Vacuum energy and repulsive Casimir forces in quantum star graphs

Casimir pistons are models in which finite Casimir forces can be calculated without any suspect renormalizations. It has been suggested that such forces are always attractive, but we present several counterexamples, notably a simple type of quantum graph in which the sign of the force depends upon the number of edges. We also show that Casimir forces in quantum graphs can be reliably computed by summing over the classical orbits, and study the rate of convergence of the periodic orbit expansion. In generic situations where no analytic expression is available, the sign and approximate magnitude of Casimir forces can often be obtained using only the shortest classical orbits.

Figure 1

A rectangular piston in two dimensions (see Ref. 3). In three dimensions there is another length $b_2$ perpendicular to the plane of the figure.

Figure 2

A star graph with a piston installed in each edge. (The pistons are actually points; the edges have no thickness.)

Figure 3

The force on a piston in a star graph with $B$ bonds of length 1, Kirchhoff boundary condition at the center, and either Neumann or Dirichlet boundary condition at each piston is computed using only the shortest periodic orbit [Eq. (24)] and compared with the exact answer. Positive values indicate repulsive forces. All quantities plotted in this and subsequent figures are dimensionless.

Figure 4

The error $\mid E_c^{L_{\max }}-E_c\mid$ in the periodic orbit expansion for the vacuum energy is shown for a star graph with four bonds of length 1.1, 1.6176, 1.2985, and 1.1159, and a Neumann piston at the end of each bond.

Figure 5

The error $\mid E_c^{L_{\max }}-E_c\mid$ is shown for the same quantum graph as in Fig. 4, but with a Dirichlet piston at the end of the first bond.