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.

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”