  Mobile impurities in cold atomic gases constitute a new platform for
investigating polaron physics. Here we show that when impurity atoms interact
with a two-dimensional Fermi gas with quadratic band touching the polaron
picture may either hold or break down depending on the particle-hole asymmetry
of the band structure. If the hole band has a smaller effective mass than the
particle band, the quasiparticle is stable and its diffusion coefficient varies
with temperature as $D(T) \propto \ln^2 T$. If the hole band has larger mass,
the quasiparticle weight vanishes at low energies due to an emergent
orthogonality catastrophe. In this case we map the problem onto a set of
one-dimensional channels and use conformal field theory techniques to obtain
$D(T)\propto T^{\nu}$ with an interaction-dependent exponent $\nu$. The
different regimes can be detected in the nonequilibrium expansion dynamics of
an initially confined impurity.
