Emergence of a Renormalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mi>N</mml:mi></mml:math> Expansion in Quenched Critical Many-Body Systems

We consider the fate of $1/N$ expansions in unstable many-body quantum systems, as realized by a quench across criticality, and show emergence ${e}^{2\ensuremath{\lambda}t}/N$ renormalized parameter ruling quantum-classical transition accounting nonperturbatively for local divergence rate $\ensuremath{\lambda}$ mean-field solutions. In terms ${e}^{2\ensuremath{\lambda}t}/N$, quasiclassical paradigmatic examples like self-trapping an integrable Bose-Hubbard dimer generic instability attractive bosonic systems toward soliton formation, are pushed to arbitrarily high orders. The agreement with numerical simulations supports general nature our results appropriately combined long-time $\ensuremath{\lambda}t\ensuremath{\rightarrow}\ensuremath{\infty}$ $N\ensuremath{\rightarrow}\ensuremath{\infty}$ regime, out reach bare $1/N$. For scrambling hyperbolic provide formal grounds conjectured multiexponential form out-of-time-ordered correlators.