Modular symmetry, the semicircle law, and quantum Hall bilayers

There is considerable experimental evidence for the existence in quantum Hall systems of an approximate emergent discrete symmetry, Γ0(2)⊂SL(2,Z). The evidence consists of the robustness of the tests of a suite of predictions concerning the transitions between the phases of the system as magnetic fields and temperatures are varied, which follow from the existence of the symmetry alone. These include the universality of and quantum numbers of …