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We study the energy transfer process in quantum battery systems consisting ofmultiple central spins and bath spins. Here with "quantum battery" we refer tothe central spins, whereas the bath serves as the "charger". For the singlecentral-spin battery, we analytically derive the time evolutions of the energytransfer and the charging power with arbitrary number of bath spins. For thecase of multiple central spins in the battery, we find the scaling-law relationbetween the maximum power $P_{max}$ and the number of central spins $N_B$. Itapproximately satisfies a scaling law relation $P_{max}\propto N_{B}^{\alpha}$,where scaling exponent $\alpha$ varies with the bath spin number $N$ from thelower bound $\alpha =1/2$ to the upper bound $\alpha =3/2$. The lower and upperbounds correspond to the limits $N\to 1$ and $N\gg N_B$, respectively. Inthermodynamic limit, by applying the Holstein-Primakoff (H-P) transformation,we rigorously prove that the upper bound is $P_{max}=0.72 B A \sqrt{N}N_{B}^{3/2}$, which shows the same advantage in scaling of a recent chargingprotocol based on the Tavis-Cummins model. Here $B$ and $A $ are the externalmagnetic field and coupling constant between the battery and the charger.

Lower and upper bounds of quantum battery power in multiple central spin systems