A. Mean field theory for H$_{E}$ In order to study the phase diagram of the Hamiltonian (5) as a function of the control parameters α and ω , it is usual to rely on a coherent state of the form | N, β 〉 = $_{e}$√ N (1+ $_{β}$2 $_{)}$( s $^{†}$+ β t $^{†}$) | 0 〉 , (8) where | 0 〉 denotes the boson vacuum. The corresponding energy surface as a function of the variational parameter β is the expectation value of H$_{E}$ (5) in the coherent state (8) E ( N, β ) = 〈 N, β | H$_{E}$ | N, β 〉 〈 N, β | N, β 〉 = N β 2 (1 + β $^{2}$)2 { 5 α − 4 + 4 βω ( α − 1) + β 2 [ α + ω $^{2}$( α − 1) ]} . (9) Minimization of the energy (9) with respect to β , for given values of the control parameters α and ω , gives the equilibrium value β$_{e}$ defining the phase of the system in the ground state. The value β$_{e}$ = 0 corresponds to the symmetric phase, and β$_{e}$ = 0 to the broken symmetry phase. This Hamiltonian has a second order Quantum Phase Transition (QPT) along the line ω = 0, and a first order QPT for ω = 0. In the latter, the critical point is defined as the situation in which the minimum in the symmetric phase and in the broken symmetry phase are degenerate and their energies are equal to zero. The study of the phase diagram has been done in several publications [10]. Here we summarize its main features. • β = 0 is always a stationary point. For ω = 0, the solution with β = 0 is a maximum for α < 4 / 5, and becomes a minimum for α > 4 / 5. In the case of α = 4 / 5, β = 0 is an inflection point. α = 4 / 5 is the point in which a minimum at β = 0 starts to develop and defines the antispinodal line. • For ω = 0 there exists a region where two minima, one spherical and one deformed, coexist. This region is defined by the point where the β = 0 minimum appears (antispinodal point) and the point where the β = 0 minimum appears (spinodal point). The spinodal line is defined by the implicit equation, 3 α 3 α − 4 = A B ( 1 − ( 1 + B $_{A}$) 3 2 ) , (10) 5