Crossover of Scaling Behaviors of Work Cumulants in a Driven Gaussian Field Theory

We derive the finite-temperature characteristic function of work (CFW) for a driven $O(N)$ Gaussian field theory and obtain closed-form expressions for all zero-temperature excess-work cumulants. For gapped protocols, we use adiabatic perturbation theory (APT) to derive the ${\tau }_Q^{-2}$ scaling for protocols with a nonzero first derivative at either boundary, where ${\tau }_Q$ is the protocol duration; smoother boundaries lead to faster decay. For power-law protocols approaching the critical point with exponent $p$, we determine the competition between critical excitations and the regular contribution. The $n$th cumulant’s critical contribution follows Kibble–Zurek (KZ) scaling ${\tau }_Q^{-p(d+n)\mathrm{/}(p+2)}$ for $d+n<2p+4$, acquires a logarithmic correction ${\tau }_Q^{-2p}\log {\tau }_Q$ at $d+n=2p+4$, and scales as ${\tau }_Q^{-2p}$ above this condition, where $d$ is the spatial dimension. This analytical study of the APT–KZ crossover in a solvable model provides a basis for studying their competition in interacting field theories.