Past exam of the physics of Nanoelectronic Systems course of the University of Cambridge / 2016 / Paper 1 / 3 / a

The fractional quantum Hall effect occurs when interactions produce an incompressible correlated state within a partially occupied Landau level. The orbital kinetic energy is already quantized; the important degeneracy is among many possible arrangements of electrons within the level. Repulsion can lift that degeneracy and create a many-body excitation gap. The interaction scale is of order $e^2/(4\pi ϵ{\ell }_B)$, where magnetic length ${\ell }_B=(\hslash /(e|B|){)}^{1/2}$; it is not a gap predicted by independent-electron fractional occupation alone.

A composite fermion description incorporates strong correlations by attaching $2p$ statistical flux quanta to each electron, with positive integer $p$. The attachment is a representation of correlated wavefunction phases and zeros, not a literal solenoid attached to a particle. Attaching an even number preserves fermionic exchange character. In a mean-field approximation for positive applied field,The particles consequently experience a reduced effective field. Define the signed effective filling ${\nu }^{\ast }=n_s{\varphi }_0/B^{\ast }$ and electronic filling $\nu =n_s{\varphi }_0/B$. Their relation isWhen $q$ effective levels are filled with the same field orientation, ${\nu }^{\ast }=q$, giving $\nu =q/(2pq+1)$. Reversed effective-field orientation has ${\nu }^{\ast }=-q$, giving $\nu =q/(2pq-1)$. Thus the Jain sequence isFor $p=1$, examples are $1/3$, $2/5$, $3/7$ on one branch and $2/3$, $3/5$ on the reversed branch. Fractional electronic plateaus can be understood as integer effective Hall states of these composite objects. The original proposal explicitly makes this connection.^[ref]

At a gapped fractional state, longitudinal dissipation becomes very small and the Hall magnitude approachesDisorder can localize quasiparticles and allow the plateau to persist over a filling interval, provided it does not overwhelm the interaction gap. The Laughlin wavefunction at $\nu =1/(2p+1)$ is a particularly simple correlated example; its charged excitations carry fractions of the electron charge. More generally the basic quasiparticle charge in the simple Jain sequence has magnitude $e/(2pq\pm 1)$. Such excitations are distinct from noninteracting bare electrons occupying a fraction of a level.

At $\nu =1/(2p)$, the mean effective field $B^{\ast }$ vanishes. This permits a compressible composite-fermion Fermi-sea description, rather than an ordinary integer effective-level gap. The external field remains large: cancellation is in the effective description. Not every fractional state is explained by the simplest integer composite fermion sequence, and the effective masses and gaps reflect interactions rather than simply the GaAs band mass. The utility of the construction is that it organizes many observed fractions and their nearby excitations into one understandable pattern.