Vacuum in a Strong Magnetic Field as a Hyperbolic Metamaterial

As demonstrated by Chernodub, vacuum in a strong magnetic field behaves as Abrikosov vortex lattice in a type-II superconductor. We investigate electromagnetic behavior of vacuum in this state and demonstrate that vacuum behaves as a hyperbolic metamaterial. If the magnetic field is constant, low frequency extraordinary photons experience this medium as a ($3+1$) Minkowski spacetime in which the role of time is played by the spatial $z$ coordinate. Variations of the magnetic field curve this spacetime, and may lead to formation of “electromagnetic black holes.” Since hyperbolic metamaterials behave as diffractionless “perfect lenses,” and large enough magnetic fields probably existed in the early Universe, the demonstrated hyperbolic behavior of early vacuum may have imprints in the large scale structure of the present-day Universe.

Figure 1

(a) Schematic view of the “wired” hyperbolic metamaterial geometry: an array of aligned conductive wires is embedded into a dielectric host. (b) According to Chernodub [2], a similar structure spontaneously develops in vacuum when it is subjected to a high magnetic field. The pictorial diagram is drawn schematically to provide a visual representation of anisotropic character of superconductivity of the $\rho$-meson condensate: it superconducts only in the direction along the magnetic field, while conductivity in the perpendicular direction is zero at $T=0$. Notice that the Meissner effect is absent due to anisotropy of vacuum superconductivity. (c) Hyperbolic dispersion relation of extraordinary photons propagating inside a hyperbolic metamaterial is illustrated as a surface of the constant frequency in $k$ space. When the $z$ coordinate is “timelike,” $k_z$ behaves as effective “energy.”

Figure 2

Numerical simulations of the electromagnetic field behavior at the $B=B_c$ interface located at $z=0$. The field plot showing the absolute value of the electric field (a) and its cross section (b) along the $y=0$ direction [the red line in (a)] illustrates electric field divergence at the interface.