Geometry of random potentials: Induction of 2D gravity in Quantum Hall plateau transitions

TL;DR

This paper establishes a geometric framework linking random potential landscapes in Quantum Hall systems to discrete surfaces, providing insights into the geometric nature of plateau transitions and their relation to 2D gravity.

Contribution

It introduces a novel map from random potentials to discrete surfaces, connecting potential critical points with surface geometry and interpreting network models in terms of Fermi level-dependent parameters.

Findings

Map from potentials to discrete surfaces established

Edge state networks linked to surface geometry

Parameter related to Fermi level influences transition dynamics

Abstract

In the context of the Integer Quantum Hall plateau transitions, we formulate a specific map from random landscape potentials onto 2D discrete random surfaces. Critical points of the potential, namely maxima, minima and saddle points uniquely define a discrete surface $S$ and its dual $S^*$ made of quadrangular and $n-$gonal faces, respectively, thereby linking the geometry of the potential with the geometry of discrete surfaces. The map is parameter-dependent on the Fermi level. Edge states of Fermi lakes moving along equipotential contours between neighbour saddle points form a network of scatterings, which define the geometric basis, in the fermionic model, for the plateau transitions. The replacement probability characterizing the network model with geometric disorder recently proposed by Gruzberg, Kl\"umper, Nuding and Sedrakyan, is physically interpreted within the current framework as a parameter connected with the Fermi level.