  We investigate the existence and the properties of fully separable (fully
factorized) ground states in quantum spin systems. Exploiting techniques of
quantum information and entanglement theory we extend a recently introduced
method and construct a general, self-contained theory of ground state
factorization in frustration free quantum spin models defined on lattices in
any spatial dimension and for interactions of arbitrary range. We show that,
quite generally, non exactly solvable translationally invariant models in
presence of an external uniform magnetic field can admit exact, fully
factorized ground state solutions. Unentangled ground states occur at finite
values of the Hamiltonian parameters satisfying well defined balancing
conditions between the applied field and the interaction strengths. These
conditions are analytically determined together with the type of magnetic
orderings compatible with factorization and the corresponding values of the
fundamental observables such as energy and magnetization. The method is applied
to a series of examples of increasing complexity, including
translationally-invariant models with short, long, and infinite ranges of
interaction, as well as systems with spatial anisotropies, in low and higher
dimensions. We also illustrate how the general method, besides yielding a large
series of novel exact results for complex models in any dimension, recovers, as
particular cases, the results previously achieved on simple models in low
dimensions exploiting direct methods based on factorized mean-field ansatz.
