Separability and ground state factorization in quantum spin systems

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.