We investigate the properties of accretion flows onto a black hole (BH) with a mass of $M_{rm BH}$ embedded in an initially uniform gas cloud with a density of $n_{infty}$ in order to study rapid growth of BHs in the early Universe. In previous work, the conditions for super-Eddington accretion from outside the Bondi radius were studied by assuming that radiation produced at the vicinity of the central BH has a single-power-law spectrum $ u^{-alpha}$ at $h u geq 13.6~{rm eV}$ ($alpha sim 1.5$). However, radiation spectra depends on the BH mass and accretion rate. Here, we perform two-dimensional multi-frequency radiation hydrodynamical simulations taking into account more realistic radiation spectra associated with the properties of nuclear accretion disks. We find that the condition for a transition to super-Eddington accretion is alleviated for a wide range of masses ($10lesssim M_{rm BH}/M_{odot} lesssim 10^6$) because photoionization for accretion disk spectra are less efficient than those for single-power-law spectra. For disk spectra, the transition to super-Eddington is more likely to occur for lower BH masses because the radiation spectra become too hard to ionize the gas. Even when accretion flows are exposed to anisotropic radiation, the effect due to radiation spectra shrinks the ionized region and likely leads to the transition to a wholly neutral accretion phase. Finally, by generalizing our simulation results, we construct a new analytical criterion required for super-Eddington accretion; $(M_{rm BH}/10^5~M_{odot}) (n_{infty}/10^4~{rm cm^{-3}}) gtrsim 2.4~ (langleepsilonrangle /100~{rm eV})^{-5/9}$, where $langleepsilonrangle$ is the mean energy of ionizing radiation from the central BH.
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