Designer curved-space geometry for relativistic fermions in weyl metamaterials

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 and inversion 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 time-reversal-And inversion-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.