Emergent critical phase in a 2D frustrated Heisenberg model

It is well-known that a discrete Ising ($Z_2$) order parameter emerges in the frustrated square lattice $J_1$-$J_2$-Heisenberg model, which may be broken at finite temperature. We ask whether a different discrete symmetry $Z_q$ with $q>2$ may be found in other frustrated Heisenberg models, giving rise to a different finite temperature phase transition. Indeed, we identify an emergent $Z_6$ symmetry at low temperatures in a frustrated Heisenberg model on a 2D lattice that contains both the sites of the triangular and its dual honeycomb lattice. Our analysis combines a spin-wave expansion, susceptible to short-distance physics, with renormalization group arguments of the corresponding long-wavelength non-linear sigma model. Our results are even more appealing since the $Z_6$ clock model has a rich finite temperature phase diagram with two distinct Berezinskii-Kosterlitz-Thouless (BKT) phase transitions separated by a massless critical phase. We also discuss possible realizations of this phenomenon using cold-atoms in optical lattices.