  We present a study of "nodal semimetal" phases, in which non-degenerate
conduction and valence bands touch at points (the "Weyl semimetal") or lines
(the "line node semimetal") in three-dimensional momentum space. We discuss a
general approach to such states by perturbation of the critical point between a
normal insulator (NI) and a topological insulator (TI), breaking either time
reversal (TR) or inversion symmetry. We give an explicit model realization of
both types of states in a NI--TI superlattice structure with broken TR
symmetry. Both the Weyl and the line-node semimetals are characterized by
topologically-protected surface states, although in the line-node case some
additional symmetries must be imposed to retain this topological protection.
The edge states have the form of "Fermi arcs" in the case of the Weyl
semimetal: these are chiral gapless edge states, which exist in a finite region
in momentum space, determined by the momentum-space separation of the bulk Weyl
nodes. The chiral character of the edge states leads to a finite Hall
conductivity. In contrast, the edge states of the line-node semimetal are "flat
bands": these states are approximately dispersionless in a subset of the
two-dimensional edge Brillouin zone, given by the projection of the line node
onto the plane of the edge. We discuss unusual transport properties of the
nodal semimetals, and in particular point out quantum critical-like scaling of
the DC and optical conductivity of the Weyl semimetal, and similarities to the
conductivity of graphene in the line node case.
