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Large‑q scaling with r = O(q) in SYK chains and full competition between transport and on‑site interactions

Develop a consistent large‑q scaling framework for complex SYK chains with r/2‑body hopping terms where r scales as O(q), in order to capture full competition between transport and on‑site q/2‑body interactions. Specifically, modify the Kadanoff–Baym equations to accommodate r = O(q), identify the appropriate time rescaling for finite‑time charge dynamics, and determine whether nontrivial charge flow occurs at finite times under this scaling regime.

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Background

The authors analyze charge dynamics in SYK chains with on‑site q/2‑body interactions and r/2‑body hopping. In the large‑q limit with r = O(q0), they obtain a closed wave‑like equation for local charge density and show how transport competes with on‑site interactions.

They point out that extending this analysis to r = O(q) is nontrivial and would require adjusting the Kadanoff–Baym structure and the time scaling to enable genuine competition in the large‑q regime. Establishing such a framework would clarify whether charge flow persists at finite times and how transport interacts with on‑site dynamics when r grows with q.

References

The case of $ r={(q) $ scaling for full competition remains an open problem. The Kadanoff-Baym structure will have to consider a different scaling for $\dot{\mathcal{Q}}_i$ in Eq. intermediate step for kb equation scaling with q and a proper time scaling will have to be introduced to allow for a true competition, otherwise without time rescaling ($t \neq q{3 / 2} \tau$), no charge flow occurs for any finite $t=\mathcal{O}\left(q0\right)$.

— Introduction to Sachdev-Ye-Kitaev Model: A Strongly Correlated System Perspective (2507.07195 - Jha, 9 Jul 2025) in Subsection “Dynamics of Local Charge Density for r={(q^0)}”, Section “Quench to a Chain: Charge Dynamics and Thermalization”