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 Schrödinger-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. © 2014 EDP Sciences.