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.
Highlights
Electrons in two spatial dimensions at low temperature and subject to a strong transverse magnetic field exhibit a set of remarkable and robust phenomena known as the quantum Hall (QH) effect
We have taken a major step toward a realistic holographic model of the fractional quantum Hall effect, extending the work of [25] and [3]
Based on strong evidence from condensed matter physics, real-world quantum Hall states seem to be governed by a modular group action on a two-dimensional subspace of couplings at low temperatures
Summary
Electrons in two spatial dimensions at low temperature and subject to a strong transverse magnetic field exhibit a set of remarkable and robust phenomena known as the quantum Hall (QH) effect. The fractional quantum Hall data can be explained by the action of the subgroup Γ0(2) ⊂ SL(2, ), it is sufficient to focus on the more symmetric case of an SL(2, )-invariant theory.4 Using this duality, one can map the relatively well understood electrically charged black holes into dyonic solutions, where the dyonic charges correspond to the charge density and magnetic field of the QH system. While these brane systems elegantly model the phase transition between a QH fluid and the ungapped transition states nearby, they only incorporate a single or perhaps a few QH states with particular filling fractions They are not imbued with an SL(2, ) duality and so do not incorporate all the related observed phenomena..
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
