  We solve the Bogoliubov-de Gennes equations self-consistently for the
$d$-wave order parameter in nanoscale $d$-wave systems with [110] surfaces and
show that spontaneous time-reversal symmetry (TRS) breaking occurs at low
temperatures due to a spontaneously induced complex order parameter of extended
$s$-wave symmetry. The Andreev surface bound states, which are protected by a
one-dimensional (1D) topological invariant in the presence of TRS, are gapped
by the emergence of this new order parameter. The extended $s$-wave order
parameter is localized within a narrow region near the surfaces, which is
consistent with the fact that topological protection of the gapless Andreev
surface states is characterized by the 1D topological invariant. In this
TRS-breaking phase, not only is the complex order parameter induced, but also
the $d$-wave order parameter itself becomes complex. Furthermore, the
disappearance of topological protection brings about novel vortex phenomena
near the surfaces. We show that vortex-antivortex pairs are formed in the
extended $s$-wave order parameter along the surfaces if the side length of a
nanoisland or the width of an infinitely long nanoribbon is relatively large.
