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The nature of the thermal melting process by which triangular-lattice Isingantiferromagnets lose their low-temperature ferrimagnetic three-sublatticeorder depends on the range of the interactions: It changes character whensecond and third neighbour ferromagnetic interactions become comparable to thenearest-neighbour antiferromagnetic coupling. We present a detailed numericalcharacterization of the corresponding threshold at which two-step melting ofthree-sublattice order gives way to a direct first-order transition at whichthis order is lost. The multicritical behaviour at this threshold is argued tobe in the universality class of the $Z_6$ parafermion conformal field theorywith central charge $c=5/4$. The presence of this multicritical thresholdinfluences the melting behaviour and long-wavelength properties over a fairlylarge range of parameters, and at temperatures that are of the same order asthe exchange interactions. It is therefore of potential experimental relevancein the context of easy-axis triangular lattice antiferromagnets that displaysuch low temperature ordering.

Melting of three-sublattice order in triangular lattice Ising  antiferromagnets: Power-law order, $Z_6$ parafermionic multicriticality, and  weakly first order transitions