  The Heisenberg antiferromagnet on a two-dimensional triangular lattice is a
paradigmatic problem in frustrated magnetism. Even in the classical limit, its
properties are far from simple. The "120 degree" ground state favoured by the
frustrated antiferromagnetic interactions contains a hidden chiral symmetry,
and supports two distinct types of excitation. And famously, three distinct
phases, including a collinear one-third magnetisation plateau, are stabilised
by thermal fluctuations in applied magnetic field. The questions of
symmetry-breaking raised by this model are deep and subtle, and after more than
thirty years of study, many of the details of its phase diagram remain
surprisingly obscure. In this paper we use modern Monte Carlo simulation
techniques to determine the finite-temperature phase diagram of the classical
Heisenberg antiferromagnet on a triangular lattice in applied magnetic field.
At low to intermediate values of magnetic field, we find evidence for a
continuous phase transition from the paramagnet into the collinear one-third
magnetisation plateau, belonging to the three-state Potts universality class.
We also find evidence for conventional Berezinskii-Kosterlitz-Thouless
transitions from the one-third magnetisation plateau into the canted "Y-state",
and into the 2:1 canted phase found at high fields. However, the phase
transition from the paramagnet into the 2:1 canted phase, while continuous,
does not appear to fall into any conventional universality class. We argue that
this, like the chiral phase transition discussed in zero field, deserves
further study as an interesting example of a finite-temperature phase
transition with compound order-parameter symmetry. We comment on the relevance
of these results for experiments on magnetic materials with a triangular
lattice.
