Ground state separability and criticality in interacting many-particle systems

We analyze exact ground state (GS) separability in general $N$ particle systems with two-site couplings. General necessary and sufficient conditions for full separability, in the form of one and two-site eigenvalue equations, are first derived. The formalism is then applied to a class of $SU(n)$-type interacting systems, where each constituent has access to $n$ local levels, and where the total number parity of each level is preserved. Explicit factorization conditions for parity-breaking GS's are obtained, which generalize those for $XYZ$ spin systems and correspond to a fundamental GS multilevel parity transition where the lowest $2^{n-1}$ energy levels cross. We also identify a multicritical factorization point with exceptional high degeneracy proportional to $N^{n-1}$, arising when the total occupation number of each level is preserved, in which any uniform product state is an exact GS. Critical entanglement properties (like full range pairwise entanglement) are shown to emerge in the immediate vicinity of factorization. Illustrative examples are provided.