Fractional Quantum Hall Fluids

In the clean limit, fractional quantum Hall fluids arise when repulsive interactions select a homogeneous incompressible state within a partially filled Landau level. A disorder-connected Hall plateau with a mobility gap need not be exactly translation invariant. The defining data include fractional charge and statistics, quantized Hall response, topology-dependent sectors, and an anomalous edge. The Laughlin state supplies the cleanest derivation; other fractions require additional components, hierarchy structure, or pairing.

Required background. Abelian Chern–Simons theory supplies the infrared gauge action; Chern–Simons level quantization fixes large-gauge consistency; Kubo response supplies the Hall transport limit.

Helpful background. Linking and braiding supplies the worldline interpretation of statistics.

The Laughlin state

For $N$ spin-polarized electrons in the lowest Landau level at filling $\nu=1/m$, with odd $m$, the disk wavefunction is

$$\Psi_m(z_1,\ldots,z_N)= \prod_{i<j}(z_i-z_j)^m \exp\!\left[-\sum_i\frac{|z_i|^2}{4\ell_B^2}\right].$$

Odd $m$ enforces fermionic antisymmetry. The polynomial has degree $m(N-1)$ in each coordinate, giving the sphere relation $N_\phi=m(N-1)$ and shift $\mathcal S=m$. The zeros keep particles apart and make $\Psi_m$ an exact densest zero-energy ground state of suitable short-range pseudopotential Hamiltonians Laughlin 1983.

Incompressibility is dynamical: it requires a neutral and charged excitation gap in the thermodynamic limit. The trial polynomial alone does not determine the Coulomb gap or its survival under Landau-level mixing, finite thickness, and disorder.

Fractional charge and statistics

A quasihole at $\eta$ is created by $\prod_i(z_i-\eta)\Psi_m$. Adiabatic Berry transport and plasma screening give positive electric charge

$$Q_{\rm qh}=\frac{e}{m}$$

relative to the electron fluid. A counterclockwise exchange of two quasiholes contributes the statistical phase $\theta=\pi/m$, after the ordinary electromagnetic Aharonov–Bohm phase is subtracted Arovas, Schrieffer, and Wilczek 1984. A full braid gives $2\theta$.

The infrared action can be written

$$S=\int\left(\frac{m}{4\pi}a\,da-\frac{e}{2\pi}A\,da\right),$$

with wedge products implicit. Integrating out $a$ gives Hall response magnitude $\sigma_{xy}=e^2/(mh)$ for the stated field orientation, $m$ torus sectors, and one chiral boson edge. The overall action sign tracks spacetime orientation; charge and Hall signs must be translated together.

Physical limits

Disorder localizes quasiparticles and stabilizes a Hall plateau over a density interval, but sufficiently strong disorder closes the mobility gap. Finite temperature produces activated longitudinal transport only below scales where variable-range hopping, edge conduction, and inhomogeneity are controlled. Spin, valley, or layer components can change the topological order at the same filling. Therefore a fraction labels density relative to flux, not a unique phase.

Exercise

For a Laughlin $1/5$ state, state the minimal quasihole charge, counterclockwise exchange angle, and torus ground-state count.

Solution

$Q=e/5$, $\theta=\pi/5$ modulo $2\pi$, and the ideal topological field theory has five torus sectors. Finite systems split this manifold exponentially and can mix sectors if translation or topology is not preserved.

References

• Daniel Arovas, John R. Schrieffer, and Frank Wilczek, “Fractional Statistics and the Quantum Hall Effect,” Physical Review Letters 53 (1984) 722–723, doi:10.1103/PhysRevLett.53.722.
• Robert B. Laughlin, “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations,” Physical Review Letters 50 (1983) 1395–1398, doi:10.1103/PhysRevLett.50.1395.