Holographic graphene in a cavity

The effective strength of electromagnetism interactions can be controlled by confining the fields to a cavity and these effects might be used to push graphene into a strongly coupled regime. We study the similar D3/probe D5 system on a compact space and discuss the gravity dual for a cavity between two mirrors. We show that the introduction of a conformal symmetry breaking length scale introduces a mass gap on a single D5 sheet. Bilayer configurations display exciton condensation between the sheets. There is a first-order phase transition away from the exciton condensate if a strong enough magnetic field is applied. We finally map out the phase structure of these systems in a cavity with the presence of mirror reflections of the probes—a mass gap may form through exciton condensation with the mirror image.

Figure 1

The embedding (extending to infinite $\rho$) of a single D5 brane for a massless quark in the AdS-soliton background is shown for $r_0=1$. The dual of the compact space therefore closes off on the smaller circle shown—note the D5 brane embedding is regular avoiding this line. We also show the solution of (21) as the larger circle with the tension chosen so that the interval between the walls has radius $R^2\pi /r_0$—the D5 embedding hits this cut off on the space.

Figure 2

U-shaped $\mathrm{D}5/\overline{D}5$ configurations in the AdS-soliton background. The space has circumference $\pi$ in the $z$-direction-we repeat the space to show configurations wrapping both ways around the circle. Note that configurations which reach down to $r_0=1$ (left-hand vertical line) describe defects separated by $\pi /2$ in $z$. We also plot the solution of (21) (in bold) with the tension chosen so that the interval between the walls is $\pi$-the probe configurations hit this boundary. If the figure is viewed as that for an interval between two mirrors then configurations corresponding to exciton condensation with the mirror partner exist if the probe lies within $\pi /4$ of the mirror.

Figure 3

The regularized energy of two D5s in the configuration of Fig 1 (top line which is independent of $z$) and that of the U-shaped joined embedding (lower line) against separation in $z$ (which cannot exceed $\pi /2$). The U shaped configurations are always preferred.

Figure 4

Phase diagram of bilayer D5s in AdS with one compact direction and an applied $B$ field. Phase A is U-shaped configurations. In phase B the probes separate and take up configurations similar to Fig 1.

Figure 5

The phase diagram of the bilayer theory in an interval between two mirrors of separation $\pi$. $d_1$ and $d_2$ measure the distance from one mirror to the first and second defect respectively. We have marked the lines $d_{1/2}=\pi /4,3\pi /4$ because these are the separations within which condensation with the mirror image are possible. In phase A both D5s condense with their mirror images. In phase B the two D5s form a U-shaped configuration. In phase C the probe nearest the mirror displays exciton condensation with its mirror partner whilst the other probe takes up the lone configuration of Fig. 1.