  In this paper, we investigate the interaction between early dark energy (EDE)
and scalar field dark matter, proposing a coupled scalar fields model to
address the Hubble tension and $S_8$ tension. While the EDE model successfully
alleviates the Hubble tension, it exacerbates the $S_8$ tension. To mitigate
the negative impact of EDE, we introduce the interaction between EDE and dark
matter. Specifically, we replace cold dark matter with scalar field dark
matter, given its capability to suppress structure growth on small scales. We
constrained the new model using cosmological observations including the
temperature and polarization anisotropy power spectra data of cosmic microwave
background radiation (CMB) from \textit{Planck} 2018 results, baryon acoustic
oscillations (BAO) measurements extracted from 6dFGS, SDSS and BOSS, the
Pantheon sample of type Ia supernovae (SNIa), the local distance-ladder data
(SH0ES), and the Dark Energy Survey Year-3 data. Employing Markov Chain Monte
Carlo method, we find that this novel model yields best-fit values of $H_0$ and
$S_8$ equal to $71.13$ km/s/Mpc and $0.8256$, respectively. Compared to the
$\Lambda$CDM model, the new model alleviates the Hubble tension but still fails
to resolve the $S_8$ tension. However, we obtain a smaller value of $S_8$
compared to the result of $0.8316$ obtained for EDE model, which mitigates to
some extent the shortcoming of the EDE model.
