Out-of-equilibrium scaling of the energy density along the critical relaxational flow after a quench of the temperature

We study the out-of-equilibrium behavior of statistical systems along critical relaxational flows arising from instantaneous quenches of the temperature $T$ to the critical point $T_c$, starting from equilibrium conditions at time $t=0$. In the case of soft quenches, i.e., when the initial temperature $T$ is assumed sufficiently close to $T_c$ (to keep the system within the critical regime), the critical modes develop an out-of-equilibrium finite-size-scaling (FSS) behavior in terms of the rescaled time variable $\Theta =t/L^z$, where $t$ is the time interval after quenching, $L$ is the size of the system, and $z$ is the dynamic exponent associated with the dynamics. However, the realization of this picture is less clear when considering the energy density, whose equilibrium scaling behavior (corresponding to the starting point of the relaxational flow) is generally dominated by a temperature-dependent regular background term or mixing with the identity operator. These issues are investigated by numerical analyses within the three-dimensional lattice $N$-vector models, for $N=3$ and 4, which provide examples of critical behaviors with negative values of the specific-heat critical exponent $\alpha$, implying that also the critical behavior of the specific heat gets hidden by the background term. The results show that, after subtraction of its asymptotic critical value at $T_c$, the energy density develops an asymptotic out-of-equilibrium FSS in terms of $\Theta$ as well, whose scaling function appears singular in the small-$\Theta$ limit.