paper

Krylov Winding and Emergent Coherence in Operator Growth Dynamics

arXiv:2509.25331 · doi:10.1103/bt23-4y1t

Abstract

The operator wavefunction provides a fine-grained description of quantum chaos and of the irreversible growth of simple operators into increasingly complex ones. Remarkably, at finite temperature this wavefunction can acquire a phase that increases linearly with the operator's size, a phenomenon called \emph{size winding}. Although size winding occurs naturally in a holographic setting, the emergence of a coherent phase in a scrambled operator remains mysterious from the standpoint of a thermalizing quantum many-body system. In this work, we elucidate this phenomenon by introducing the related concept of \textit{Krylov winding}, whereby the operator wavefunction acquires a phase which winds linearly with the Krylov index. We show that Krylov winding is a generic feature of quantum chaotic systems and is a direct consequence of the universal operator growth bound hypothesis. It gives rise to size winding under two additional conditions: (i) a low-rank mapping between the Krylov and size bases, which ensures phase alignment among operators of the same size, and (ii) the saturation of the ``chaos-operator growth'' bound (with the Lyapunov exponent and the growth rate), which ensures a linear phase dependence on size. For systems which do not saturate this bound, with , the winding with Pauli size becomes \emph{superlinear}, behaving as . We illustrate these results with two classes of microscopic models: the Sachdev-Ye-Kitaev (SYK) model and its variants, and a disordered -local spin model.

9 pages, 5 figures for main text + 6 pages, 4 figures for supplemental material; published version

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