Hall Conductivity of Flavor Fields from AdS/CFT

TL;DR

The paper studies the Hall conductivity of massive ${N}=2$ flavor fields at finite density in a strongly coupled ${N}=4$ SYM plasma using AdS/CFT, modeling flavors with probe D7-branes in AdS-Schwarzschild and turning on external electric and magnetic fields. It derives the conductivity tensor from the on-shell DBI action, identifies an effective worldvolume horizon on the D7-brane, and analyzes the drag regime at large mass where the carriers behave as quasi-particles with a universal drag product $\mu M = \dfrac{\pi}{2}\sqrt{\lambda}\,T^2$; the results reproduce known limits and extend to a broad class of Dp/Dq probe-brane systems. The main contributions are explicit formulas for $\sigma_{xx}$ and $\sigma_{xy}$ in terms of dimensionless parameters $e$, $b$, $\rho$, and the function ${\cal F}(e,b)$, the interpretation of the Hall response as arising from flavor degrees of freedom at finite density, and the demonstration of a universal drag behavior tied to a D7-brane worldvolume horizon. These findings illuminate nonperturbative transport in holographic flavor sectors and suggest broad applicability to other holographic defect theories via the Dp/Dq generalization.

Abstract

We use the AdS/CFT correspondence to compute a conductivity associated with massive N=2 supersymmetric hypermultiplet fields at finite baryon density, propagating through an N=4 supersymmetric SU(Nc) Yang-Mills plasma in the large Nc, large 't Hooft coupling limit. We do so by introducing external electric and magnetic fields coupled to baryon number and computing the resulting induced current, from which we extract the conductivity tensor. At large hypermultiplet mass we compute the drag force on the charge carriers. We also compute the product of the drag coefficient with the kinetic mass, and find that the answer is unchanged from the zero density case. The gravitational dual is a probe D7-brane, with a nontrivial worldvolume gauge field configuration, in an AdS-Schwarzschild background. We identify an effective horizon on the D7-brane worldvolume analogous to the worldsheet horizon observed for strings moving in the same background. We generalize our results to a class of theories described by probe D-branes in various backgrounds.

Figures (2)

• Figure 1: Cartoons of D7-brane embeddings, with the coordinates $z$ and $\theta$ indicated. We can imagine that the D3-branes sit at the origin. The semicircle about the origin represents the horizon at $z = z_H$. The boundary is $z=0$. $\theta$ runs from $0$ to $\frac{\pi}{2}$. The horizontal axis is a direction transverse to the D3-branes but parallel to the D7-branes, i.e. one of $X_4, X_5, X_6$ or $X_7$. The vertical axis is $X_8$. (a.) A Minkowski embedding. (b.) A black hole embedding. (c.) A black hole embedding with a "spike" in the $m \rightarrow \infty$ limit.
• Figure 2: Cartoon of the trailing string. The AdS boundary $z=0$ is at the top. The AdS-Schwarzschild horizon $z=z_H$ is at the bottom. The dashed line is the position where the D7-brane ends. The worldsheet horizon on the string, $z_{WH}$, is indicated.