  After summarizing briefly some numerical results for four-dimensional
supersymmetric SU(2) Yang-Mills quantum mechanics, we review a recent study of
systems with an infinite number of colours. We study in detail a particular
supersymmetric matrix model which exhibits a phase transition, strong-weak
duality, and a rich structure of supersymmetric vacua. In the planar and strong
coupling limits, this field theoretical system is equivalent to a
one-dimensional XXZ Heisenberg chain and, at the same time, to a gas of
$q$-bosons. This not only reveals a hidden supersymmetry in these well-studied
models; it also maps the intricate pattern of our supersymmetic vacua into that
of the now-popular ground states of the XXZ chain.
