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Intermediate-mass black holes (with $\geq\!10^5\,M_\odot$) are promisingcandidates 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 inmetal-enriched atomic-cooling halos (ACHs), which relaxes the DCBH formationcriterion. On the other hand, Hirano et al. (2021) showed that the magneticeffects promote the DCBH formation in metal-free ACHs. We perform a set ofmagnetohydrodynamical simulations to investigate star formation in themagnetized ACHs with metallicities $Z/Z_\odot = 0$, $10^{-5}$, and $10^{-4}$.Our simulations show that the mass accretion rate onto the protostars becomeslower in metal-enriched ACHs than that of metal-free ACHs. However, manyprotostars form from gravitationally and thermally unstable metal-enriched gasclouds. Under such circumstances, the magnetic field rapidly increases as themagnetic field lines wind up due to the spin of protostars. The region with theamplified magnetic field expands outwards due to the orbital motion ofprotostars and the rotation of the accreting gas. The amplified magnetic fieldextracts the angular momentum from the accreting gas, promotes the coalescenceof the low-mass protostars, and increases the mass growth rate of the primaryprotostar. We conclude that the magnetic field amplification is always realizedin the metal-enriched ACHs regardless of the initial magnetic field strength,which affects the DCBH formation criterion. In addition, we find aqualitatively different trend from the previous unmagnetized simulations inthat the mass growth rate is maximal for the extremely metal-poor ACHs with$Z/Z_\odot = 10^{-5}$.

Magnetic Effects Promote Supermassive Star Formation in Metal-enriched Atomic-cooling Halos