Integrable modification of the critical Chalker-Coddington network model

Abstract

We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity $n$. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an $\mathrm{SU}{(2)}_{r}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{r}/\mathrm{SU}{(2)}_{2r}$ Wess-Zumino-Witten (WZW) coset model for the particular values $n=\ensuremath{-}2\mathrm{cos}[\ensuremath{\pi}/(r+2)]$, where $r$ is a positive integer. The truncated Chalker-Coddington model is the $n=0$ point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.