# Third-order topological quantum phase transition in p+ip superfluid

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?

• What does it take for a phase transition to be third order? – Ryan Thorngren Dec 4 '13 at 0:55
• In general, the order of the phase transition tells us how bad the non-analyticity of the thermodynamics potential is at the phase transition point. The third order phase transition has a discontinuity in the third derivative of the thermodynamic potential, while the first and the second derivatives are continuous. – Sergej Moroz Dec 4 '13 at 18:26
• What does the phase diagram look like? I'm trying to figure out why they're not generic. – Ryan Thorngren Dec 5 '13 at 5:49
• At T=0 the system is a chiral superfluid for any strength of the interatomic attraction. The BEC and BCS regimes are however topologically different and are separated by a quantum phase transition. – Sergej Moroz Dec 7 '13 at 2:36
• Hmm. So any smooth perturbation of the thermodynamic potential will not change the order of the phase transition. A subquestion could be whether the difference between mean field potentials and true potentials is smooth. – Ryan Thorngren Dec 8 '13 at 19:36