Topological Defect Formation beyond the Kibble-Zurek Mechanism in Crossover Transitions with Approximate Symmetries

The formation of topological defects during continuous second-order phase transitions is well described by the Kibble-Zurek mechanism (KZM). However, when the spontaneously broken symmetry is only approximate, such transitions become smooth crossovers, and the applicability of KZM in these scenarios remains an open question. In this Letter, we address this problem by analyzing both a weakly coupled Ginzburg-Landau model and a strongly coupled holographic setup, each featuring pseudospontaneous breaking of a global U(1) symmetry. In the slow quench regime, we observe a breakdown of the universal power-law scaling predicted by the Kibble-Zurek Mechanism. Specifically, the defect density acquires an exponential correction dependent on the quench rate, following a universal form dictated by the source of explicit symmetry breaking. Although these dynamics extend beyond the scope of the traditional KZM, we demonstrate that a generalized framework, which incorporates the effects of explicit symmetry breaking into the dynamical correlation length, remains valid and accurately captures the nonequilibrium defect formation across the entire range of quench rates.