# π-flux square lattice model (left) and corresponding first BZ (right)

**Figure 1.** π-flux square lattice model (left) and corresponding first BZ (right). Direct and reciprocal lattice vectors are shown. Hopping amplitudes on the central red line in the left figure are complex with phase π; the hopping amplitudes on black bonds have instead a real phase equal to 1.

**Abstract**

We study the superfluid properties of attractively interacting fermions hopping in a family of 2D and 3D lattices in the presence of synthetic gauge fields having π-flux per plaquette. The reason for such a choice is that the π-flux cubic lattice displays Dirac points and that decreasing the hopping coefficient in a spatial direction (say, *t _{z}*), these Dirac points are unaltered: it is then possible to study the 3D–2D interpolation towards the π-flux square lattice. We also consider the lattice configuration providing the continuous interpolation between the 2

*D*π-flux square lattice and the honeycomb geometry. We investigate by a mean-field analysis the effects of interaction and dimensionality on the superfluid gap, chemical potential and critical temperature, showing that these quantities continuously vary along the patterns of interpolation. In the two-dimensional cases at zero temperature and half-filling, there is a quantum phase transition occurring at a critical (negative) interaction

*U*presenting a linear critical exponent for the gap as a function of |

_{c}*U*−

*U*|. We show that in three dimensions, this quantum phase transition is again retrieved, pointing out that the critical exponent for the gap changes from 1 to 1/2 for each finite value of

_{c}*t*.

_{z}