  We explore exchange coupling of a pair of spins in a double dot and in an
optical lattice. Our algorithm uses the frequency of exchanges in a bosonic
path integral, evaluated with Monte Carlo. This algorithm is simple enough to
be a "black box" calculator, yet gives insights into the role of correlation
through two-particle probability densities, visualization of instantons, and
pair correlation functions. We map the problem to Hubbard model and see that
exchange and correlation renormalize the effective parameters, dramatically
lowering U at larger separations.
