  Supersymmetric lattice models of constrained fermions are known to feature
exotic phenomena such as superfrustration, with an extensive degeneracy of
ground states, the nature of which is however generally unknown. Here we
address this issue by considering a superfrustrated model, which we deform from
the supersymetric point. By numerically studying its two-parameter phase
diagram, we reveal a rich phenomenology. The vicinity of the supersymmetric
point features period-4 and period-5 density waves which are connected by a
floating phase (incommensurate Luttinger liquid) with smoothly varying density.
The supersymmetric point emerges as a multicritical point between these three
phases. Inside the period-4 phase we report a valence-bond solid type ground
state that persists up to the supersymmetric point. Our numerical data for
transitions out of density-wave phases are consistent with the
Pokrovsky-Talapov universality class. Furthermore, our analysis unveiled a
period-3 phase with a boundary determined by a competition between single and
two-particle instabilities accompanied by a doubling of the wavevector of the
density profiles along a line in the phase diagram.
