Triggering the formation of direct collapse black holes by their
  congeners
  Direct collapse black holes (DCBHs) are excellent candidates as seeds of
supermassive black holes (SMBHs) observed at $z \gsim 6$. The formation of a
DCBH requires a strong external radiation field to suppress $\rm H_2$ formation
and cooling in a collapsing gas cloud. Such strong field is not easily achieved
by first stars or normal star-forming galaxies. Here we investigate a scenario
in which the previously-formed DCBH can provide the necessary radiation field
for the formation of additional ones. Using one-zone model and the simulated
DCBH Spectral Energy Distributions (SEDs) filtered through absorbing gas
initially having column density $N_{\rm H}$, we derive the critical field
intensity, $J_{\rm LW}^{\rm crit}$, to suppress $\rm H_2$ formation and
cooling. For the SED model with $N_{\rm H}=1.3\times10^{25}$ cm$^{-2}$,
$8.0\times10^{24}$ cm$^{-2}$ and $5.0\times10^{24}$ cm$^{-2}$, we obtain
$J_{\rm LW}^{\rm crit}\approx22$, 35 and 54, all much smaller than the critical
field intensity for normal star-forming galaxies ($J_{\rm LW}^{\rm crit}\simgt
1000$). X-ray photons from previously-formed DCBHs build up a high-$z$ X-ray
background (XRB) that may boost the $J_{\rm LW}^{\rm crit}$. However, we find
that in the three SED models $J_{\rm LW}^{\rm crit}$ only increases to
$\approx80$, 170 and 390 respectively even when $\dt{\rho}_\bullet$ reaches the
maximum value allowed by the present-day XRB level ($0.22, 0.034,
0.006~M_\odot$yr$^{-1}$Mpc$^{-3}$), still much smaller than the galactic value.
Although considering the XRB from first galaxies may further increase $J_{\rm
LW}^{\rm crit}$, we conclude that our investigation supports a scenario in
which DCBH may be more abundant than predicted by models only including
galaxies as external radiation sources.
