content_image/343860.jpg
<img src='content_image/343860.jpg'>

of the first order ESQPT on the decoherence factor will be discussed in the next subsection. 

In Fig. 5 we show the results for the decoherence factor for N = 1000 bosons and α = 1 / 2, for two ω values, ω = 0 (left panels) and ω = 1 $_{/}$√$_{2 (rigth panels). The objective}$ of this figure is to show the effect of the continuous ESQPT on the decoherence factor. Solving Eq. (17) for the value of α = 1 / 2, the continuous ESQPT ( E (2) c = 0) takes place at λ (2) c = 0 . 75 for ω = 0 (left panel) and at λ (2) c = 1 . 17 for ω = 1 $_{/}$√$_{2 (right panel). Several}$ features deserve to be discussed. First of all, we can see that the TDA calculation works pretty well for small and large values of λ . In particular, the shape of the envelope, which remains unaffected by the increase of λ for λ ≫ λ (2) c , is very well described by the TDA calculation (see panels for λ = 4 and λ = 8 in Fig. 5). Since this approximation mainly 

