  Weyl semimetals are recently discovered materials supporting emergent
relativistic fermions in the vicinity of band-crossing points known as Weyl
nodes. The positions of the nodes and the low-energy spectrum depend
sensitively on the time-reversal (TR) and inversion (I) symmetry breaking in
the system. We introduce the concept of Weyl metamaterials where the particles
experience a 3d curved geometry and gauge fields emerging from smooth spatially
varying TR and I breaking fields. The Weyl metamaterials can be fabricated from
semimetal or insulator parent states where the geometry can be tuned, for
example, through inhomogeneous magnetization. We derive an explicit connection
between the effective geometry and the local symmetry-breaking configuration.
This result opens the door for a systematic study of 3d designer geometries and
gauge fields for relativistic carriers. The Weyl metamaterials provide a route
to novel electronic devices as highlighted by a remarkable 3d electron lens
effect.
