Phase transitions in the antiferromagnetic XY model with a kagomé lattice

The ground state of the antiferromagnetic $\mathrm{XY}$ model with a kagom\'e lattice is known to be characterized by a well developed accidental degeneracy. As a consequence the phase transition in this system consists in unbinding of pairs of fractional vortices. Addition of the next-to-nearest neighbor (NNN) interaction leads to stabilization of the long-range order in chirality (or staggered chirality). We show that the phase transition, related with destruction of this long-range order, can happen as a separate phase transition below the temperature of the fractional vortex pairs unbinding only if the NNN coupling is extremely weak, and find how the temperature of this transition depends on coupling constants. We also demonstrate that the antiferromagnetic ordering of chiralities and, accordingly, the presence of the second phase transition are induced by the free energy of spin wave fluctuations even in absence of the NNN coupling.