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 ($\nu \approx 2.593$) and exact-diagonalisation studies ($\nu =2.58(3)$). By increasing the amount of data $500$ fold we can now measure closer to the critical point and find an improved estimate $\nu \approx 2.51$. This deviates only $\sim 3\mathrm{\%}$ from the previous two values and is already better than the $\sim 7\mathrm{\%}$ 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 $\nu$.