ω = 0 α = 0 α = 0 . 4 α = 0 . 6 α = 0 . 7 γ = 0 . 247 ± 0 . 003 γ = 0 . 248 ± 0 . 003 γ = 0 . 248 ± 0 . 001 γ = 0 . 245 ± 0 . 003 α = 1 / 2 ω = 0 . 2 ω = 0 . 5 ω = 1 $_{/}$√$_{2}$ ω = 1 γ = 0 . 255 ± 0 . 006 γ = 0 . 259 ± 0 . 003 γ = 0 . 264 ± 0 . 008 γ = 0 . 284 ± 0 . 001 TABLE II: Finite size scaling exponents γ for the cases depicted in Fig. 7 FIG. 8: | r ( t ) | for α = 1 / 2, ω = 1 $_{/}$√$_{2 and three different values of}$ λ . In all cases N = 10000. 0 0.5 1 0 0.5 1 0 0.5 1  0  5  10  15  20 | r(t) | t =1.10 =1.05 =1.00 B. Decoherence factor for the first order ESQPT However, the numerical estimates seem to increase for larger values of ω ; in particular, for the case ω = 1, the result for exponent γ is significatively larger than γ = 1 / 4. For the case ω = 0, the Hamiltonian considered produce, in addition to the continuous ESQPT studied in the preceding subsection, a first order ESQPT at energy E (1) c . This critical energy can be estimated calculating the local minima in the energy surface H ( φ, ξ ), as it is shown in Fig. 4. Inserting this value in Eq. (17) a critical coupling λ (1) c is obtained. For the case α = 1 / 2 and ω = 1 $_{/}$√$_{2 the first order EQSPT is obtained at}$ λ (1) c ≈ 1 . 05. In Fig. 8 we show the exact result for the decoherence factor | r ( t ) | for α = 1 / 2, ω = 1 $_{/}$√$_{2,}$ and three different values of λ around λ = λ (1) c ≈ 1 . 05. The most significative result is that 21