0 0.2 0.4 0.6 0.8 1 -50 -25 0 25 50 E $_{c}$ <n$_{t}$> FIG. 1: (Color online). Energy levels for the environment Hamiltonian (4) with N = 50. The arrow (red online) shows the jump that the coupling with the central qubit produces in the environment. ground state of the environment | g$_{0}$ 〉 . At t = 0 we switch on the interaction between the system and the environment, and let the system evolve under the complete Hamiltonian. By instantaneously switching on this interaction, the energy of the environment increases, and its state gets fragmented into a region with average energy equal to E = 〈 g$_{0}$ | H$_{1}$ ( α ) | g$_{0}$ 〉 . Therefore, if 〈 g$_{0}$ | H$_{1}$ ( α ) | g$_{0}$ 〉 = 0, the coupling with the central qubit induces the environment to jump into a region arround the critical energy E$_{c}$ . This is illustrated in Fig. 1. Starting from a state in the parity broken phase with α < α$_{c}$ , the coupling with the qubit, H$_{λ}$$_{1}$ = λn$_{t}$ increases the energy of the environment up to the critical point E$_{c}$ . Resorting to the coherent state approach [12], we can obtain a critical value of the coupling strength In Fig. 2 we show the modulus of the decoherence factor | r ( t ) | for α = 0 and several values of λ (see caption). In four of the five cases we can see a similar pattern, fast oscillations plus a smooth decaying envelope. For λ ≳ λ$_{∗}$ , this envelope is weakly dependent on λ . As we increase λ , the frequency of the short period oscillations increases linearly ν ≈ λ/ 3, but the main trend of the curve remains the same. The shape of the envelope can be fitted to a Gaussian decay for short times, and to a power-law decay for longer times; a similar behavior was identified in [5] if the Hamiltonian parameter is close to or larger than the critical value. However, the most striking feature of Fig. 2 is the panel corresponding to λ = 2, for which | r ( t ) | quickly decays to zero and then randomly oscillates around a small value. We note that this particular case constitutes a singular point for both the shape of λ$_{c}$ ( α ) = 1 2 (4 − 5 α ) , α < 4 5 . (7) 5