  Bell correlations are among the most exotic phenomena through which quantum
mechanics manifests itself. Their presence signals that the system can violate
the postulates of local realism, once believed to be the nonnegotiable property
of the physical world. The importance of Bell correlations from this
fundamental point of view is even straightened by their applications -- ranging
from quantum cryptography through quantum metrology to quantum computing. Hence
it is of growing interest to characterize the ``Bell content'' of complex,
scalable many-body systems. Here we perform the detailed analysis of the
character and strength of many-body Bell correlations in interacting
multi-qubit systems with particle-exchange symmetry. Such configuration can be
mapped onto an effective Schr\"odinger-like equation, which allows for precise
analytical predictions. We show that in the vicinity of the quantum critical
point, these correlations quickly become so strong that only a fraction of
qubits remains uncorrelated. We also identify the threshold temperature, which,
once overpassed, empowers thermal fluctuations that destroy Bell correlations
in the system. We hope that the approach presented here, due to its
universality, could be useful for the upcoming research on genuinely
nonclassical Bell-correlated complex systems.
