Quantized Casimir Force

We investigate the Casimir effect between two-dimensional electron systems driven to the quantum Hall regime by a strong perpendicular magnetic field. In the large-separation ($d$) limit where retardation effects are essential, we find (i) that the Casimir force is quantized in units of $3\hslash c{\alpha }^2/8{\pi }^2{d}^4$ and (ii) that the force is repulsive for mirrors with the same type of carrier and attractive for mirrors with opposite types of carrier. The sign of the Casimir force is therefore electrically tunable in ambipolar materials such as graphene. The Casimir force is suppressed when one mirror is a charge-neutral graphene system in a filling factor $\nu =0$ quantum Hall state.

Figure 1

Feynman diagrams for the ground state correlation energy between mirrors, where green wavy lines represent the photon Green function $\mathcal{D}$ and filled blue bubbles represent the current-current correlation function $\Pi$. (a) Leading-order diagram for the longitudinal contribution $E_l$ to the ground state correlation energy. The transverse contribution $E_t$ is given by replacing ${\mathcal{D}}_l$, ${\Pi }_l\to {\mathcal{D}}_t$, ${\Pi }_t$. (b) Leading-order diagram for the Hall contribution $E_H$; this diagram is to be counted twice because an equivalent diagram results from interchanging ${\mathcal{D}}_l$ and ${\mathcal{D}}_t$ and using ${\Pi }_H\equiv {\Pi }_{xy}=-{\Pi }_{yx}$. (c) Diagram for the current-current correlation function. The first diagram on the right represents the diamagnetic contribution, whereas the second represents the paramagnetic contribution.

Figure 2

Casimir force $F$ normalized by the perfect metal value $F_0=-\hslash c{\pi }^2/240d^4$ as a function of separation $d$ in a strong magnetic field. Solid lines correspond to numerical results, and dashed lines correspond to the analytic large $d$ limit in Eq. (3). The case with opposite-sign carrier densities on the two graphene sheets $n_L=1.2\times {10}^{12}{\mathrm{cm}}^{-2}=-n_R$ is depicted by the red (grey) curve, whereas the case with the same-sign carrier density $n_L=n_R=1.2\times {10}^{12}{\mathrm{cm}}^{-2}$ is depicted by the black (dark) curve. At these carrier densities, the magnetic fields $B=25$, 8.3 T correspond to filling factors $|\nu |=2$, 6. These calculations are valid for $d\gg l_B$, where $l_B=257/\sqrt{B[T]}\AA$ is the magnetic length, and weak disorder.

Figure 3

(a) Casimir force versus separation for magnetic field strengths $B=10$, 20, 30 T between two charge-neutral graphene sheets. (b) Casimir energy between a charge-neutral graphene sheet and a half-space composed of an insulator with a dielectric constant $ϵ$, normalized by the perfect metal value $E_0=-\hslash c{\pi }^2/720d^3$. In the large $d$ regime where the suppression effect becomes prevalent, the insulator is characterized by its low-frequency dielectric behavior, and we take $ϵ\simeq 4$, corresponding to ${\mathrm{SiO}}_2$ or hexagonal boron nitride ($\mathrm{h}\mathrm{\text{−}}\mathrm{BN}$).