Non-local spacetime supersymmetry on the lattice

We show that several well-known one-dimensional quantum systems possess a hidden non-local supersymmetry. The simplest example is the open XXZ spin chain with Δ = -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 spacetime supersymmetry holds for the t-J model at its integrable ferromagnetic point, where the spacetime supersymmetry and the Hamiltonian it yields coexist with a global u(1 vertical bar 2) graded Lie algebra symmetry. Possible generalizations to other algebras are discussed