Figure 1

(Color online) A sketch of the finite-temperature crossover phase diagram close to the quantum critical point. Crossover lines $T\sim {\left|h-h_c\right|}^{\nu z}$ separating the semiclassical regions from the quantum critical region are shown. The latter is traversed by the system during the quench in a time $t_{QC}$.Reuse & Permissions

Figure 2

(Color online) A cartoon of the spins-baths coupling [Eq. (17)] for $l=3$, i.e., each bath coupled to three spins.Reuse & Permissions

Figure 3

Lowest-order diagrams contributing to the self-consistent Born approximation: dashed lines correspond to the noninteracting bath Green’s function $g\doteq -i⟨{\mathcal{T}}_{\gamma }{X}_q\left(t_1\right)X_q\left(t_2\right)⟩$ while solid lines to the interacting system Green function $G$. (a) corresponds to ${\Sigma }_k\left(t_1,t_2\right)=\frac{i}{N}{\sum }_q{g}_{k-q}\left(t_1,t_2\right){\tau }^z{G}_q\left(t_1,t_2\right){\tau }^z$, while (b) is the polaronic shift contribution ${\Sigma }_k^{\delta }\left(t_1,t_2\right)=-\frac{i}{N}\delta \left(t_{1,}{t}_2\right){\tau }^z{\int }_{\gamma }d\overline{t}g\left(t_1,\overline{t}\right){\sum }_q\text{Tr}\left[{\tau }^z{G}_q\left(\overline{t},\overline{t}\right)\right]$Reuse & Permissions

Figure 4

(Color online) Relaxation rate $1/\tau$ as a function of $T$ and $h$ for $N=400$ (here $\gamma =1$, and $s=1$).Reuse & Permissions

Figure 5

(Color online) Relaxation rate $1/\tau$ as a function of $T$ obtained from the exact diagonalization of $\mathcal{R}$ (symbols) and the approximation in Eq. (37) (dashed lines). Lines are relative to $\gamma =0.3$, 0.5, and 1 (from top to bottom) and $h=1$, $s=1$.Reuse & Permissions

Figure 6

(Color online) Ising model $\left(\gamma =1\right)$ with Ohmic baths $\left(s=1\right)$. Relative gap between the two longest relaxation times $\left({\lambda }_2^{-1}-{\lambda }_1^{-1}\right)/{\lambda }_1^{-1}$ (${\lambda }_{1,2}$ being the two lowest eigenvalues of $\mathcal{R}$) as a function of $T$ and $h$; crossover lines $T=\Delta =\left|h-1\right|$ are plotted for comparison.Reuse & Permissions

Figure 7

(Color online) Density of excitation $\mathcal{E}$ (circles) versus quench velocity for different system sizes $N=26$, 50, 100, 200, 400, and 800 from bottom to top, according to the arrow; the points corresponding to $N=800$ and $N=400$ are indistinguishable. Parameters are set to $\alpha =0.01$, $T=0.1$ $\gamma =1$ and $s=1$ and the quench is halted at $h_f=0.8$. Dotted line is the coherent contribution ${\mathcal{E}}_{\text{coh}}$ obtained for $\alpha =0$ and stars represent the incoherent contribution ${\mathcal{E}}_{\text{inc}}$ due to thermal excitation (see text); both curves are relative to $N=800$; for ${\mathcal{E}}_{\text{inc}}$ the same curve is obtained already at $N\sim 30$.Reuse & Permissions

Figure 8

(Color online) Populations of the excited states ${\mathcal{P}}_k$ for $N=400$, $v=0.0017$, and the same parameter values of Fig. 7. Left: dynamics of two low-energy modes $k_1\sim \pi$ (solid line) and $k_2\sim 0.9\pi$ (dashed line) as a function of $h\left(t\right)$ obtained by solving the kinetic equation (instantaneous thermal equilibrium values are plotted for reference as dotted lines). The inset shows the energy levels near the critical point and the scale of temperature; marked levels refer to $k_1$ and $k_2$. The energy gap closes at $k=\pi$. Right: distribution of ${\mathcal{P}}_k$ as a function of $k$ at $h\left(t\right)=1$. Stars represent the excitations created incoherently and triangles represent the excitations produced coherently when there is no coupling to the bath. The two excitation mechanisms act on different energy scales: the lowest energy modes are coherently populated, the highest ones thermally excited.Reuse & Permissions

Figure 9

(Color online) Density of energy (lowest panel) and of excitations versus quench velocity $v$ for $h_f=0.8$ and 1. Parameters are set to $\gamma =0.7$, $s=1.5$, $\alpha =0.01$, and $T=0.15$ or 0.1 (upper and lower curves of each panel). Circles are obtained by solving the kinetic equation; dotted lines are the coherent contributions ${\mathcal{E}}_{\text{coh}}$ and $E_{\text{coh}}$ evaluated by solving the kinetic equation for $\alpha =0$; a fit gives correctly ${\mathcal{E}}_{\text{coh}}\propto \sqrt{v}$ and $E_{\text{coh}}\propto v$, consistently with the KZ scaling law for excitations [Eq. (5)] and the modified scaling we derived for the energy density [Eq. (6)]. Solid and dashed lines are Eqs. (3, 4) using for the incoherent contributions the expressions (39, 40) and their linearized forms Eq. (41) and Eq. (42), respectively.Reuse & Permissions

Figure 10

(Color online) System and bath parameters are fixed as in Fig. 9. Upper panel: data collapse of $v{\mathcal{E}}_{\text{inc}}$ and $v{E}_{\text{inc}}$ as a function of $T$ obtained from the kinetic equation; data refer to ${10}^{-3}\lesssim v\lesssim {10}^{-2}$ (data relative to $h_f=1$ for ${\mathcal{E}}_{\text{inc}}$ are rescaled by a factor 2, since in this case only half quantum critical region is crossed). Lower panel: scaling of $v_{\text{cross}}$ is obtained equating ${\mathcal{E}}_{\text{inc}}={\mathcal{E}}_{\text{coh}}$ and analogously for $E$. The fits confirm the scaling predicted by Eqs. (10, 11) and Eqs. (12, 13), that, for the specific case of $s=1.5$ considered, are shown in their corresponding plots.Reuse & Permissions

Figure 11

(Color online) Density of excitations versus quench velocity $v$ for $h_f=0.8,0.6,0.4,0.2,0$. Parameters are set to $\gamma =1$, $s=1$, $\alpha =0.01$, and $T=0.1$. Upper solid line is the scaling law [Eq. (3)] using the expression (39). Decreasing the value of $h_f$, the crossing of the semiclassical region, after the critical point, becomes more relevant at low $v$ and scaling no longer holds strictly.Reuse & Permissions