  Contemporary understanding of correlations in quantum many-body systems and
in quantum phase transitions is based to a large extent on the recent intensive
studies of entanglement in many-body systems. In contrast, much less is known
about the role of quantum nonlocality in these systems, mostly because the
available multipartite Bell inequalities involve high-order correlations among
many particles, which are hard to access theoretically, and even harder
experimentally. Standard, "theorist- and experimentalist-friendly" many-body
observables involve correlations among only few (one, two, rarely three...)
particles. Typically, there is no multipartite Bell inequality for this
scenario based on such low-order correlations. Recently, however, we have
succeeded in constructing multipartite Bell inequalities that involve two- and
one-body correlations only, and showed how they revealed the nonlocality in
many-body systems relevant for nuclear and atomic physics [Science 344, 1256
(2014)]. With the present contribution we continue our work on this problem. On
the one hand, we present a detailed derivation of the above Bell inequalities,
pertaining to permutation symmetry among the involved parties. On the other
hand, we present a couple of new results concerning such Bell inequalities.
First, we characterize their tightness. We then discuss maximal quantum
violations of these inequalities in the general case, and their scaling with
the number of parties. Moreover, we provide new classes of two-body Bell
inequalities which reveal nonlocality of the Dicke states---ground states of
physically relevant and experimentally realizable Hamiltonians. Finally, we
shortly discuss various scenarios for nonlocality detection in mesoscopic
systems of trapped ions or atoms, and by atoms trapped in the vicinity of
designed nanostructures.
