Casimir effect on the lattice: U(1) gauge theory in two spatial dimensions

We propose a general numerical method to study the Casimir effect in lattice gauge theories. We illustrate the method by calculating the energy density of zero-point fluctuations around two parallel wires of finite static permittivity in Abelian gauge theory in two spatial dimensions. We discuss various subtle issues related to the lattice formulation of the problem and show how they can successfully be resolved. Finally, we calculate the Casimir potential between the wires of a fixed permittivity, extrapolate our results to the limit of ideally conducting wires and demonstrate excellent agreement with a known theoretical result.

Figure 1

The Casimir problem in two spatial dimensions.

Figure 2

The geometry of the Casimir problem in two spatial dimensions. The shadowed planes indicate the plaquettes ${\mathcal{P}}_{\mathcal{S}}$ where the boundary condition (19) is implemented.

Figure 3

Theoretically calculated zero-point energy (75) between two parallel wires separated by the distance $R$ on $L_s=24$ symmetric lattice. The inset shows the energy with the $R=0$ and $R=L_{s}a$ points excluded.

Figure 4

Excess in expectation values of the components of the field strength tensor squared (76) vs the coordinate $x_1$ normal to the wires at the lattice coupling $\beta =3$ and at fixed permittivity $ϵ=6$. The positions of the wires are marked by the vertical pink lines, the distances between the wires are $R/a=1$, 2, 8 for left, middle and right plots, respectively.

Figure 5

The integral excess in the expectation values of electromagnetic fields (34) and (35) vs permittivity $ϵ$ for $L=24$, $\beta =3$ and $R=4$. The statistical errors are smaller than the size of the symbols. The lines are the fits by the function (77).

Figure 6

Integral expectation values of electromagnetic fields (34) and (35) vs the distance between the wires $R$ in the perfect-metal limit ($ϵ\to \infty$) for the lattice $L_s=24$ at $\beta =3$.

Figure 7

The zero-point potential $V_{\mathrm{Cas}}$ between perfectly conducting wires separated by the distance $R$ in units of the continuum coupling constant $g$ at various lattice coupling constants $\beta$. The lines represent the fits by Eq. (78) with zero, one and two first points removed at each $\beta$.

Figure 8

Fits of the numerical data for the zero-point energy by the lattice potential (81) on the lattice $L_s=24$ at various $\beta$. The wires are perfectly conducting.

Figure 9

The ratio of numerically simulated vs. theoretically calculated prefactors of the Casimir energy at various lattice coupling constants $\beta$ for perfectly conducting wires.

Figure 10

The strength factor $C$ of the zero-point energy (83) as the function of permittivity $ϵ$ of the wires. The solid line corresponds to the function (84).