AI 中文总结
该研究揭示谱测度的全局拓扑可编码更精细的Krylov动力学不变量，通过正交多项式方法推导了单割谱的增长律与修正，验证了带隙谱的准周期振荡及能隙闭合处的相变行为，建立了基于谱拓扑的算符增长普适性分类。
AI 中文摘要
Lanczos系数的主导渐近增长由谱尾控制，并提供了Krylov动力学的粗略分类。我们表明，谱测度的**全局拓扑**，特别是连通分量的数量、能隙结构以及能隙闭合转变处的行为，编码了更精细的动力学不变量层级，这是基于谱尾的论证无法观测到的。利用正交多项式的Riemann-Hilbert表述和Deift-Zhou最速下降法，我们重现了单割测度的Freud增长律$b_n\sim n^{1/β}$，并从端点数据确定了其亚领头修正。带隙谱会产生准周期的Lanczos振荡，其频率仅由谱带的填充分数决定，因此可从带边预测。我们在SSH链及其次近邻形变中验证了这一点。在能隙闭合转变处，振荡振幅由Painlevé II的Hastings-McLeod解控制，在临界点处按$n^{-1/3}$衰减，并在带隙相和合并相之间插值，从而使谱曲线的拓扑变化表现为一种Krylov相变。我们还证明，虽然在SYK的共形极限中，算符标度维度在主导速率$α= πT$中不可见，但可以从亚领头偏移$b_0 = πT(Δ- \frac{1}{2})$中提取出来。这些结果建立了算符增长中普适性的精细化概念，其分类依据是谱拓扑，而非仅依赖谱尾。
英文摘要
The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the \textit{global topology} of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws $b_n\sim n^{1/β}$ for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlevé II, decaying as $n^{-1/3}$ at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate $α= πT$, it can be extracted from the subleading offset $b_0 = πT(Δ- \frac{1}{2})$. These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.
Comments43+12 pages
