  We investigate the formation of topological defects in the course of a
dynamical phase transition with different boundary conditions in a ring from
AdS/CFT correspondence. According to the Kibble-Zurek mechanism, quenching the
system across the critical point to symmetry-breaking phase will result in
topological defects -- winding numbers -- in a compact ring. By setting two
different boundary conditions, i.e., Dirichlet and Neumann boundary conditions
for the spatial component of the gauge fields in the AdS boundary, we achieve
the holographic superfluid and holographic superconductor models, respectively.
In the final equilibrium state, different configurations of the order parameter
phases for these two models indicate a persistent superflow in the holographic
superfluid, however, the holographic superconductor lacks this superflow due to
the existence of local gauge fields. The two-point correlation functions of the
order parameter also behave differently. In particular, for holographic
superfluid the correlation function is a cosine function depending on the
winding number. The correlation function for the holographic superconductor,
however, decays rapidly at short distances and vanishes at long distance, due
to the random localities of the gauge fields. These results are consistent with
our theoretical analysis.
