Analytic Spread Complexity from Level Statistics: From Chaos to Integrability

• URL: http://arxiv.org/abs/2608.07412v1
• Date: Fri, 07 Aug 2026 16:59:35 GMT
• Title: Analytic Spread Complexity from Level Statistics: From Chaos to Integrability
• Abstract summary: We develop an analytic framework that relates spread complexity directly to local spectral statistics.<n>We show that the Krylov kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum.<n>We identify the spectral origin of the complexity peak and its late-time behaviour, and provide a direct connection between Krylov dynamics and spectral statistics.
• Score: 1.497481482212619
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Spread complexity has emerged as a useful probe of quantum chaos, yet the microscopic spectral origin of its characteristic finite-time peak remains incompletely understood. We develop an analytic framework that relates spread complexity directly to local spectral statistics. Starting from an energy-space representation of the Krylov kernel, we show that the kernel is approximately banded, leading to a rapidly convergent diagonal expansion dominated by nearby levels in the ordered spectrum. Motivated by this structure, we propose an approximate kernel-universality hypothesis: after unfolding, the Krylov kernel is well approximated by that of a uniform lattice. Combining this universal kernel with local spectral statistics yields a simple analytic expression for spread complexity in terms of the Fourier transforms of the $k$-th nearest-neighbour spacing distributions. In particular, at leading order, the finite-time peak is controlled by the Fourier transform of the nearest-neighbour spacing distribution. The resulting framework describes both chaotic random-matrix ensembles and the integrable Poisson limit, identifies the spectral origin of the complexity peak and its late-time behaviour, and provides a direct connection between Krylov dynamics and spectral statistics.

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