relies on the position of the first and the second excited states of H$_{E}$ , we can conclude that the information contained in the low energy spectrum is enough to have a good idea about the properties of the highest excited levels of the environmental Hamiltonian. Note that switching on the interaction between the central qubit and the environment entails an effective increase of the environmental energy roughly given by Δ E = 〈 g$_{0}$ | H$_{1}$ ( λ ) | g$_{0}$ 〉 − E$_{0}$ , and therefore a large value of λ implies that the state of the environment jumps from the ground state to a mixed high-energy state. 

On the other hand, as it is clearly shown in the left panels corresponding to λ = λ (2) c = 0 . 75 and λ = λ$_{∗}$ = 1 . 5, and the right panels λ = λ (2) c = 1 . 17 and λ = λ$_{∗}$ = 1 . 75, the TDA calculations fails for intermediate values of λ . These two values correspond to the critical couplings λ (2) c and λ$_{∗}$ , corresponding to the excited state and the ground state quantum phase transitions, given by Eqs. (17) and (15) respectively. The reason why the Tamm- Dankoff approximation does not work for these values is straightforward. The ESQPT entails a singularity in the energy spectrum far above the first excited state, which gives rise to the main contribution in the TDA calculation. On the other hand, the ground state QPT does not affect the decoherence suffered by the central qubit because the coupling λ makes the environment to jump far above the critical point which entails a singularity in the gap between the ground and the first excited states. However, as the TDA calculation for r ( t ) strongly depends on this gap, it is spuriously affected by the QPT induced by the critical coupling λ$_{∗}$ . 

Finally, the best agreement between the Tamm-Dankoff approximation and the exact calculation happens for λ = 0 . 3, far below λ (2) c . Not only the envelope of the decoherence factor is well reproduced, but also the positions of the local maximum are well placed. In this case, the small coupling makes the environment to jump from the ground state to a mixed low-energy state. Therefore, it is reasonable to assume that the description provided by the TDA, which only takes in consideration the first excited state and a global measure of the anharmonicites of the spectrum, is a better approximation for small values of λ . 

## 1. Analysis of the critical behavior of the decoherence at the continuous ESQPT 

As it is shown in Fig. 5, the decoherence of the central qubit behaves in a singular way for a critical coupling λ (2) c , which makes the environment to jump to the critical energy 

