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Modular invariance, universality and crossover in the quantum Hall effect

An analytic form for the conductivity tensor in crossover between two quantum Hall plateaux is derived, which appears to be in good agreement with existing experimental data. The derivation relies on an assumed symmetry between quantum Hall states, a generalisation of the law of corresponding states from rational filling factors to complex conductivity, which has a mathematical expression in terms of an action of the modular group on the upper-half complex conductivity plane. This symmetry implies universality in quantum Hall crossovers. The assumption that the $β$-function for the complex conductivity is a complex analytic function, together with some experimental constraints, results in an analytic expression for the crossover, as a function of the external magnetic field.