Composite Fermions and Hierarchical Hall States

Composite-fermion and hierarchy constructions reorganize strongly correlated electrons into quasiparticles moving in a reduced effective magnetic field. They explain prominent sequences of fractions and supply trial states, but flux attachment is a gauge transformation, not literal binding of measurable flux tubes. Lowest-Landau-level projection and the emergent gauge constraint remain essential.

Required background. Landau-level projection supplies magnetic translations and the projected Hilbert space; fractional Hall fluids supplies Laughlin correlations and quasiparticle data.

Helpful background. Parton constructions supplies the enlarged-Hilbert-space constraint.

Jain sequences from effective flux

Attach $2s$ units of statistical flux to each electron. At mean field the composite fermion sees

$$B^*=B-2s\phi_0 n, \qquad \phi_0=\frac{h}{e}.$$

If $p$ composite-fermion Landau levels are filled, $n=p|B^*|/\phi_0$. For $B^*$ parallel to $B$ this gives

$$\nu=\frac{n\phi_0}{B}=\frac{p}{2sp+1},$$

and the opposite effective-field orientation gives $p/(2sp-1)$. Microscopic trial states are

$$\Psi_{p/(2sp+1)}=\mathcal P_{\rm LLL}\, \Phi_p\prod_{i<j}(z_i-z_j)^{2s},$$

where $\Phi_p$ fills $p$ integer Landau levels and $\mathcal P_{\rm LLL}$ is not optional. Projection can change short-distance correlations and must be specified in numerical comparisons Jain 1989.

Gauge fluctuations and half filling

The Chern–Simons gauge field enforces flux attachment. Its fluctuations generate singular interactions and ensure that composite-fermion currents are not directly the physical electron current. At $\nu=1/2$ with two attached flux quanta, the mean effective field vanishes and a composite-fermion Fermi surface emerges. Landau-level particle–hole symmetry motivates a Dirac composite-fermion formulation, while Landau-level mixing can break that symmetry Son 2015.

The Fermi wave vector, geometric response, and pairing instabilities must be inferred from gauge-invariant electron observables. Mean-field composite fermions are an organizing description, not asymptotic free particles.

Partons and hierarchy data

In a parton construction, an electron operator is written as a product, for example $c=f_1f_2\cdots f_r$, and each parton occupies an integer Hall band. The decomposition enlarges the Hilbert space; a local gauge constraint projects back to physical states. Integrating out gapped partons produces a multicomponent Chern–Simons theory whose $K$ matrix determines the actual charge and statistics. Different decompositions can represent the same order, and an unprojected mean-field Chern number is not a physical invariant.

Spin, valley, layer polarization, Landau-level mixing, and finite width can select different hierarchy states at the same $\nu$. The correct claim includes the shift, quasiparticle data, and edge signature, not only the filling fraction.

Exercise

Use $s=1$ and $p=3$ to find the two Jain fractions corresponding to parallel and antiparallel $B^*$.

Solution

The parallel sequence gives $\nu=p/(2p+1)=3/7$. The opposite sequence gives $\nu=p/(2p-1)=3/5$. Their edge structures and shifts differ even though both arise from three filled composite-fermion levels.

References

• Jainendra K. Jain, “Composite-Fermion Approach for the Fractional Quantum Hall Effect,” Physical Review Letters 63 (1989) 199–202, doi:10.1103/PhysRevLett.63.199.
• Dam Thanh Son, “Is the Composite Fermion a Dirac Particle?” Physical Review X 5 (2015) 031027, doi:10.1103/PhysRevX.5.031027.