Superfluid phases of $^{3}$He in a periodic confined geometry

We report theoretical and computational results on the phases of superfluid $^3$He confined by a two-dimensional periodic array of square boundaries (``posts'') with maximal pair-breaking on the boundaries and translational invariance in the third dimension. We obtain a phase diagram by numerically minimizing the Ginzburg-Landau free energy functional. Results are reported for the pressure range $P=0-34\,\mbox{bar}$, based on material parameters that include strong-coupling corrections that account for the bulk $^3$He phase diagram, and for lattice periods, $L \le 30\xi_{0}$ and post dimensions, $0.5\xi_{0}\le d < L$. At all pressures we find a transition from the normal state to a periodic polar phase with $T_{c1}