Quantum Hall Effect in a Holographic Model

TL;DR

This work constructs a holographic model of strongly coupled 2+1d fermions using a D7-brane probe in the D3-brane background and stabilizes embeddings with worldvolume fluxes. It identifies two embedding classes: Minkowski (MN) embeddings, which avoid the horizon and realize quantum Hall states with discrete filling fractions, and black-hole (BH) embeddings, which describe metallic states with nonzero longitudinal conductivity. The MN states exhibit a topologically protected, temperature-independent Hall conductivity given by $oldsymbol{\sigma_{xy}= u/(2\,pi)}$, where the filling fraction $oldsymbol{ u}$ is discretized by flux quantization on the wrapped $S^2$'s; varying the magnetic field drives a continuous MN→BH transition to a metallic phase. Thermodynamics shows a first-order phase transition between the quantum Hall and metallic states at a critical temperature, highlighting a holographic realization of quantum Hall physics in a strongly coupled system with a rich phase structure. The results illuminate how topological flux quantization and holographic transport combine to yield robust quantum Hall behavior in a brane construction, while outlining avenues to connect to edge states and fractional quantum Hall phenomena.

Abstract

We consider a holographic description of a system of strongly coupled fermions in 2+1 dimensions based on a D7-brane probe in the background of D3-branes, and construct stable embeddings by turning on worldvolume fluxes. We study the system at finite temperature and charge density, and in the presence of a background magnetic field. We show that Minkowski-like embeddings that terminate above the horizon describe a family of quantum Hall states with filling fractions that are parameterized by a single discrete parameter. The quantization of the Hall conductivity is a direct consequence of the topological quantization of the fluxes. When the magnetic field is varied relative to the charge density away from these discrete filling fractions, the embeddings deform continuously into black-hole-like embeddings that enter the horizon and that describe metallic states. We also study the thermodynamics of this system and show that there is a first order phase transition at a critical temperature from the quantum Hall state to the metallic state.

Figures (7)

• Figure 1: Brane configuration for (a) massless fermions and (b), (c) massive fermions.
• Figure 2: MN embeddings: (a) at fixed $d=0.01$, $\Delta_{+}=-1$, and with $r_{T}=0, 0.079, 0.1, 0.2, 0.3$ from bottom to top, and (b) at fixed $r_{T}=0.05734$, $\Delta_{+}=-1$, and with $d=0,0.00015,0.00017,0.0002,0.0005,0.001,0.0017$ from top to bottom.
• Figure 3: The mass gap ($r_0$) as a function of the magnetic field (at $r_T=0.01$). Fitting to $r_0 = \alpha + \beta \sqrt{b} + \gamma \, b$ we find $\alpha = 5.695\times 10^{-6}$, $\beta = 1.176$, and $\gamma = 6.719$.
• Figure 4: BH embeddings at fixed $r_{T}=0.01$, $d=0$, $b=0$, $\Delta_+=-1$, and with $f_1=1/\sqrt{2}$ (black), $0.5387$ (red), $0.4$ (green), $0.308$ (blue), and $0$ (brown).
• Figure 5: BH embeddings with $f_1=0$, $d=0.01$, $b=0.0044$, $\Delta_{+}=-1$, and (a) $r_{T}=0.06$ (b) $r_T=0.07$.