Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy

TL;DR

<3-5 sentence high-level summary> This work builds three holographic realizations of the fractional quantum Hall effect in string theory. Model I embeds edge states in the ABJM Chern-Simons theory using D4 and D8 branes to reproduce Laughlin-type Hall conductivities and to model interfaces between different filling fractions. Model II shows that holography of pure Chern-Simons theory realizes the level-rank duality and allows computation of topological entanglement entropy within AdS/CFT. Model III extends to hierarchical FQHE by leveraging resolved C^2/Z_n orbifolds to generate a network of U(1) gauge fields and a K-matrix description, linking continued fractions to string-theoretic geometric data. Collectively, these constructions illuminate how holographic duality can capture edge physics, topological invariants, and hierarchy in FQHE systems with potential connections to condensed-matter realizations.

Abstract

We present three holographic constructions of fractional quantum Hall effect (FQHE) via string theory. The first model studies edge states in FQHE using supersymmetric domain walls in N=6 Chern-Simons theory. We show that D4-branes wrapped on CP^1 or D8-branes wrapped on CP^3 create edge states that shift the rank or the level of the gauge group, respectively. These holographic edge states correctly reproduce the Hall conductivity. The second model presents a holographic dual to the pure U(N)_k (Yang-Mills-)Chern-Simons theory based on a D3-D7 system. Its holography is equivalent to the level-rank duality, which enables us to compute the Hall conductivity and the topological entanglement entropy. The third model introduces the first string theory embedding of hierarchical FQHEs, using IIA string on C^2/Z_n.

Figures (6)

• Figure 1: Experimental setup of the quantum Hall effect: a sample of a two-dimensional electron gas is placed on the $xy$-plane, with a magnetic field $B_z$ perpendicular to the plane and an electric field $E_y$ along the $y$-direction. The quantum Hall current flows along the $x$-direction, perpendicular to the $\vec{E}$-field.
• Figure 2: Edge state from intersecting $M$ D4-branes wrapped on $\mathbb{CP}^1$ with $N$ D2-branes: the gauge group on D2-branes changes from $U(N)_{k}\times U(N)_{-k}$ to $U(N-M)_{k}\times U(N)_{-k}$ when crossing the edge.
• Figure 3: Edge state from intersecting $l$ D8-branes wrapped on $\mathbb{CP}^3$ with $N$ D2-branes: the gauge group on D2-branes changes from $U(N)_{k}\times U(N)_{-k}$ to $U(N)_{k-l}\times U(N)_{-k}$ when crossing the edge.
• Figure 4: A bending brane that models the pair of edge states perpendicular to $\vec{E}$-field.
• Figure 5: An infinite brane that models the single edge that is parallel to the $\vec{E}$-field.