Order of the nematic-to-antiferromagnetic transition

Determine whether the transition between the spin-nematic phase and the antiferromagnetic phase is necessarily continuous, or whether it can instead be first order, beyond what can be established by the effective field theory.

Background

The continuum theory analyzes competing quadrupolar spin-nematic and dipolar antiferromagnetic orders. In the regime of strong nematic order, the authors map the critical vector mode to an effective Ising theory and therefore expect a continuous Ising transition in both two and three dimensions. However, the conclusion explicitly acknowledges that this effective description does not rigorously rule out a first-order transition, leaving the order of the transition unresolved beyond the stated approximation.