Order and supersymmetry at high filling zero-energy states on the triangular lattice

We perform exact diagonalization studies in $d=2$ dimensions for the Fendley and Schoutens model of hard-core and nearest-neighbor excluding fermions that displays an exact non-relativistic supersymmetry. Using clusters of all possible shapes up to 46 sites, we systematically study the behavior of the ground state phase diagram as a function of filling. We focus on the highly degenerate zero-energy states found at fillings between $1/7$ and $\sim 1/5$. At the lower end of that interval, at filling $1/7$, we explicitly show that the ground states are gapped crystals. Consistent with previous suggestions, we find that the extensive entropy of zero states peaks at a filling of $\sim 0.178$. At the higher end of the interval, we find zero energy ground states at fillings above $1/5$, contrary to previous expectations; which display non-trivial amplitude degeneracies.