  We show that several well-known one-dimensional quantum systems possess a
hidden nonlocal supersymmetry. The simplest example is the open XXZ spin chain
with \Delta=-1/2. We use the supersymmetry to place lower bounds on the ground
state energy with various boundary conditions. For an odd number of sites in
the periodic chain, and with a particular boundary magnetic field in the open
chain, we can derive the ground state energy exactly. The supersymmetry thus
explains why it is possible to solve the Bethe equations for the ground state
in these cases. We also show that a similar space-time supersymmetry holds for
the t-J model at its integrable ferromagnetic point, where the space-time
supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded
Lie algebra symmetry. Possible generalizations to other algebras are discussed.
