  We analyze the ground state of a strongly interacting fermion chain with a
supersymmetry. We conjecture a number of exact results, such as a hidden
duality between weak and strong couplings. By exploiting a scale free property
of the perturbative expansions, we find exact expressions for the order
parameters, yielding the critical exponents. We show that the ground state of
this fermion chain and another model in the same universality class, the XYZ
chain along a line of couplings, are both written in terms of the same
polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide
consistency checks for large N. These polynomials satisfy a recursion relation
related to the Painlev\'e VI differential equation, and using a scale-free
property of these polynomials, we derive a simple and exact formula for their
limit as N goes to infinity.
