  We apply a spin-coherent states formalism to study the central-spin model
with monochromatic bath and symmetric coupling (the Mermin model); in
particular, we derive analytic expressions for the density of states in the
thermodynamic limit when the number of bath spins is taken to infinity. From
the thermodynamic limit spectra we show the phase diagram for the system can be
divided into four regions, partitioned on the one hand into a symmetric
(non-degenerate) phase or a broken symmetry (degenerate) phase, and on the
other hand by the case of overlapping or non-overlapping energy surfaces. The
nature and position of singularities appearing in the energy surfaces change as
one moves from region to region. Our spin-coherent states formalism naturally
leads us to the Majorana representation, which is useful to transform the
Schr\"odinger equation into a Ricatti-like form that can be solved in the
thermodynamic limit to obtain closed-form expressions for the density of
states. The energy surface singularities correspond with critical points in the
density of states. We then use our results to compute expectation values for
the system that help to characterize the nature of the quantum phase transition
between the symmetric and broken phases.
