Three-state Potts nematic order in stacked frustrated spin models with SO(3) symmetry

We propose stacked two-dimensional lattice designs of frustrated and SO(3)-symmetric spin models consisting of antiferromagnetic (AFM) triangular and ferromagnetic (FM) sixfold-symmetric sublattices that realize emergent ${\mathbb{Z}}₃$ Potts nematic order. Considering bilinear-biquadratic spin interactions, our models describe an SO(3)-symmetric triangular lattice AFM subject to a fluctuating magnetization arising from the FM coupled sublattice. We focus on the classical AFM-FM windmill model and map out the zero- and finite-temperature phase diagrams using Monte Carlo simulations and analytical calculations. We discover a state with composite Potts nematic order above the ferrimagnetic three-sublattice up-up-down ground state and relate it to Potts phases in SO(3)-broken Heisenberg and Ising AFMs in external magnetic fields. Finally, we show that the biquadratic exchange in our model is automatically induced by thermal and quantum fluctuations in the purely bilinear Heisenberg model, easing the requirements for realizing these lattice designs experimentally.