The structures of order parameters which determine the bounds of the phase states in the framework of the $CP^{1}$ Ginzburg-Landau model were considered. Using the formulation of this model in terms of the gauged order parameters (the unit vector ${b
f n}$, density $rho^{2}$ and momentum of particles ${bf c}$) we found that some universal properties of phases and field configurations are determined by the Hopf invariant, $Q$ and its generalizations. At a sufficiently high level of doping it was found that beyond the superconducting phase the charge distributions in the form of loops may be more preferable than those in the form of stripes. It was shown that in the phase with its mutual linking number $L<Q$ the transition to an inhomogeneous superconducting state with non-zero total momentum of pairs takes place. The universal mechanism of the topological coherence breaking of the superconducting state due to a decrease of the charge density was discussed.
We consider the form of the charge density nano-scale configurations in underdoped states of planar antiferromagnetic insulators in the framework of a soft variant of Faddeev-Niemi model. It is shown that there is such a level of doping and the tempe
rature range, where charge density distributions in the form of closed quasi-one-dimensional structures are more preferable.
We consider the structure of multi-meron knot action in the Yang-Mills theory and in the CP^1 Ginzburg-Landau model. Self-dual equations have been obtained without identifying orientations in the space-time and in the color space. The dependence of t
he energy bounds on topological parameters of coherent states in planar systems is also discussed. In particular, it is shown that a characteristic size of a knot in the Faddeev-Niemi model is determined by the Hopf invariant.
