  We consider the ground states of the ferromagnetic XXZ chain with spin up
boundary conditions in sectors with a fixed number of down spins. This forces
the existence of a droplet of down spins in the system. We find the exact
energy and the states that describe these droplets in the limit of an infinite
number of down spins. We prove that there is a gap in the spectrum above the
droplet states. As the XXZ Hamiltonian has a gap above the fully magnetized
ground states as well, this means that the droplet states (for sufficiently
large droplets) form an isolated band. The width of this band tends to zero in
the limit of infinitely large droplets. We also prove the analogous results for
finite chains with periodic boundary conditions and for the infinite chain.
