We investigate disordered one- and two-dimensional Heisenberg spin lattices
across a transition from integrability to quantum chaos from both a statistical
many-body and a quantum-information perspective. Special emphasis is devoted to
quantitatively exploring the interplay between eigenvector statistics,
delocalization, and entanglement in the presence of nontrivial symmetries. The
implications of basis dependence of state delocalization indicators (such as
the number of principal components) is addressed, and a measure of {\em
relative delocalization} is proposed in order to robustly characterize the
onset of chaos in the presence of disorder. Both standard multipartite and {\em
generalized entanglement} are investigated in a wide parameter regime by using
a family of spin- and fermion- purity measures, their dependence on
delocalization and on energy spectrum statistics being examined. A distinctive
{\em correlation between entanglement, delocalization, and integrability} is
uncovered, which may be generic to systems described by the two-body random
ensemble and may point to a new diagnostic tool for quantum chaos. Analytical
estimates for typical entanglement of random pure states restricted to a proper
subspace of the full Hilbert space are also established and compared with
random matrix theory predictions.Comment: 17 pages, 10 figures, revised versio
