One-dimensional flows in the quantum Hall system

We construct the c function whose gradient determines the RG flow of the conductivities (ifσxy and σxx) for a quantum Hall system, subject to two assumptions. (1) We take the flow to be invariant with respect to the infinite discrete symmetry group, ΓH, recently proposed by several workers to explain the ‘superuniversality’ of the delocalization exponents in these systems. (2) We also suppose the flow to be ‘quasi-holomorphic’ (which we make precise) in the sense that it is as close as possible to a one-dimensional flow in the complex parameter σxy + iσxx. These assumptions together with the known asymptotic behaviour for large σxx, completely determine the c function, and so the phase diagram, for these systems. A complete description of the RG flow also requires a metric in addition to the c function, and we identify the features which are required for this by the RG. A similar construction produces the c function for other systems enjoying an infinite discrete symmetry, such as for supersymmetric QED.