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where A = (4 − 3 α +2 ( α − 1) ω $^{2}$)2 and B = 36 ω $^{2}$( α − 1) $^{2}$. For example, for ω = 1 $_{/}$√$_{2,}$ α ≃ 0 . 822559. 

• In the coexistence region, the critical point is defined by the condition that both min- ima (spherical and deformed) are degenerate. At the critical point the two degenerated minima are at β$_{e}$ = 0 and β$_{e}$ = ω/ 2, and their energy is equal to zero. The critical line is therefore defined as 2 

For example, for ω = 1 $_{/}$√$_{2,}$ α$_{c}$ = 9 / 11. 

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• According to the previous analysis, for ω ̸ = 0 there appears a first-order phase tran- sition, while for ω = 0 there is an isolated point of second-order phase transition at α = 4 / 5. In this case, antispinodal, spinodal and critical points collapse to a single point. 

In Fig. 1 we present a schematic view of the phase diagram for the environment Hamil- tonian H$_{E}$ (5) in the ω − α plane. 

The Hamiltonian (5) also displays an Excited State Quantum Phase Transition (ESQPT), which is analogous to a standard quantum phase transition, but taking place at some excited critical energy E$_{c}$ of the system. We can distinguish between different kinds of ESQPTs. As it is stated in [11], in the thermodynamic limit a crossing of two levels at E = E$_{c}$ determines 

