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A quasihole adds one zero at $\eta$ per electron; its local deficit carries positive electric charge $e/m$.^[ref]

Here is a transport counting outline, conditional on an incompressible quantum Hall fluid. Inserting one magnetic flux quantum $\Delta \Phi =h/e$ slowly compared with $\hslash /\Delta$, where $\Delta$ is the excitation energy gap, produces a circulating electric field by Faraday's law. The transverse sheet conductivity sends a radial electric current whose time integral has magnitudeThe opposite electric charge moves to the boundary, so the total electric charge remains conserved. The counted deficit belongs to a collective quasiparticle; no individual electron has been cut into pieces. The argument assumes the gapped fluid and its Hall response, and is not a consequence of the finite two-electron normalization integral.

The fractional quantum Hall effect is quantized transverse sheet conductivity at fractional filling factors, produced by interacting electrons. The Laughlin wavefunction describes the sequence $\nu =1/m$, with odd $m$, including $\nu =1/3$.^[ref]

For an original ideal readout example, take a Hall bar on the $\nu =1/3$ plateau with negligible longitudinal sheet conductivity. The quantum Hall conductivity tensor inversion givesHere $R_K$ is the von Klitzing constant; $R_H$ denotes transverse electrical resistance, rather than the bulk Hall coefficient. Thus a fractional sheet conductivity corresponds to a larger transverse electrical resistance. The fraction labels a collective quantum state: it does not mean that each electron has acquired one third of its usual electric charge.

An incompressible quantum Hall fluid has a bulk spectral gap against charged excitations; disorder can localize excess quasiparticles.^[ref]

Robert Laughlin's 1983 variational method uses pairwise zeros to suppress close electron encounters within the lowest Landau level. For aligned spin, odd $m$ gives fermionic exchange symmetry.^[ref]

Here $\mathcal{N}$ enforces wavefunction normalization, ${\ell }_B$ is the magnetic length, and complex coordinates $z_j=x_j-i{y}_j$ describe electrons in a magnetic field $B\hat{\mathbf{z}}$, $B>0$, with the symmetric gauge. Reversing the orientation conjugates the polynomial without changing probability densities. The Gaussian function confines the wavefunction.

The exchange check is especially simple for two electrons. Writing $w=z_1-z_2$, exchange sends $w\mapsto -w$, so $w^m\mapsto (-1{)}^m{w}^m$. An aligned two-electron spin state is symmetric, requiring the spatial wavefunction to change sign. Hence $m=1,3,5,\dots$ works, whereas $m=2$ would give the wrong sign. At a close encounter, the probability density vanishes as $|w{|}^{2m}$: $m=3$ imposes a sixth-order zero, stronger than the quadratic zero at $m=1$. This is the local geometric mechanism for avoiding a large repulsive Coulomb potential energy.

A useful large-$N$ counting outline follows directly from the polynomial. Each complex coordinate has maximum degree $m(N-1)$. A lowest Landau level orbital proportional to $z^k{e}^{-|z{|}^2/(4{\ell }_B^2)}$ has its areal probability density maximum at $r^2=2k{\ell }_B^2$, by differentiating its logarithm. This suggests droplet area $A\simeq 2\pi mN{\ell }_B^2$, neglecting the boundary correction. Dividing $N/A$ by the degeneracy of a Landau level per area, $1/(2\pi {\ell }_B^2)$, gives filling factor $\nu \simeq 1/m$. This counting is a motivation for the fraction, not an independent proof that the wavefunction minimizes the interacting Hamiltonian.

Robert B. Laughlin proposed correlated many-electron wave functions explaining important states of the fractional quantum Hall effect. The variational construction strongly suppresses configurations in which electrons approach one another while keeping them within the lowest Landau level. Its excited states behave as localized disturbances carrying a fraction of the electron electric charge. The theory connects a microscopic correlated state with an incompressible quantum fluid, rather than interpreting the fractional plateau as an ordinary filled electron band.^[ref]

A physicist.^[ref]