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Emergent Criticality and Friedan Scaling in a 2D Frustrated Heisenberg Antiferromagnet

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.