Knots and links in the order parameter distributions of strongly correlated systems

Research on the coherent distribution of order parameters determining phase existence regions in the two-component Ginzburg–Landau model is reviewed. A major result of this research, obtained by formulating this model in terms of gauged order parameters (the unit vector field $\mathbf n$, the density $\rho^2$, and the particle momentum $\mathbf c$), is that some of the universal phase and field configuration properties are determined by topological features related to the Hopf invariant $Q$ and its generalizations. For sufficiently low densities, a ring-shaped density distribution may be favored over stripes. For an $L<Q$ phase ($L$ being the mutual linking index of the $\mathbf n$ and $\mathbf c$ field configurations), a gain in free energy occurs when a transition to a nonuniform current state occurs. A universal mechanism accounting for decorrelation with increasing charge density is discussed. The second part of the review is concerned with implications of non-Abelian field theory for knotted configurations. The key properties of semiclassical configurations arising in the Yang–Mills theory and the Skyrme model are discussed in detail, and the relation of these configurations to knotted distributions is scrutinized.