  When a second-order phase transition is crossed at fine rate, the evolution
of the system stops being adiabatic as a result of the critical slowing down in
the neighborhood of the critical point. In systems with a topologically
nontrivial vacuum manifold, disparate local choices of the ground state lead to
the formation of topological defects. The universality class of the transition
imprints a signature on the resulting density of topological defects: It obeys
a power law in the quench rate, with an exponent dictated by a combination of
the critical exponents of the transition. In inhomogeneous systems the
situation is more complicated, as the spontaneous symmetry breaking competes
with bias caused by the influence of the nearby regions that already chose the
new vacuum. As a result, the choice of the broken symmetry vacuum may be
inherited from the neighboring regions that have already entered the new phase.
This competition between the inherited and spontaneous symmetry breaking
enhances the role of causality, as the defect formation is restricted to a
fraction of the system where the front velocity surpasses the relevant sound
velocity and phase transition remains effectively homogeneous. As a
consequence, the overall number of topological defects can be substantially
suppressed. When the fraction of the system is small, the resulting total
number of defects is still given by a power law related to the universality
class of the transition, but exhibits a more pronounced dependence on the
quench rate. This enhanced dependence complicates the analysis but may also
facilitate experimental test of defect formation theories.
