Topological Casimir effect in Maxwell electrodynamics on a compact manifold

We study the topological Casimir effect, in which extra vacuum energy emerges as a result of the topological features of the theory, rather than due to the conventional fluctuations of the physical propagating degrees of freedom. We compute the corresponding topological term in quantum Maxwell theory defined on a compact manifold. Numerically, the topological effect is much smaller than the conventional Casimir effect. However, we argue that the topological Casimir effect is highly sensitive to an external magnetic field, which may help to discriminate it from the conventional Casimir effect. It is quite amazing that the external magnetic field plays the role of the $\theta$ state, similar to a $\theta$ vacuum in QCD, or $\theta =\pi$ in topological insulators.

Figure 1

The topological pressure on the 4d system of parallel plates as a function of $\tau \equiv 2\beta L_3/e^2{L}_1{L}_2$. Pressure is measured in units $\frac{2}{L_1^2{L}_2^2{e}^2}$.

Figure 2

Topological Casimir pressure is plotted in units $\frac{2}{L_1^2{L}_2^2{e}^2}$ for different $\tau$. A clear $2\pi$ periodicity is seen and local extrema are between odd and even integer multiples of $\pi$.

Figure 3

A numerical plot of the induced magnetic field in units $\frac{1}{L_1{L}_{2}e}$ as a function of ${\theta }_{\mathrm{eff}}$. The oscillatory behavior becomes more pronounced for large $\tau$.

Figure 4

Numerical plot of magnetic susceptibility ${\chi }_{\mathrm{mag}}$ as a function of ${\theta }_{\mathrm{eff}}$ for different values of $\tau$. It oscillates with ${\theta }_{\mathrm{eff}}$ as it should, and it does not vanish even at zero external magnetic field at ${\theta }_{\mathrm{eff}}=0$ due to topological fluctuations, similar to well-studied cases of 2d QED and 4d QCD.