Density of states and quantum phase transition in the thermodynamic limit of the Mermin central-spin model

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.