AI 中文总结
该研究建立二维Toda晶格与非厄米哈密顿量的Arnoldi约化间的对应，揭示全纯Krylov几何性质，提出用Arnoldi框架联络实现精确等谱逆绝热输运的方法。
AI 中文摘要
尽管一般非厄米哈密顿量的Arnoldi约化会得到上Hessenberg矩阵，而非厄米Lanczos理论的三对角形式，但我们证明在其对角和次对角系数中存在一个闭合的Toda扇区。对于固定的有限维哈密顿量和全纯形变的循环态矢量，Krylov Gram行列式是有限二维Toda晶格的τ函数，其Flaschka变量与这些Arnoldi系数完全一致。因此，Toda动力学在该扇区上闭合，无需确定其余上Hessenberg元。同一扇区的次对角部分也具有直接的几何意义：次对角系数的平方既决定了全纯Krylov子空间的Fubini–Study度量，也决定了Berry曲率，而Arnoldi分解失效时丢失的子空间相关几何量则不再有定义。在循环区域的光滑实路径上，Arnoldi框架联络进一步提供了精确等谱输运的厄米三对角生成元。当该生成元与Arnoldi矩阵相加时，会抵消瞬时本征空间之间的跃迁，并且只要矩阵可对角化且谱非简并，就能实现逆绝热驱动。
英文摘要
Although Arnoldi reduction of a generally non-Hermitian Hamiltonian yields an upper Hessenberg matrix rather than the tridiagonal form of Hermitian Lanczos theory, we show that a closed Toda sector survives in its diagonal and subdiagonal coefficients. For a fixed finite-dimensional Hamiltonian and a cyclic state vector deformed holomorphically, the Krylov Gram determinants are $τ$ functions of the finite two-dimensional Toda lattice, whose Flaschka variables coincide exactly with these Arnoldi coefficients. The Toda dynamics therefore closes on this sector without determining the remaining upper Hessenberg entries. The subdiagonal part of the same sector also has a direct geometric meaning: the squared subdiagonal coefficients determine both the Fubini--Study metric and the Berry curvature of holomorphic Krylov subspaces, whereas the geometric quantities associated with subspaces lost at Arnoldi breakdown cease to be defined. Along a smooth real path in the cyclic region, the Arnoldi-frame connection further provides a Hermitian tridiagonal generator of exact isospectral transport. When added to the Arnoldi matrix, this generator cancels transitions between instantaneous eigenspaces and realizes counterdiabatic driving whenever the matrix is diagonalizable with a nondegenerate spectrum.
Comments34 pages, 5 figures, 1 table. Submission to SciPost Physics
