Quantum aging and dynamical universality in the long-range $O(N\to\infty)$ model

Quantum quenches to or near criticality give rise to the phenomenon of \textit{aging}, manifested by glassy-like dynamics at short times and far from equilibrium. The recent surge of interest in the dynamics of quantum many-body systems has rejuvenated interest in this phenomenon. Motivated by the ubiquitous long-range interactions in emerging experimental platforms, it is vital to study quantum aging in such settings. In this work, we investigate the dynamical universality and aging in the $d$-dimensional $O(N)$ model with the long-range coupling $1/x^{d+\sigma}$ and in the mean-field limit $N\to\infty$ that allows an exact treatment. An immediate consequence of long-range coupling is the emergence of nonlinear light cones. We focus on the correlation and response functions, and identify a rich scaling behavior depending on how the corresponding space-time positions are located relative to each other, via a \textit{local light cone}, and to the time of the quench via a global \textit{quench light cone}. We determine the initial-slip exponent that governs the short-time dependence of two-point functions. We highlight the new qualitative features of aging due to the long-range coupling, in particular in the region outside the light cones. As an important consequence of long-range coupling, the correlation function decays as $1/x^{d+\sigma}$ outside the quench light cone while increasing polynomially with the total time after quench. This is while, for short time differences, the two-time response function "equilibrates" at \textit{all} distances even outside this light cone. Our analytic findings are in excellent agreement with exact numerics, and provide a useful benchmark for modern experimental platforms with long-range interactions.