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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 ~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 ~0.178. At the higher end of the interval, we find zero energy ground states at fillings above 1/5, in contradiction to previous numerical studies and analytical suggestions; these display non-trivial amplitude degeneracies.