Critical points of 2d disordered Dirac fermions: the Quantum Hall Transitions revisited

TL;DR

This paper revisits the quantum Hall transition by proposing a two-stage renormalization group flow involving super spin-charge separation, leading to new fixed points and analytical predictions for multifractal exponents that align with numerical data.

Contribution

It introduces a novel two-stage RG flow approach with superalgebra techniques to resolve previous lack of fixed points in disordered Dirac fermion theories.

Findings

Derived explicit forms of actions with scalar fields and symplectic fermions.

Calculated multifractal exponents q(1-q)/4 and q(1-q)/8 for QHT and SQHT.

Results closely match existing numerical estimates.

Abstract

We propose a resolution of the renormalization group flow for the disordered Dirac fermion theories describing the quantum Hall transition (QHT) and spin Quantum Hall transition (SQHT), which previously revealed no perturbative fixed points at 1-loop and higher. The approach involves carrying out the flow in 2 stages, the first stage utilizing a new form of super spin-charge separation to flow to gl(1|1)_N and osp(2|2)_{-2N} supercurrent algebra theories, where N is the number of copies. This leads to the unconventional feature that at the critical point the exponents depend on the original number of copies N. In the second stage, additional forms of disorder are incorporated as dimension zero logarithmic operators, and the resulting actions have explicit forms in terms of two scalar fields and a symplectic fermion. Under some assumptions, the multi-fractal exponents are computed with the result q(1-q)/4 and q(1-q)/8 for the QHT and SQHT respectively, and are within a few percent of numerical estimates.