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 simplest version of a central-spin model, the so-called Mermin model, in which a given (central) spin 1/2 interacts symmetrically with a "bath" composed of N spins 1/2, characterized by a common frequency. The symmetric interaction allows one to restrict the problem to the fully symmetric sector of the Hilbert space associated with these N spins and therefore to treat these as one large N/2 spin. In particular, we derive analytic expressions for the integrated density of states in the thermodynamic limit when the number of bath spins is taken to infinity. From the thermodynamic limit spectrum we show the phase diagram for the system can be divided into four regions, partitioned, on the one hand, into a symmetric (nondegenerate) phase or a broken symmetry (degenerate) phase, and, on the other hand, by the case of overlapping or nonoverlapping 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 Schrödinger equation into a Ricatti-like form that can be solved in the thermodynamic limit to obtain closed-form expressions for the integrated 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.