Casimir effect for Dirac lattices

We consider polarizable sheets, which recently received some attention, especially in the context of the dispersion interaction of thin sheets like graphene. These sheets are modeled by a collection of delta function potentials and resemble zero-range potentials, which are known in quantum mechanics. We develop a theoretical description and apply the so-called TGTG formula to calculate the interaction of two such lattices. Thereby, we make use of the formulation of the scattering of waves off such sheets provided earlier. We consider all limiting cases, providing a link to earlier results. Also, we discuss the relation to the pairwise summation method.

Figure 1

The vacuum energy per unit as a function of distance between lattices for different values of the coupling. Here lattices are exactly opposite to one another, $c=0$. From left to right $g/a=0.01$, 0.05, 0.1. Inset: The same for two parallel Dirac chains.

Figure 2

The factor $\eta =E(c\ne 0)/E(c=0)$ is plotted as a function of $c/a$ for $g/a=-0.1$ and different separations $\beta =b/a$ of the chains. From bottom to top $\beta =0.1$, 0.2, 0.4, 0.6.

Figure 3

Possible transitions from one-dimensional lattices (chains) and two-dimensional lattices of delta functions for large and small separation. For convenience, the two-dimensional lattices are shown for $c=0$, i.e., without displacement. For large separation, the lattices turn into continuous plates and the chains into wires. For small separation both turn into the closest points. Also shown are the transitions from spheres and cylinders to plates and wires.

Figure 4

The difference between exact result (48) (dashed line) and pairwise summation of individual Casimir-Polder interactions for chains (73), $g/a=0.1$.