Conservation laws in the quantum Hall Liouvillian theory and its generalizations

TL;DR

This paper explores the conservation laws and geometric structures in the quantum Hall Liouvillian theory, including its generalizations to noninteracting bosonic models with extended symmetries, and investigates their implications for localization and scaling near the quantum Hall transition.

Contribution

It introduces large-N generalizations of the Liouvillian theory, analyzes their conservation laws, and compares numerical results to the original fermionic-based theory.

Findings

Large-N bosonic models preserve key geometric conservation laws.

Logarithmic corrections to diffusive behavior emerge at order 1/N.

Scaling at the quantum Hall plateau transition depends on Landau level geometry.

Abstract

It is known that the localization length scaling of noninteracting electrons near the quantum Hall plateau transition can be described in a theory of the bosonic density operators, with no reference to the underlying fermions. The resulting ``Liouvillian'' theory has a $U(1|1)$ global supersymmetry as well as a hierarchy of geometric conservation laws related to the noncommutative geometry of the lowest Landau level (LLL). Approximations to the Liouvillian theory contain quite different physics from standard approximations to the underlying fermionic theory. Mean-field and large-N generalizations of the Liouvillian are shown to describe problems of noninteracting bosons that enlarge the $U(1|1)$ supersymmetry to $U(1|1) \times SO(N)$ or $U(1|1) \times SU(N)$. These noninteracting bosonic problems are studied numerically for $2 \leq N \leq 8$ by Monte Carlo simulation and compared to the original N=1 Liouvillian theory. The $N>1$ generalizations preserve the first two of the hierarchy of geometric conservation laws, leading to logarithmic corrections at order 1/N to the diffusive large-N limit, but do not preserve the remaining conservation laws. The emergence of nontrivial scaling at the plateau transition, in the Liouvillian approach, is shown to depend sensitively on the unusual geometry of Landau levels.