We consider models for the plateau transition in the integer quantum Hall effect. Starting from the network model, we construct a mapping to the Dirac Hamiltonian in two dimensions. In the general case, the Dirac Hamiltonian has randomness in the mass, the scalar potential and the vector potential.Separately, we show that the network model can also be associated with a nearest neighbour, tight-binding Hamiltonian. 73.40.Hm, 71.50.+t, 72.15.Rn Typeset using REVT E X Anderson localisation is central to understanding of the integer quantum Hall effect (IQHE) [1]. In particular, the plateau transitions, between different quantised values for the Hall conductance, reflect delocalisation transitions in each Landau level. Scaling ideas [2] provide a framework for understanding these transitions, and are supported by the results of experiment [3] and of numerical simulation [1]. Progress towards an analytical theory of the critical point, however, remains limited. The simplest starting point for such a theory is to neglect electron-electron interactions and consider a single particle moving in a magnetic field with a disordered impurity potential. In pioneering work, Pruisken and collaborators [4] obtained from this a field-theoretic description, in terms of a σ-model. More recently, in response to the difficulties of extracting quantitative results from the σ-model, several alternative formulations have been explored: Read [5], Lee [6] and Zirnbauer [7] have investigated spin chains; Lee and Wang [8] have considered the replica limit of Hubbard chains; and Ludwig and collaborators [9] have discussed the Dirac equation. The correspondence between Dirac fermions in two space dimensions, and non-relativistic charged particles moving in a magnetic field, stems from the fact that time-reversal symmetry is broken both by a mass term in the two-dimensional Dirac equation [9,10], and by a magnetic field in the Schrödinger equation. Moreover, as emphasised by Ludwig et al, the Hall conductance of Dirac fermions, with fixed Fermi energy, has a jump of e 2 /h, if the fermion mass is tuned through zero. The critical behaviour at this transition depends on the symmetries of the Hamiltonian. The Dirac equation with only a random vectorpotential is particularly amenable to analysis [9,11], since the zero-energy eigenstates are known explicitly [12]. Critical properties are controlled by a line of fixed points, and turn out to be different from those expected at plateau transitions in the IQHE. The line of fixed points, however, is unstable against additional randomness, either in the mass or in the scalar potential, and flow is conjectured [9] to be towards a generic quantum Hall fixed
