Intertwined Swirling Polarization States in BaTiO$_3$ with Embedded BaZrO$_3$ Nanoregions
Ferroelectric materials embedded with dielectric inclusions offer a unique platform for exploring novel topological polar textures. Using first-principles-based atomistic simulations, we investigate the polarization behavior of a BaTiO$_3$ matrix containing segregated BaZrO$_3$ nanoregions. We demonstrate that the polar texture in three-dimensionally ordered arrays of dielectric inclusions is governed by their size and spacing, revealing three distinct regimes. At large separations, the nanocomposite exhibits bulk-like BaTiO3 phase transitions, while at smaller spacings, interconnected swirling polarization patterns give rise to vortex supercrystal states. We analyze the stabilization mechanisms of these states and show that each regime is characterized by distinct switching behavior. Furthermore, we find that nanocomposites with randomly distributed dielectric inclusions exhibit swirling polarization textures, giving rise to an amorphous network of entangled vortices. Our findings provide new insights into the physics of relaxor ferroelectrics, are consistent with recent experimental observations, and open up new possibilities for designing materials with emergent topological functionalities.