Polynomial Prethermal Lifetimes in Non Smoothly Driven Quantum Systems
Matteo Gallone, Beatrice Langella
TL;DR
This work establishes that quantum many-body lattices driven quasi-periodically with finite regularity exhibit a polynomial-in-frequency prethermal lifetime when the driving frequencies satisfy a Diophantine condition. The authors develop a finite-step non-convergent normal-form scheme, augmented by analytic smoothing of $C^p$-regular local operators and Lieb-Robinson bounds, to construct an effective local Hamiltonian $H_{\mathrm{eff}}$ close to $H_0$ that governs dynamics on local observables up to a controlled error. They prove explicit bounds: a prethermal heating bound $|t|\lesssim\lambda^{-(b - p(1-b)/\tau + \epsilon)}$ with $b\in(0,p/\tau)$ and an effective description valid for $|t|\lesssim\lambda^{(-b + p/\tau(1-b) - \epsilon)/(d+1)}$, both scaling polynomially with the driving frequency, and they show almost-optimality by constructing quasi-resonant sequences that drive heating on comparable times. Moreover, they demonstrate the breakdown of the prethermal regime through an explicit almost-resonant construction, where magnetization deviates significantly on times $t_m$ in $[C_1\lambda_m^{p/\tau},C_2\lambda_m^{p/\tau+\epsilon}]$, matching the predicted lifetime. Overall, the paper extends prethermalization theory to finitely differentiable quasi-periodic drives and provides a rigorous framework for computing an effective Hamiltonian and assessing the robustness and limits of the prethermal plateau.
Abstract
We study the dynamics of a quantum many-body lattice system with a local Hamiltonian subjected to a quasi-periodic driving with finite regularity. For sufficiently large driving frequencies, we prove that the system remains in a prethermal state for times growing polynomially with the frequency, and we show the optimality of this bound by constructing an explicit example that nearly saturates it. Within this prethermal regime, the dynamics is captured by an effective time-independent local Hamiltonian close to the undriven one. The proof relies on a non convergent normal form scheme, combined with original smoothing techniques for finitely differentiable local operators, and Lieb-Robinson bounds.