Optimized adiabatic-impulse protocol preserving Kibble-Zurek scaling with attenuated anti-Kibble-Zurek behavior

We propose an optimized adiabatic-impulse (OAI) protocol that substantially reduces the evolution time for crossing a quantum phase transition while preserving Kibble-Zurek (KZ) scaling. Near criticality, the control parameter is ramped linearly across the critical point at a rate characterized by a quench time $τ_Q$. Away from criticality, the evolution remains adiabatic and is tuned close to the threshold of adiabatic breakdown, as quantified by an adiabatic coefficient $ζ$ that scales as $τ_Q^α$. As a consequence, the total evolution time exhibits a sublinear power-law dependence on $τ_Q$, and the conventional linear quench is recovered in the limit $α\rightarrow\infty$. We apply the OAI protocol to the transverse Ising chain and numerically determine the minimal $ζ$ required for KZ scaling. We further investigate the nonequilibrium dynamics in the presence of a noisy field that can induce anti-Kibble-Zurek (AKZ) behavior. Within the OAI protocol, noise-induced defects is significantly attenuated due to the shorter evolution time. The optimal quench time at which the defect density is minimized obeys an altered universal power-law scaling with the noise strength. Finally, we generalize the OAI protocol to the nonlinear quenches and numerically demonstrate a marked reduction in noise-induced defects.

💡 Research Summary

**
The manuscript introduces an “Optimized Adiabatic‑Impulse” (OAI) protocol that dramatically shortens the total evolution time required to drive a many‑body quantum system across a second‑order quantum phase transition, while preserving the universal Kibble‑Zurek (KZ) scaling of defect formation. The authors start from the standard adiabatic‑impulse approximation (AIA), which separates the dynamics into two adiabatic regimes far from the critical point and an impulse regime near it. In the OAI scheme, the control parameter ϵ(t) is engineered so that, away from criticality, the instantaneous driving timescale |ϵ/˙ϵ| is kept proportional to the relaxation time τ(t)=|ϵ|^{-zν} via an “adiabatic coefficient” ζ. Crucially, ζ is not a constant but scales as a sub‑linear power of the quench time τ_Q, ζ∝τ_Q^α with α<1. This choice forces the adiabatic condition |ϵ/˙ϵ|≈ζ τ(t) to hold over a much larger interval than in a linear quench, while the central impulse region remains linear (ϵ≈−t/τ_Q) to retain the KZ mechanism. As a result, the total protocol duration T_α≈2θ scales as τ_Q^{α/(1+zν)}, i.e., sub‑linearly with τ_Q, whereas the defect density continues to obey the KZ power law n∝τ_Q^{-β} with β=zν/(1+zν). In the limit α→∞ the protocol reduces to the conventional linear quench.

The authors benchmark the OAI protocol on the paradigmatic one‑dimensional transverse‑field Ising chain (z=ν=1). They derive an explicit time‑dependence of the transverse field g(t) (Eq. 19) that implements the OAI schedule, and solve the resulting Bogoliubov‑de Gennes equations numerically. By varying ζ (ζ=4, 8, 16, 32) they demonstrate that the final kink density approaches the KZ prediction n_KZ=π^{-1}τ_Q^{-1/2} as ζ increases, while the total evolution time is reduced by roughly 30 % compared with a pure linear quench at the same τ_Q. Importantly, the defect density becomes essentially independent of the initial field g_i, confirming that the OAI protocol isolates the universal critical dynamics from non‑universal initial conditions.

The paper then addresses the anti‑Kibble‑Zurek (AKZ) effect that arises when a stochastic white‑noise term γ(t)V is added to the Hamiltonian. In the presence of noise, the defect density acquires an extra contribution proportional to the total evolution time, leading to an increase of defects for slower quenches. Because OAI shortens the evolution time, the noise‑induced term b γ T_α is strongly suppressed. Analytically, the authors obtain a modified defect expression n = a τ_Q^{-β} + b′ γ τ_Q^{α/(1+zν)} and identify an optimal quench time τ̃_{OAI} that minimizes n. This optimal τ̃ scales universally with the noise strength as τ̃_{OAI}∝γ^{-