Krylov Winding and Emergent Coherence in Operator Growth Dynamics
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arXiv
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2025
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| _version_ | 1866914468422746112 |
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| author | Perugu, Rishik Kobrin, Bryce Flynn, Michael O. Scaffidi, Thomas |
| author_facet | Perugu, Rishik Kobrin, Bryce Flynn, Michael O. Scaffidi, Thomas |
| contents | 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 $λ_L \leq 2 α$ (with $λ_L$ the Lyapunov exponent and $α$ the growth rate), which ensures a linear phase dependence on size. For systems which do not saturate this bound, with $h = λ_L / 2α<1$, the winding with Pauli size $\ell$ becomes \emph{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Krylov Winding and Emergent Coherence in Operator Growth Dynamics Perugu, Rishik Kobrin, Bryce Flynn, Michael O. Scaffidi, Thomas Quantum Physics Statistical Mechanics Strongly Correlated Electrons High Energy Physics - Theory 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 $λ_L \leq 2 α$ (with $λ_L$ the Lyapunov exponent and $α$ the growth rate), which ensures a linear phase dependence on size. For systems which do not saturate this bound, with $h = λ_L / 2α<1$, the winding with Pauli size $\ell$ becomes \emph{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. |
| title | Krylov Winding and Emergent Coherence in Operator Growth Dynamics |
| topic | Quantum Physics Statistical Mechanics Strongly Correlated Electrons High Energy Physics - Theory |
| url | https://arxiv.org/abs/2509.25331 |