  We study a two-dimensional frustrated Heisenberg antiferromagnet on the
windmill lattice consisting of triangular and dual honeycomb lattice sites. In
the classical ground state the spins on different sublattices are decoupled,
but quantum and thermal fluctuations drive the system into a coplanar state via
an "order from disorder" mechanism. We obtain the finite temperature phase
diagram using renormalization group approaches. In the coplanar regime, the
relative U$(1)$ phase between the spins on the two sublattices decouples from
the remaining degrees of freedom, and is described by a six-state clock model
with an emergent critical phase. At lower temperatures the system enters a
$\mathbb{Z}_6$ broken phase with long-range phase correlations. We derive these
results by two distinct renormalization group approaches to two-dimensional
magnetism: by Wilson-Polyakov scaling and by Friedan's geometric approach to
nonlinear sigma models where the scaling of the spin-stiffnesses is governed by
the Ricci flow of a 4D metric tensor.
