content_image/365859.jpg
<img src='content_image/365859.jpg'>

limit N → ∞ for α = 1 / 2 and ω = 1 $_{/}$√$_{2. The curves drawn in the base of the figure are}$ contour curves for fixed values of the energy H ( φ, ξ ) /N = E . Gray curves (red online) represent different values of E around E (2) c for the continuous phase transition. The solid gray (red online) line represents the critical point E$_{c}$ = 0; this is the only value for which the contour curve is non-analytic. On the other hand, black curves (blue online) represent different values of E around the critical energy at which the first-order ESQPT takes place, E (1) c . In this case, the critical value is the one at which the island around ξ = π appears, that correspond to a local minimum in the energy surface. This entails the appearance of another region in the ( φ, ξ ) plane for which the equation H ( φ, ξ ) /N = E has a solution, and consequently the density of states ρ ( E ) suddenly increases. 

## B. Coupling to a single qubit 

Since we are interested in relating the phenomenon of decoherence in a single qubit with the structure of phases and critical regions in the environment as defined by the Hamiltonian (1), we propose as a coupling Hamiltonian H$_{λ}$$_{i}$ = λ$_{i}$ ̂ n$_{t}$ . Choosing λ$_{0}$ = 0 if the qubit is on 

