  We study the structure of the spectrum of the infinite XXZ quantum spin
chain, an anisotropic version of the Heisenberg model. The XXZ chain
Hamiltonian preserves the number of down spins (or particle number), allowing
to represent it as a direct sum of N-particle interacting discrete
Schr\"odinger-type operators restricted to the fermionic subspace. In the Ising
phase of the model we use this representation to give a detailed determination
of the band and gap structure of the spectrum at low energy. In particular, we
show that at sufficiently strong anisotropy the so-called droplet bands are
separated from higher spectral bands uniformly in the particle number. Our
presentation of all necessary background is self-contained and can serve as an
introduction to the mathematical theory of the Heisenberg and XXZ quantum spin
chains.
