We study the structure of the spectrum of the infinite XXZ quantum spin chain, ananisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves thenumber of down spins (or particle number), allowing to represent it as a direct sum ofN-particleinteracting discrete Schrödinger-type operators restricted to the fermionic subspace. Inthe Ising phase of the model we use this representation to give a detailed determinationof the band and gap structure of the spectrum at low energy. In particular, we show thatat sufficiently strong anisotropy the so-called droplet bands are separated from higherspectral bands uniformly in the particle number. Our presentation of all necessarybackground is self-contained and can serve as an introduction to the mathematical theoryof the Heisenberg and XXZ quantum spin chains.
