  The Kibble-Zurek mechanism (KZM) describes the non-equilibrium dynamics and
topological defect formation in systems undergoing second-order phase
transitions. KZM has found applications in fields such as cosmology and
condensed matter physics. However, it is generally not suitable for describing
first-order phase transitions. It has been demonstrated that transitions in
systems like superconductors or charged superfluids, typically classified as
second-order, can exhibit weakly first-order characteristics when the influence
of fluctuations is taken into account. Moreover, the order of the phase
transition (i.e., the extent to which it becomes first rather than second
order) can be tuned. We explore quench-induced formation of topological defects
in such tunable phase transitions and propose that their density can be
predicted by combining KZM with nucleation theory.
