  Lattices of topological defects, such as Abrikosov lattices and domain wall
lattices often arise as metastable ground states in higher-dimensional field
theoretical models. We demonstrate that such lattice states may be described as
extra-dimensional metamaterials via higher-dimensional effective medium theory.
A 4+1 dimensional extension of Maxwell electrodynamics with a compactified
time-like dimension has been considered as an example. It is demonstrated that
from the point of view of macroscopic electrodynamics an Abrikosov lattice
state in such a 4+1 dimensional spacetime may be described as a uniaxial
hyperbolic metamaterial. Extraordinary photons perceive this medium as a 3+1
dimensional Minkowski spacetime in which one of the original spatial dimensions
(the optical axis of the metamaterial) plays the role of a new time-like
coordinate. Since metric signature of this effective space-time depends on the
Abrikosov lattice periodicity, the described model may be useful in studying
metric signature transitions. A particular kind of metric signature transition
understood as a macroscopic medium effect may emulate cosmological inflation.
