  A variational approach for constructing an effective particle description of
the low-energy physics of one-dimensional quantum spin chains is presented.
Based on the matrix product state formalism, we compute the one- and
two-particle excitations as eigenstates of the full microscopic Hamiltonian. We
interpret the excitations as particles on a strongly-correlated background with
non-trivial dispersion relations, spectral weights and two-particle S matrices.
Based on this information, we show how to describe a finite density of
excitations as an interacting gas of bosons, using its approximate
integrability at low densities. We apply our framework to the Heisenberg
antiferromagnetic ladder: we compute the elementary excitation spectrum and the
magnon-magnon S matrix, study the formation of bound states and determine both
static and dynamic properties of the magnetized ladder.
