AI 中文总结
研究利用经典兰索斯算法结合相空间辛结构定义克里洛夫复杂度，通过量子力学方法展示\(\hbar \to 0\)极限过渡，定义相关时间和复杂度，应用于LMG和FP模型，揭示LMG不稳定性可由微正则克里洛夫复杂度解决。
AI 中文摘要
在这项工作中，我们开发并研究了经典的兰索斯算法，该算法使我们能够利用相空间的辛结构定义克里洛夫复杂度：泊松括号起到量子对易子的作用，相空间积分提供定义兰索斯递归所需的内积。我们使用相空间中的量子力学通用方法表明，通常量子力学克里洛夫框架的\(\hbar \to 0\)极限平滑地过渡到经典框架。在具有明确半经典极限的理论中，我们表明经典克里洛夫复杂度在足够早的时间准确地近似量子复杂度，因此是早期混沌动力学的一个有用特征。我们定义了一个克里洛夫 - 埃伦费斯特时间，它量化了经典和量子复杂度的最终发散，对应于克里洛夫链的特征深度\(n\sim n_*(\hbar)\)，在时域中转化为通用混沌系统中众所周知的尺度\(t_*\sim\lambda_K^{-1}\log(1/\hbar)\)。我们还在经典和量子环境中定义了微正则克里洛夫复杂度，这允许对复杂度进行细粒度的逐能壳研究。我们将此框架应用于利普金 - 梅什科夫 - 格利克（LMG）和费因戈尔德 - 佩雷斯（FP）模型，它们是已知在热力学极限下经典化的集体自旋系统。特别是，虽然FP模型在某些耦合值范围内具有谱混沌，但LMG模型已知表现出早期鞍点主导的混沌。我们的分析表明，LMG中的不稳定性通过微正则克里洛夫复杂度得到解决，该复杂度在远离不稳定性的谱窗口中由哈密顿量的可积结构控制，无论是在早期还是晚期。
英文摘要
In this work, we develop and study the classical Lanczos algorithm allowing us to define Krylov complexity using the symplectic structure of phase space: Poisson brackets take on the role of the quantum commutators and phase-space integrals furnish the inner product needed to define the Lanczos recursion. We show, using general methods of quantum mechanics in phase space, that the $\hbar \to 0$ limit of the usual quantum mechanical Krylov framework smoothly goes over into the classical one. In theories with well-defined semiclassical limits, we show that classical Krylov complexity accurately approximates quantum complexity at early enough times, and thus is a useful characteristic of early-time chaotic dynamics. We define a Krylov-Ehrenfest time, which quantifies the eventual divergence of classical and quantum complexities, corresponding to a characteristic depth of the Krylov chain, $n\sim n_*(\hbar)$, which in the time domain translates to the well-known scale, $t_*\simλ_K^{-1}\log(1/\hbar)$, in generic chaotic systems. We additionally define microcanonical Krylov complexities, both in the classical and quantum setting, which allows one a fine-grained study of complexity, energy shell by energy shell. We apply this framework to the Lipkin-Meshkov-Glick (LMG) and Feingold-Peres (FP) models, which are collective spin systems known to classicalize in the thermodynamic limit. In particular, while the FP model features spectral chaos for some range of coupling values, the LMG model is known to exhibit early-time saddle-dominated scrambling. Our analysis shows that the instability in LMG is resolved by the microcanonical Krylov complexity, which is controlled by the integrable structure of the Hamiltonian in spectral windows away from the instability, both at early and late times.
Comments74 pages, 27 figures
