  The Heisenberg chain with a weak link is studied, as a simple example of a
quantum ring with a constriction or defect. The Heisenberg chain is equivalent
to a spinless electron gas under a Jordan-Wigner transformation. Using density
matrix renormalization group and quantum Monte Carlo methods we calculate the
spin/charge stiffness of the model, which determines the strength of the
`persistent currents'. The stiffness is found to scale to zero in the weak link
case, in agreement with renormalization group arguments of Eggert and Affleck,
and Kane and Fisher.
