  Most electronic properties of metals are determined solely by the low-energy
states around the Fermi level, and for topological metals/semimetals, these
low-energy states become distinct because of their unusual energy dispersion
and emergent pseudospin degree of freedom. Here, we propose a class of
materials which are termed as quadratic contact point (QCP) semimetals. In
these materials, the conduction and valence bands contact at isolated points in
the Brillouin zone, around which the band dispersions are quadratic along all
three directions. We show that in the absence/presence of spin-orbit coupling,
there may exist triply/quadruply-degenerate QCPs that are protected by the
crystalline symmetry. We construct effective models to characterize the
low-energy fermions near these QCPs. Under strong magnetic field, unlike the
usual 3D electron gas, there appear unconventional features in the Landau
spectrum. The QCP semimetal phase is adjacent to a variety of topological
phases. For example, by breaking symmetries via Zeeman field or lattice strain,
it can be transformed into a Weyl semimetal with Weyl and double Weyl points, a
Z2 topological insulator/metal, or a Dirac semimetal. Via first-principles
calculations, we identify realistic materials Cu2Se and RhAs3 as candidates for
QCP semimetals.
