  The majority of magnetic materials possess some degree of magnetic
anisotropy, either at the level of a single ion, or in the exchange
interactions between different magnetic ions. Where these exchange interactions
are also frustrated, the competition between them and anisotropy can stabilize
a wide variety of new phases in applied magnetic field. Motivated by the
hexagonal delafossite 2H-AgNiO 2, we study the Heisenberg antiferromagnet on a
layered triangular lattice with competing first- and second-neighbour
interactions and single-ion easy-axis anisotropy. Using a combination of
classical Monte Carlo simulation, mean-field analysis, and Landau theory, we
establish the magnetic phase diagram of this model as a function of temperature
and magnetic field for a fixed ratio of exchange interactions, but with values
of easy-axis anisotropy D extending from the Heisenberg (D =0) to the Ising
(D=&#8734) limits. We uncover a rich variety of different magnetic phases.
These include several phases which are magnetic supersolids (in the sense of
Matsuda and Tstuneto or Liu and Fisher), one of which may already have been
observed in AgNiO 2. We explore how this particular supersolid arises through
the closing of a gap in the spin-wave spectrum, and how it competes with rival
collinear phases as the easy-axis anisotropy is increased. The finite
temperature properties of this phase are found to be different from those of
any previously studied magnetic supersolid.
