A holographic model for the fractional quantum Hall effect

Abstract

Experimental data for fractional quantum Hall systems can to a large extent be explained by assuming the existence of a Γ 0 (2) 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.

Pinning of holographic sliding stripes

“…We will only briefly review the relevant aspects of the model here; for more details, see [27]. It is noteworthy to mention other closely related holographic constructions [36][37][38][39][40][41][42][43][44], which have significantly contributed to the understanding of the current setting.…”

Flux and Hall states in ABJM with dynamical flavors

“…These models are endowed with an SL(2, Z) duality and, as a consequence, they capture some observed features of QH physics. However, it is very difficult to engineer these types of models to have a mass gap; [8] is so far the only example of a gapped model in this class.…”

“…Moreover, the equation for the scalar field X m becomes [69]: 8) where the source for the scalar X m is:…”

Revisiting Wilson loops for nonrelativistic backgrounds

“…On the other hand, it is important to notice that not all curvature singularities affect physical quantities [41][42][43], therefore, these spaces are not severely ill-defined.…”