A two-dimensional spinless non-relativistic p+ip superfluid undergoes a quantum phase transition between the BCS (weakly-coupled) and BEC (strongly-coupled) regimes. This transition is driven by massless fermions and believed to be of the third order. Most easily it can be shown, but relating the problem to the Dirac equation in 2+1 dimensions (Read/Green 2000) and calculating its energy
$$ E \sim \int d^2 p \sqrt{m^2+p^2} $$
which contains the non-analytic piece of the form $\sim |m|^3$. This is not a rigorous argument since the non-relativistic superfluid is not completely equivalent to the Dirac problem, but the third order phase transition is also found rigorously in the mean-field treatment of the spinless p+ip superfluid, e.g.
http://arxiv.org/abs/1008.3406
Since third order phase transitions are rare and usually require fine-tuning, I am wondering now if the order of the phase transition is preserved if one goes beyond mean-field, i.e., can the order of the phase transition change if we include bosonic fluctuations of Cooper pairs? Is there a general argument which forbids or allows that?
In summary, is there a simple general argument that protects the order of the phase transition in this model?