Jain states (also called composite-fermion states) are a broad class of many-body quantum states ^[1] that explain most experimentally observed fractions in the Fractional quantum Hall effect. They arise from the composite-fermion theory introduced by Jainendra Kumar Jain, in which strongly interacting two-dimensional electrons in a large magnetic field reorganize into weakly interacting quasiparticles known as composite fermions ^{[2] [3]} .

Contents

• History
• Flux attachment
• Effective magnetic field
• Filling factor sequence
• Jain wavefunctions
• References

The theory maps complicated interacting electrons at fractional filling factors to nearly non-interacting composite fermions at integer filling factors, thereby converting the fractional quantum Hall effect (FQHE) into an effective integer quantum Hall effect of emergent particles.

Jain states successfully predicts the hierarchy of filling fractions

${\displaystyle \nu ={\frac {n}{2pn\pm 1}}}$,

which includes most experimentally observed plateaus such as 2/5, 3/7, 4/9, and their particle-hole conjugates.

History

The first theoretical explanation of the FQHE was provided in 1983 by Robert B. Laughlin, who proposed a variational wavefunction describing filling factors of the form ${\displaystyle \nu =1/(2m+1)}$.

In 1989, Jain proposed the composite-fermion picture, in which electrons bind an even number of magnetic flux quanta. This approach naturally generated a large hierarchy of fractions that matched experiments with high accuracy. The resulting states are now called Jain states.

Filling factor sequence

If composite fermions fill ${\displaystyle n}$ effective Landau levels, the electron filling factor becomes

${\displaystyle \nu ={\frac {n}{2pn\pm 1}}}$.

The "+" sequence gives

1/3, 2/5, 3/7, 4/9, ...

and the "−" sequence gives

2/3, 3/5, 4/7, ...

These fractions account for most observed Hall plateaus.

Jain wavefunctions

The many-body Jain wavefunctions ^[4] are constructed in three steps:

1. Start with an integer quantum Hall Slater determinant ${\displaystyle \Phi _{n}}$.
2. Attach flux using a Jastrow factor ${\displaystyle \prod _{i<j}(z_{i}-z_{j})^{2p}}$.
3. Project to the lowest Landau level using ${\displaystyle {\mathcal {P}}_{\mathrm {LLL} }}$.

The resulting state is given by

$${\displaystyle \Psi _{\nu ={\frac {n}{2pn+1}}}={\mathcal {P}}_{\mathrm {LLL} }\left[\Phi _{n}\prod _{i<j}(z_{i}-z_{j})^{2p}\right].}$$

For ${\displaystyle n=1}$, this reduces to the Laughlin wavefunction, showing that Laughlin states are the first members of the Jain hierarchy.

References