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arXiv:2609.36476

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

Yanbo Qiao, Ruohan Xu, H. T. Quan

cond-mat.stat-mechquant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations
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Abstract

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 $τ_Q^{-2}$ scaling for protocols with a nonzero first derivative at either boundary, where $τ_Q$ 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 $τ_Q^{-p(d+n)/(p+2)}$ for $d+n<2p+4$, acquires a logarithmic correction $τ_Q^{-2p}\logτ_Q$ at $d+n=2p+4$, and scales as $τ_Q^{-2p}$ 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.

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We derive the finite-temperature characteristic function of work (CFW) for a driven O(N)O(N)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 τQ−2τ_Q^{-2}τQ−2​ scaling for protocols with a nonzero first derivative at either boundary, where τQτ_QτQ​ is the protocol duration; smoother boundaries lead to faster decay. For power-law protocols approaching the critical point with exponent ppp, we determine the competition between critical excitations and the regular contribution. The nnnth cumulant's critical contribution follows Kibble--Zurek (KZ) scaling τQ−p(d+n)/(p+2)τ_Q^{-p(d+n)/(p+2)}τQ−p(d+n)/(p+2)​ for d+n<2p+4d+n<2p+4d+n<2p+4, acquires a logarithmic correction τQ−2plog⁡τQτ_Q^{-2p}\logτ_QτQ−2p​logτQ​ at d+n=2p+4d+n=2p+4d+n=2p+4, and scales as τQ−2pτ_Q^{-2p}τQ−2p​ above this condition, where ddd 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.
We investigate the statistics of the work performed during a quench across a quantum phase transition using the adiabatic perturbation theory when the system is characterized by independent quasiparticles and the "single-excitation" approximation is assumed. It is shown that all the cumulants of work exhibit universal scaling behavior analogous to the Kibble-Zurek scaling for the average density of defects. Two kinds of transformations are considered: quenches between two gapped phases in which a critical point is traversed, and quenches that end near the critical point. In contrast to the scaling behavior of the density of defects, the scaling behavior of the cumulants of work are shown to be qualitatively different for these two kinds of quenches. However, in both cases the corresponding exponents are fully determined by the dimension of the system and the critical exponents of the transition, as in the traditional Kibble-Zurek mechanism (KZM). Thus, our study deepens our understanding about the nonequilibrium dynamics of a quantum phase transition by revealing the imprint of the KZM on the work statistics.
Using Monte Carlo simulations, we investigate the dynamical properties of the Baxter-Wu (BW) model under linear quenches. For the linear cooling process, the scaling behavior of the excess defect density in the critical region aligns well with the predictions of the Kibble-Zurek (KZ) mechanism. However, the scaling behavior of the excess defect density after exiting the impulse regime does not follow from a simple interplay between the KZ mechanism and the coarsening dynamics; the system undergoes a decay close to a power-law form with an exponent that is significantly different from the coarsening exponent observed in instantaneous quenching. For the linear heating process, we show that, if the system starts from its ground state, the relevant exponents describing the KZ mechanism are identical to those in the cooling scenario. We find that the system does not directly enter the adiabatic regime after leaving the impulse regime but instead passes through a crossover regime with an exponential decay of the excess defect density. If the initial state is ordered but not the ground state of the system, the defect density exhibits a good scaling behavior, but the relevant exponents do not conform to the predictions of the KZ mechanism.
The Kibble--Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the quantum critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble--Zurek mechanism in terms of instantaneous eigenstates in a genuinely interacting field theory. For different protocols, we demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evidence for specific scaling properties of various quantities. In particular, we argue that the higher cumulants of the excess heat exhibit universal scaling in generic interacting models for a slow enough ramp.
The Kibble-Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the quantum critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble-Zurek mechanism in terms of instantaneous eigenstates in a genuinely interacting field theory. For different protocols, we demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evidence for specific scaling properties of various quantities. In particular, we argue that the higher cumulants of the excess heat exhibit universal scaling in generic interacting models for a slow enough ramp.
We study the effects of symmetry-breaking perturbations in the out-of-equilibrium quantum dynamics of many-body systems slowly driven by a time-dependent symmetry-preserving parameter, across the quantum critical regime associated with a continuous quantum transition (CQT). For this purpose, we analyze the out-of-equilibrium dynamics arising from generalized Kibble-Zurek (KZ) protocols, within a dynamic renormalization-group framework allowing for finite-size systems. We show that the time dependence of generic observables develops an out-of-equilibrium finite-size scaling (FSS) behavior, arising from the interplay between the timescale ts{t}_{s}ts​ of the parameter variations in the KZ protocol, the size LLL of the system, and the strength hhh of the symmetry-breaking perturbation, in the limit of large ts{t}_{s}ts​ and LLL. Moreover, scaling arguments based on the first-order adiabatic approximation of slow variations in quantum systems allow us to characterize the approach to the adiabatic regimes for some limits of the model parameters (for example, when we take ${t}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ before $L\ensuremath{\rightarrow}\ensuremath{\infty}$), predicting asymptotic power-law suppressions of the nonadiabatic behaviors in the adiabatic limits. This out-of-equilibrium FSS is supported by numerical analyses for the paradigmatic quantum Ising chain along generalized KZ protocols, with a time-dependent transverse field crossing its CQT, in the presence of a static longitudinal field breaking the Z2{\mathbb{Z}}_{2}Z2​ symmetry.
We study the universality of work statistics of a system quenched through a quantum critical surface. By using the adiabatic perturbation theory, we obtain the general scaling behavior for all cumulants of work. These results extend the studies of Kibble-Zurek mechanism scaling of work statistics from an isolated quantum critical point to a critical surface. As an example, we study the scaling behavior of work statistics in the two-dimensional (2D) Kitaev honeycomb model featured with a critical line. By utilizing the trace formula for quadratic fermionic Hamiltonian, we obtain the exact characteristic function of work of the 2D Kitaev model at zero temperature. The results confirm our prediction.