Stability and interactions of hopfions in confined chiral systems: Role of the background phase

We present a systematic study of isolated hopfions in confined layers of chiral magnets and cholesteric liquid crystals with strong surface anchoring and a film thickness fixed to one spiral pitch. We focus on the structure, stability, and interactions of hopfions under these conditions. Using numerical simulations combined with scaling-energy analysis, we demonstrate that hopfions possess well-defined equilibrium radii and energy minima in several background phases, including homogeneous states, undulated conical phases, and cholesteric fingers of the first type. Strong easy-plane or easy-axis anisotropy reduces the hopfion size and can ultimately drive a collapse into torons, whereas moderate anisotropy stabilizes hopfions against both collapse and expansion. Interhopfion interactions are quantified and found to depend strongly on the surrounding phase. Hopfions repel each other in the homogeneous state, whereas the "soft" undulated conical phase mediates attraction. In the latter case, a shell of positive energy density formed between a hopfion and the surrounding background promotes the formation of hopfion clusters, whose energetics is governed by the number of pairwise overlaps, providing a simple geometric framework for describing their self-assembly. By analyzing hexagonally ordered clusters with increasing numbers of hopfions, we extract the interaction energy. It saturates in the thermodynamic limit and amounts to several percent (≈6%) of the single-hopfion energy. Finally, we examine the possibility of periodic hopfion crystals. Within the considered one-pitch confinement, competing modulated phases of lower energy, together with the inability of isolated hopfions to attain a negative eigenenergy relative to the background, preclude the formation of genuine hopfion lattices. These findings provide a rigorous framework for reassessing previous claims of stable hopfion lattices and indicate that experimentally observed hopfion assemblies are better interpreted as finite clusters rather than true periodic crystals. Our study elucidates the mechanisms controlling hopfion stability and assembly in confined chiral systems, offering guidance for future experimental and theoretical investigations of topological solitons.