A Field-Theoretic Framework for Work Statistics and Universal Scaling in Non-equilibrium Phase Transitions

We develop a field-theoretic framework for work statistics in $O(N)$ models driven through criticality. By analyzing the dynamic renormalization group flow of composite power operators, we find the Kibble-Zurek scaling laws as a natural consequence of the flow, and we derive the scaling of work cumulants relevant to Kibble-Zurek scaling of topological defects from first principles, bypassing heuristic freeze-out argument. This yields the universal scaling $c_n\sim {\tau }_Q^{-{\alpha }_n}$ for the $n$-th work cumulant density: isolated quantum systems exhibit a scaling of ${\alpha }_n=p(d+nz)\nu \mathrm{/}(1+pz\nu )$, whereas open quantum and classical systems undergo a dimensional collapse to ${\alpha }_n=pd\nu \mathrm{/}(1+pz\nu )$. Validated by exact Gaussian solutions and numerical simulations, our theory establishes a foundation for general work statistics far from equilibrium, thereby bridging stochastic thermodynamics and the renormalization group theory.