AC Transport at Holographic Quantum Hall Transitions

TL;DR

We study AC transport at quantum Hall critical points using holographic duals built from intersecting branes and AdS/CFT. By analyzing probe-brane actions (D3-D7 and D6 in ABJM) with Born-Infeld and Chern-Simons terms, we derive DC conductivities from horizon data, obtain analytical AC results in key limits, and perform numerical calculations at finite temperature. A striking outcome is the strong agreement with Sachdev's field-theory model and a universal high-frequency conductivity across the class of transitions, tied to the horizon tension of the brane. The work highlights the utility of holography for 2+1D quantum critical transport and motivates further study of poles and cyclotron resonances, with potential experimental relevance.

Abstract

We compute AC electrical transport at quantum Hall critical points, as modeled by intersecting branes and gauge/gravity duality. We compare our results with a previous field theory computation by Sachdev, and find unexpectedly good agreement. We also give general results for DC Hall and longitudinal conductivities valid for a wide class of quantum Hall transitions, as well as (semi)analytical results for AC quantities in special limits. Our results exhibit a surprising degree of universality; for example, we find that the high frequency behavior, including subleading behavior, is identical for our entire class of theories.

Figures (7)

• Figure 1: $\Sigma'$ as a function of $\hat{\omega}$
• Figure 2: Longitudinal conductivity $\sigma_{xx}$ and Hall conductivity $\sigma_{xy}$ a) from the D-brane model and b) from Sachdev's analysis sachdev. Real parts are depicted by the solid curves and imaginary parts by the dashed lines. The plots in a) use parameter values $\hat{\rho}=15.1,\hat{B}=1.4,\tau=0.3$ of the model.
• Figure 3: Finite temperature probe embeddings (from Davis:2008nv). The vertical axis is $x^9=r\cos \psi$, and the horizontal is $r\sin \psi$. Each embedding asymptotes to some particular $x^9$ value, $x^9=L$. The black hole embeddings shown in (b) can also be characterized by the angle $\psi_0$ at which they enter the horizon. See Davis:2008nv for discussion of the meaning of $r^{\rm crit}_+$.
• Figure 4: The real (solid) and imaginary (dashed) parts of $G(\omega)$. Note the resemblance of the real part to Fig. \ref{['fig1']}.
• Figure 5: Real and imaginary parts of the (a) longitudinal and (b) Hall conductivity, plotted over a range of negative to positive frequencies to illustrate symmetry and antisymmetry. In (a), the top (black) curve is the real part and bottom (blue) curve is the imaginary part of the longitudinal conductivity $\sigma_{xx}$, for parameter values $\hat{\rho}=0,\hat{B}=1,\tau=1$. The value at $\omega=0$ corresponds to the DC conductivity $\sigma_{xx}=1/\sqrt{1+\hat{B}^2}$, a special case of (\ref{['vp']}). In (b), the symmetric (black) curve is the real part and the antisymmetric (blue) curve is the imaginary part of the Hall conductivity $\sigma_{xy}$, for parameter values $\hat{\rho}=0.25,\hat{B}=1,\tau=1$.