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.
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