We study an impurity atom in a two-dimensional Fermi gas using variational
wave functions for (i) an impurity dressed by particle-hole excitations
(polaron) and (ii) a dimer consisting of the impurity and a majority atom. In
contrast to three dimensions, where similar calculations predict a sharp
transition to a dimer state with increasing interspecies attraction, we show
that the polaron ansatz always gives a lower energy. However, the exact
solution for a heavy impurity reveals that both a two-body bound state and
distortions of the Fermi sea are crucial. This reflects the importance of
particle-hole pairs in lower dimensions and makes simple variational
calculations unreliable. We show that the energy of an impurity gives important
information about its dressing cloud, for which both ans\"atze give inaccurate
results.Comment: 5 pages, 2 figures, minor change
