Supersymmetry and Localization in the Quantum Hall Effect

TL;DR

This paper investigates the localization transition in the integer quantum Hall effect using a supersymmetric path integral approach and DMRG, providing an exact solution in a supersymmetric subspace and analyzing critical behavior at the transition.

Contribution

It introduces a supersymmetric path integral formulation for quantum Hall localization and applies DMRG to analyze critical phenomena.

Findings

Exact solution in a supersymmetric subspace

Identification of critical behavior at the plateau transition

Mapping localization to a non-Hermitian Hamiltonian

Abstract

We study the localization transition in the integer quantum Hall effect as described by the network model of quantum percolation. Starting from a path integral representation of transport Green's functions for the network model, which employs both complex (bosonic) and Grassman (fermionic) fields, we map the problem of localization to the problem of diagonalizing a one-dimensional non-Hermitian Hamiltonian of interacting bosons and fermions. An exact solution is obtained in a restricted subspace of the Hilbert space which preserves boson-fermion supersymmetry. The physically relevant regime is investigated using the density matrix renormalization group (DMRG) method, and critical behavior is found at the plateau transition.