Quantum advantage in batteries for Sachdev-Ye-Kitaev interactions

A quantum advantage can be achieved in the unitary charging of quantum batteries if their cells are interacting. Here, we try to clarify with some analytical calculations whether and how this quantum advantage is achieved for sparse Sachdev-Ye-Kitaev (SYK) interactions and in general for fermionic interactions with disorder. To do this we perform a simple modelization of the interactions. In particular, we find that for $q$-point rescaled sparse SYK interactions the quantum advantage goes as $\mathrm{\Gamma }\sim N^{\frac{\alpha -q}{2}+1}$ for $q\ge \alpha \ge q\mathrm{/}2$ and $\mathrm{\Gamma }\sim N^{1-\frac{\alpha }{2}}$ for $q\mathrm{/}2>\alpha \ge 0$, where $\alpha$ is related to the connectivity and $N$ is the number of cells. This shows how we can get $\mathrm{\Gamma }\sim N$, i.e., an average power that scales as $N^2$ thanks to the disorder.