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 ${\lambda }_L\le 2\alpha$ (with ${\lambda }_L$ the Lyapunov exponent and $\alpha$ the growth rate), which ensures a linear phase dependence on size. For systems that do not saturate this bound, with $h={\lambda }_L/2\alpha <1$, the winding with Pauli size $\ell$ becomes superlinear, behaving as ${\ell }^{1/h}$. 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.

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• Information scrambling
• Quantum aspects of black holes
• Quantum chaos
• Quantum gravity
• Quantum statistical mechanics
• Quantum teleportation
• Quantum many-body systems
• Sachdev-Ye-Kitaev model