.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Composite fermions on a torus

AU - Pu, Songyang

AU - Wu, Ying Hai

AU - Jain, J. K.

PY - 2017/11/7

Y1 - 2017/11/7

N2 - We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the "unprojected" wave functions satisfying the correct (quasi)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called "proper states," which includes the incompressible ground states at electron filling factors ν=n2pn+1, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form ν=ν∗2pν∗+1, where n and p are integers and ν∗ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors ν=1/3, 2/5, and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.

AB - We achieve an explicit construction of the lowest Landau level (LLL) projected wave functions for composite fermions in the periodic (torus) geometry. To this end, we first demonstrate how the vortex attachment of the composite fermion (CF) theory can be accomplished in the torus geometry to produce the "unprojected" wave functions satisfying the correct (quasi)periodic boundary conditions. We then consider two methods for projecting these wave functions into the LLL. The direct projection produces valid wave functions but can be implemented only for very small systems. The more powerful and more useful projection method of Jain and Kamilla fails in the torus geometry because it does not preserve the periodic boundary conditions and thus takes us out of the original Hilbert space. We have succeeded in constructing a modified projection method that is consistent with both the periodic boundary conditions and the general structure of the CF theory. This method is valid for a large class of states of composite fermions, called "proper states," which includes the incompressible ground states at electron filling factors ν=n2pn+1, their charged and neutral excitations, and also the quasidegenerate ground states at arbitrary filling factors of the form ν=ν∗2pν∗+1, where n and p are integers and ν∗ is the CF filling factor. Comparison with exact results known for small systems for the ground and excited states at filling factors ν=1/3, 2/5, and 3/7 demonstrates our LLL-projected wave functions to be extremely accurate representations of the actual Coulomb eigenstates. Our construction enables the study of large systems of composite fermions on the torus, thereby opening the possibility of investigating numerous interesting questions and phenomena.

UR - http://www.scopus.com/inward/record.url?scp=85038863969&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevB.96.195302

DO - 10.1103/PhysRevB.96.195302

M3 - Article

AN - SCOPUS:85038863969

VL - 96

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 2469-9950

IS - 19

M1 - 195302

ER -