On truncations of the Chalker-Coddington model

TL;DR

This paper investigates truncated versions of the Chalker-Coddington model using supersymmetric reformulation, analyzing their critical properties through analytical and numerical methods to understand their relation to the original model.

Contribution

It introduces and studies a series of truncated loop models and spin chains related to the CC model, including an analytical solution for the first truncation.

Findings

First truncation exhibits qualitative features of the untruncated theory.

Quantitative discrepancies with CC model simulations.

Higher truncations show closer properties to CC but with uncertain finite-size effects.

Abstract

The supersymmetric reformulation of physical observables in the Chalker-Coddington model (CC) for the plateau transition in the integer quantum Hall effect leads to a reformulation of its critical properties in terms of a 2D non-compact loop model or a 1D non-compact $gl(2|2)$ spin chain. Following a proposal by Ikhlef, Fendley and Cardy, we define and study a series of truncations of these loop models and spin chains, involving a finite and growing number of degrees of freedom per site. The case of the first truncation is solved analytically using the Bethe-ansatz. It is shown to exhibit many of the qualitative features expected for the untruncated theory, including a quadratic spectrum of exponents with a continuous component, and a normalizable ground state below that continuum. Quantitative properties are however at odds with the results of simulations on the CC model. Higher truncations are studied only numerically. While their properties are found to get closer to those of the CC model, it is not clear whether this is a genuine effect, or the result of strong finite-size corrections.