A holographic model for the fractional quantum Hall effect

TL;DR

This paper develops a holographic model based on modular symmetry to explain fractional quantum Hall effects, capturing key features like gapped states and Hall conductivity matching filling fractions.

Contribution

It introduces an SL(2,Z)-invariant Einstein-Maxwell-axio-dilaton holographic model that reproduces fractional quantum Hall phenomena and analyzes gauge field fluctuations around black hole solutions.

Findings

Black hole solutions with Hall conductivity matching filling fractions

Preservation of the energy gap under SL(2,Z) transformations

Identification of an accessory singularity in fluctuation analysis

Abstract

Experimental data for fractional quantum Hall systems can to a large extent be explained by assuming the existence of a modular symmetry group commuting with the renormalization group flow and hence mapping different phases of two-dimensional electron gases into each other. Based on this insight, we construct a phenomenological holographic model which captures many features of the fractional quantum Hall effect. Using an SL(2,Z)-invariant Einstein-Maxwell-axio-dilaton theory capturing the important modular transformation properties of quantum Hall physics, we find dyonic diatonic black hole solutions which are gapped and have a Hall conductivity equal to the filling fraction, as expected for quantum Hall states. We also provide several technical results on the general behavior of the gauge field fluctuations around these dyonic dilatonic black hole solutions: We specify a sufficient criterion for IR normalizability of the fluctuations, demonstrate the preservation of the gap under the SL(2,Z) action, and prove that the singularity of the fluctuation problem in the presence of a magnetic field is an accessory singularity. We finish with a preliminary investigation of the possible IR scaling solutions of our model and some speculations on how they could be important for the observed universality of quantum Hall transitions.