Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States

With use of spherical geometry, a translationally invariant version of Laughlin's proposed "incompressible quantum fluid" state of the two-dimensional electron gas is formulated, and extended to a hierarchy of continued-fraction Landau-level filling factors $\nu$. Observed anomalies at $\nu =\frac{2}{5},\frac{2}{7}$ are explained by fluids deriving from a $\nu =\frac{1}{3}$ parent.