  The energy spectrum of Dicke Hamiltonians with and without the rotating wave
approximation for arbitrary atom-number is obtained analytically with the
variational method, in which the effective pseudo-spin Hamiltonian resulted
from the expectation value in the boson-field coherent state is diagonalized by
the spin-coherent-state transformation. In addition to the ground-state energy
an excited macroscopic quantum-state is found corresponding to the
south-and-north-pole gauges of the spin-coherent states respectively. Our
results of ground-state energies in exact agreement with various approaches
show that these models exhibit a zero-temperature quantum phase transition of
second-order for any number of atoms, which however was commonly considered as
a phenomenon of the thermodynamic limit with the atom-number tending to
infinite. The critical behavior of geometric phase is analyzed, which displays
no singularity at the critical point.
