Numerical Simulations of the Superfluid 3 He Order Parameter Using dyGiLa: A Homogeneous Rapid Quench to B Phase

The first-order A-B phase transition in superfluid $_{}^3$ He remains poorly understood, as homogeneous nucleation theory predicts negligible transition rates, while experiments observe rapid and reproducible phase conversion. Recent studies, including confined geometries and controlled supercooling experiments, point to the importance of non-equilibrium dynamics and localized energy deposition. In this work, we present dyGiLa, a massively parallel numerical framework for simulating the real-time evolution of the superfluid order parameter using time-dependent Ginzburg-Landau (TDGL) theory. The code evolves both the order parameter and the temperature field self-consistently, enabling the study of quenches and other non-equilibrium processes on large three-dimensional lattices. We demonstrate the capabilities of the framework through large-scale quench simulations, where we observe the dynamical emergence of the B phase, along with B-B domain walls and vortices. The onset of B-phase formation is consistent with a Kibble-Zurek-type mechanism, providing a natural pathway for phase conversion beyond homogeneous nucleation. These results represent a step toward a dynamical understanding of the A-B transition and establish superfluid $_{}^3$ He as a laboratory for studying non-equilibrium phenomena relevant to cosmological phase transitions.