Draft:Jain states

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.

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.

The many-body Jain wavefunctions^[15] 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.