Krylov Winding and Emergent Coherence in Operator Growth Dynamics

The operator wave function 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 wave function can acquire a phase that increases linearly with the operator's size, a phenomenon called 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 Krylov winding, whereby the operator wave function acquires a phase that 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 λ_{L}≤2α (with λ_{L} the Lyapunov exponent and α the growth rate), which ensures a linear phase dependence on size. For systems that do not saturate this bound, with h=λ_{L}/2α<1, the winding with Pauli size ℓ becomes superlinear, behaving as ℓ^{1/h}. We illustrate these results with two classes of microscopic models: the Sachdev-Ye-Kitaev (SYK) model and its variants, and a disordered k-local spin model.