Edge Excitations of an Incompressible Fermionic Liquid in a Disorder Magnetic Field

Abstract

The model of lattice fermions in 2+1 dimensional space is formulated, the critical states of which are lying in the basis of such physical problems, as 3D Ising Model(3DIM) and the edge excitations in the Hall effect. The action for this exitations coincides with the action of so called sign-factor model in 3DIM at one values of its parameters, and represent a model for the edge excitations, which are responsible for the plato transitions in the Hall effect, at other values. The model can be formulated also as a loop gas models in 2D, but unlikely the O(n) models, where the loop fugacity is real, here we have directed (clochwise and conterclochwise) loops and phase factors $e^{\pm 2π{p \over q} i}$ for them. The line of phase transitions in the parametric space will be found and corresponding continuum limits of this models will be constructed. It appears, that besides the ordinary critical line, which separates the dense and diluted phases of the models(like in ordinary O(n) models), there is a line, which corresponds to the full covering of the space by curves. The N=2 twisted superconformal models with SU(2)/U(1) coset model coupling constant $k={q \over p}-2$ describes this states.