-50 -25  0  25  50  0  0.2  0.4  0.6  0.8  1 E -100 -75 -50 -25  0  25  50  0  0.2  0.4  0.6  0.8  1 E (a) ω = 0 (b) ω = 1 $_{/}$√$_{2}$ FIG. 2: Energy levels of the Hamiltonian (5) as a function of α for N = 50 and two different values of ω . a first order ESQPT, while if the number of interacting levels is locally large at E = E$_{c}$ but without real crossings, the ESQPT is continuous. As the entropy of a quantum system is related to its density of states, a relationship between an ESQPT and a standard phase transition at a certain critical temperature can be established in the thermodynamic limit [12]. These kinds of phase transitions have been identified in the Lipkin model [13], in the interacting boson model [14], and in more general boson or fermion two-level pairing Hamiltonians (for a complete discussion, including a semiclassical analysis, see [15]). In all these cases, the ESQPT takes place beyond the critical value of the Hamiltonian control parameter, implying that the critical point moves from the ground state to an excited state. In Fig. 2 we show the energy eigenvalues of the environmental Hamiltonian (5) with N = 50 bosons as a function of the control parameter α for ω = 0 in left panel, and ω = 1 $_{/}$√$_{2 in right panel. In both cases, we see for}$ α < α$_{c}$ a collapse of several levels at E ≈ 0. In the right panel ( ω = 1 $_{/}$√$_{2) we can also see a second critical curve for}$ E < 0 that divides the level diagram in two regions: one in which levels behave smoothly, and another in which the level density increases and some crossings are observed. One simple way to analyze the phase diagram is by means of the density of states. To obtain an analytical approximation for this quantity, one can start from a coherent state similar to (8), in which real parameter β is replaced by the complex parameter z = tan ( φ/ 2) exp ( iξ ), in terms of which the energy is expressed as H ( φ, ξ ) = 〈 N, φ, ξ | H | N, φ, ξ 〉 . 7