  We investigate the non-equilibrium dynamics across the miscible-immiscible
phase separation in a binary mixture of Bose-Einstein condensates.
  The excitation spectra reveal that the Landau critical velocity vanishes at
the critical point, where the superfluidity spontaneously breaks down.
  We analytically extract the dynamical critical exponent $z=2$ from the Landau
critical velocity.
  Moreover, by simulating the real-time dynamics across the critical point, we
find the average domain number and the average bifurcation delay show universal
scaling laws with respect to the quench time.
  We then numerically extract the static correlation length critical exponent
$v=1/2$ and the dynamical critical exponent $z=2$ according to Kibble-Zurek
mechanism.
  The scaling exponents $(v=1/2, z=2)$ in the phase separation driven by
quenching the atom-atom interaction are different from the ones $(v=1/2, z=1)$
in the phase separation driven by quenching the Rabi coupling strength [PRL
\textbf{102}, 070401 (2009); PRL \textbf{107}, 230402 (2011)].
  Our study explores the connections between the spontaneous superfluidity
breakdown and the spontaneous defect formation in the phase separation
dynamics.
