  Intermediate-mass black holes (with $\geq\!10^5\,M_\odot$) are promising
candidates for the origin of supermassive black holes (with
$\sim\!10^9\,M_\odot$) in the early universe (redshift $z\sim6$). Chon & Omukai
(2020) firstly pointed out the direct collapse black hole (DCBH) formation in
metal-enriched atomic-cooling halos (ACHs), which relaxes the DCBH formation
criterion. On the other hand, Hirano et al. (2021) showed that the magnetic
effects promote the DCBH formation in metal-free ACHs. We perform a set of
magnetohydrodynamical simulations to investigate star formation in the
magnetized ACHs with metallicities $Z/Z_\odot = 0$, $10^{-5}$, and $10^{-4}$.
Our simulations show that the mass accretion rate onto the protostars becomes
lower in metal-enriched ACHs than that of metal-free ACHs. However, many
protostars form from gravitationally and thermally unstable metal-enriched gas
clouds. Under such circumstances, the magnetic field rapidly increases as the
magnetic field lines wind up due to the spin of protostars. The region with the
amplified magnetic field expands outwards due to the orbital motion of
protostars and the rotation of the accreting gas. The amplified magnetic field
extracts the angular momentum from the accreting gas, promotes the coalescence
of the low-mass protostars, and increases the mass growth rate of the primary
protostar. We conclude that the magnetic field amplification is always realized
in the metal-enriched ACHs regardless of the initial magnetic field strength,
which affects the DCBH formation criterion. In addition, we find a
qualitatively different trend from the previous unmagnetized simulations in
that the mass growth rate is maximal for the extremely metal-poor ACHs with
$Z/Z_\odot = 10^{-5}$.
