Poisson-Dirichlet distributions and weakly first-order spin-nematic phase transitions

Weakly first-order transitions, i.e. discontinuous phase transitions with very large correlation lengths, have become a vivid subject in condensed matter research and beyond in recent years. Therefore, establishing quantum systems in which weakly first-order phase transitions can be robustly demonstrated is of great interest. We present a quantitative characterization of generic weakly first-order thermal phase transitions out of planar spin-nematic states in three-dimensional spin-one quantum magnets, based on calculations using Poisson-Dirichlet distributions within a universal loop model formulation, combined with large-scale quantum Monte Carlo calculations. In contrast to earlier claims, the thermal melting of the nematic state is not continuous, instead we identify a weakly first-order transition. Furthermore, we obtain exact results for the order parameter distribution and cumulant ratios at the melting transition. Our findings establish the thermal melting of planar spin-nematic states as a generic platform for quantitative approaches to weakly first-order phase transitions in quantum systems with a continuous SU(2) internal symmetry.