Real-space renormalisation approach to the Chalker–Coddington model revisited: Improved statistics

The real-space renormalisation group method can be applied to the Chalker–Coddington model of the quantum Hall transition to provide a convenient numerical estimation of the localisation critical exponent, $\nu$. Previous such studies found $\nu \sim 2.39$ which falls considerably short of the current best estimates by transfer matrix ($\nu =2.593\frac{+0.005}{-0.006}$) and exact-diagonalisation studies ($\nu =2.58\left(3\right)$). By increasing the amount of data 500 fold we can now measure closer to the critical point and find an improved estimate $\nu =2.51\frac{+0.11}{-0.11}$. This deviates only $\sim$3% from the previous two values and is already better than the $\sim$7% accuracy of the classical small-cell renormalisation approach from which our method is adapted. We also study a previously proposed mixing of the Chalker–Coddington model with a classical scattering model which is meant to provide a route to understanding why experimental estimates give a lower $\nu \sim 2.3$. Upon implementing this mixing into our RG unit, we find only further increases to the value of $\nu$.