  We study quantum criticality in the infinite range Transverse-Field Ising
Model. We find subtle differences with respect to the well-known single-site
mean-field theory, especially in terms of gap, entanglement and quantum
criticality. The calculations are based on numerical diagonalization of
Hamiltonians with up to a few thousand spins. This is made possible by the
enhanced symmetries of the model, which divide the Hamiltonian into many
block-diagonal sectors. The finite temperature phase diagram and the
characteristic jump in heat capacity closely resemble the behavior in
mean-field theory. However, unlike mean-field theory where excitations are
always gapped, the excitation gap in the infinite range model goes to zero from
both the paramagnetic side and from the ferromagnetic side on approach to the
quantum critical point. Also, contrary to mean-field theory, at the quantum
critical point the Quantum Fisher Information becomes large, implying
long-range multi-partite entanglement. We find that the main role of
temperature is to shift statistical weights from one conserved sector to
another. However, low energy excitations in each sector arise only near the
quantum critical point implying that low energy quantum fluctuations can arise
only in the vicinity of the quantum critical field where they can persist up to
temperatures of order the exchange constant.
