content_image/355128.jpg
to show the main effect. Here we present a more extensive study of a similar system including both first and second order ESQPT, and more general sets of parameters. Together with the exact evolution of the system, we present a simple mean field treatment. We show that the decoherence is maximal when the interaction of the system with the environment produces second order ESQPT, while no noticeable effects are observed in the case of a first order ESQPT. For the former case, a finite-size scaling analysis allows us to postulate that the fidelity goes to zero as soon as the interaction between system and environment is switched on. We also show that mean field treatment provides a good description for the decoherence of the small system, except around the critical points. 

The paper is structured as follows. In Sect. II, we present the model for the environment and study the phase diagram and its relation with the density of energy levels. We then discuss the interaction of the environment with the system. In Sect. III we show results for the decoherence factor. Both exact numerical results for large boson number, and an analytic mean field method with simple extensions of the Tamm-Dankoff approximation are presented. In Sect. IV, results for the decoherence factor in the case of continuous and first order ESQPT, including a finite size scaling study for the decoherence factor (or fidelity), are discussed. Finally, in Sect. V we summarize giving the main conclusions of this work. 

## II. THE MODEL 

Following [3], we will consider our system composed by a spin 1 / 2 particle coupled to a spin environment by the Hamiltonian H$_{SE}$ , 

<img src='content_image/355128.jpg'>

where | 0 〉 and | 1 〉 are the two components of the spin 1 / 2 system, and λ$_{0}$ , λ$_{1}$ the couplings of each component to the environment. The three terms H$_{E}$ , H$_{λ}$$_{0}$ and H$_{λ}$$_{1}$ act on the Hilbert space of the environment; therefore, it evolves with an effective Hamiltonian depending on the state of the central spin H$_{i}$ = H$_{E}$ + H$_{λ}$$_{i}$ , i = 0 , 1. The term H$_{λ}$$_{i}$ makes it possible that the environment crosses a critical point as a consequence of the interaction with the central spin [3]. 

Considering the initial state | Ψ$_{SE}$(0) 〉 = ( a | 0 〉 + b | 1 〉 ) |E (0) 〉 , where |E (0) 〉 is the initial 

