Quantum Hall Transition in an Array of Quantum Dots

TL;DR

This paper models a 2D array of quantum dots under a magnetic field using random matrix theory to analyze the Integer Quantum Hall Effect, revealing how level statistics influence conductivity and density of states.

Contribution

It introduces a supersymmetric field theory approach to solve the quantum Hall transition in large quantum dot arrays, connecting level statistics with observable effects.

Findings

Level statistics affect the density of states.

Magnetic field induces the Integer Quantum Hall Effect.

Supersymmetric field theory provides exact solutions in the large N limit.

Abstract

A two-dimensional array of quantum dots in a magnetic field is considered. The electrons in the quantum dots are described as unitary random matrix ensembles. The strength of the magnetic field is such that there is half a flux quantum per plaquette. This model exhibits the Integer Quantum Hall Effect. For $N$ electronic states per quantum dot the limit $N\to\infty$ can be solved by a saddle point integration of a supersymmetric field theory. The effect of level statistics on the density of states and the Hall conductivity is compared with the effect of temperature fluctuations.