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We consider the $d$-dimensional transverse-field Ising model with power-lawinteractions $J/r^{d+\sigma}$ in the presence of a noisy longitudinal fieldwith zero average. We study the longitudinal-magnetization dynamics of aninitial paramagnetic state after a sudden switch-on of both the interactionsand the noisy field. While the system eventually relaxes to aninfinite-temperature state with vanishing magnetization correlations, we findthat two-time correlation functions show aging at intermediate times. Moreover,for times shorter than the inverse noise strength $\kappa$ and distances longerthan $a(J/\kappa)^{2/\sigma}$ with $a$ being the lattice spacing, we find acritical scaling regime of correlation and response functions consistent withthe model A dynamical universality class with an initial-slip exponent$\theta=1$ and dynamical critical exponent $z=\sigma/2$. We obtain our resultsanalytically by deriving an effective action for the magnetization fieldincluding the noise in a non-perturbative way. The above scaling regime isgoverned by a non-equilibrium fixed point dominated by the noise fluctuations.

Aging dynamics in quenched noisy long-range quantum Ising models