coupled to an environment which induces a pumping mechanism, as in Eq. : $L_n^{(1)} = c^{\dagger}_n$. The various data sets denote different values of the dissipative coupling $\kappa$, as listed in the legend. Here we simulated the annealing protocol of Eq.  for chains of $L=10^3$ sites. Black squares denote data for $\kappa=0$, which obey a power-law behaviour for $\tau > 1$ with the KZ scaling exponent $\gamma = 0.5$ (dashed line).[]{data-label="fig:ExcT_FreeF-Pump"}](ExcT_FreeF-Pump.pdf){width="\columnwidth"}

The excess energy per site $\varepsilon$ during the annealing protocol is thus obtained via a numerical integration of the linearized Liouville equations for each $k$ mode . For numerical convenience, we restricted the initial point of the annealing procedure  to $t_{\rm in} = -5 \tau$, and checked that the results are not appreciably affected by this choice [@Caneva_2007]. We studied systems up to $L=10^3$ sites and annealing times up to $\tau = 10^3$; a fourth-order Runge-Kutta integration procedure with time step $dt = 10^{-2}$ has been employed.

Fig. \[fig:ExcT\_FreeF-Pump\] shows the behaviour of the excess energy at the end of the annealing protocol, $\varepsilon(0)$, for various values of the dissipation strength $\kappa$, as a function of the annealing time $\tau$. In the absence of dissipation ($\kappa=0$), we recover the 