Effect of accreting tidally disrupted stars on the spin evolution of
  $\sim 10^6M_{\odot}$ black holes
  Accretion of tidally disrupted stars (TDSs) is expected to contribute
significantly to the growth of massive black holes (MBHs) with mass $\sim 10^6
M_{\odot}$ in galactic centers. In this paper, we quantitatively investigate
the effect of the TDS accretion on the spin evolution of these relatively small
MBHs, by also considering the accretion of gas-clouds with (many) chaotic
episodes. We find that the accretion of TDSs can play an important role or even
a dominant role in shaping the spin distribution of $\sim 10^6 M_{\odot}$ MBHs,
depending on the contribution fraction ($f_{\rm TDE}$) of the TDS accretion to
the MBH growth. If $f_{\rm TDE}$ is as large as $\gtrsim 0.9$, most $\sim
10^6M_{\odot}$ MBHs have low spins ($|a| \lesssim 0.3$); if $f_{\rm TDE}$ is as
small as $\lesssim 0.1$, most $\sim 10^6 M_{\odot}$ MBHs have high spins ($|a|
\gtrsim 0.7$). We also find that (1) the fraction of highly spinning $\sim 10^6
M_{\odot}$ MBHs in the TDS accretion states is smaller than that in the
gas-cloud (AGN) accretion states, which is a consequence of more rapid spin
decrease during the period of consecutive TDS accretion than the spin increase
during the AGN periods when the spin is large; (2) the fraction of retrograde
spin accretion in the TDS accretion states is almost the same as that of
prograde spin accretion, while it is negligible in the gas-cloud (AGN)
accretion states. Current scarce sample of AGNs ($\sim 10^6 M_{\odot}$) with
spin measurements hints an insignificant contribution from TDS accretion to MBH
growth. Future measurements on spins of $\sim 10^6 M_{\odot}$ MBHs may provide
stronger constraints on the importance of both AGN and TDS accretion states in
their growth history.
