Casimir force variability in one-dimensional QED systems

The Casimir force between two extended charge sources, embedded in a background of one-dimensional massive Dirac fermions, is explored by means of original contour integration techniques. For identical sources with the same (positive) charge, we find that in the nonperturbative region the Casimir interaction between them can reach sufficiently large negative values and simultaneously reveal the features of a long-range force in spite of nonzero fermion mass. For large distances $s$ between sources we recover that their mutual interaction is governed primarily by the structure of the discrete spectrum of a single source, through which it can be tuned to give an attractive, a repulsive, or an almost compensated Casimir force with various rates of the exponential decay, quite different from the standard $exp(-2ms)$ law. A quite different behavior of the Casimir force is found for the system of two extended sources with the opposite charge. In particular, in this case, there is no possibility for a long-range interaction between sources. The asymptotics of the Casimir force follows the standard $exp(-2ms)$ law. Moreover, for small separations between sources the Casimir force, being calculated completely nonperturbatively, for symmetric and antisymmetric cases turns out to be of different sign and also opposite to the classic electrostatic force for such Coulomb sources. By means of the same (dubbed $ln\text{[Wronskian]}$) techniques, the case of two pointlike charge sources is also considered in a self-consistent manner with similar results for the variability of the Casimir force.

Figure 1

The WK contours in the complex energy plane, used for the representation of the vacuum charge density and vacuum energy via contour integrals.

Figure 2

Dependence of the Casimir interaction energy between two wells on the distance $d$ between them for $a=1$: (a) and (b) for $V_0=4.08$; (c) and (d) for $V_0=10$.

Figure 3

Dependence of the Casimir interaction energy between two wells on the distance $d$ between them for $a=1$: (a) and (b) for $V_0=7.4$; (c) and (d) for $V_0=8$.

Figure 4

Different types of the Casimir interaction energy between two $\delta$ wells as a function of the distance $d$ between them for (a) $C=1$, (b) $C=10$, (c) and (d) $C=3$, (e) and (f) $C=5$.

Figure 5

Energy levels in the case $a=1$ and $V_0=4.08$ for the (a) antisymmetric configuration, (b) single-well (black and red lines), and barrier (orange solid and dashed lines) configurations.

Figure 6

Behavior of positive energy levels in the antisymmetric case of the barrier-well type in dependence on $V_0$ for (a) $d=2$ and $a=1$; (b) $d=2$ and $a=2$.

Figure 7

Renormalized vacuum charge density in the antisymmetric case for the following sets of the system parameters: (a) $d=2$, $a=1$, and $V_0=8$; (b) $d=2$, $a=2$, and $V_0=2$.

Figure 8

Behavior of ${\mathcal{E}}_{\mathrm{int}}^R\left(d\right)$ in the antisymmetric case for $a=1$ and (a) $V_0=4.08$, (b) $V_0=7.4$, (c) $V_0=10$, (d) and (e) $V_0=8$.

Figure 9

Behavior of the levels in the barrier-well system without antisymmetry of the potential dependent on the well depth $V_1$ with fixed height of the barrier $V_2=-2$ for (a) and (b): $d=2$ and $a=1$; (c) and (d): $d=2$ and $a=2$.

Figure 10

Different types of the Casimir interaction energy in the antisymmetric case between the $\delta$ barrier and $\delta$ well as a function of the distance $d$ between them for (a) and (b) $C=1$, (c) $C=3$, (d) $C=5$, and (e) $C=10$.

Figure 11

Auxiliary contours for dealing with the integrals (a) $J_1$ and $J_3$ and (b) $J_2$.