Structures of Order Parameters in Inhomogeneous Phase States of Strongly Correlated Systems

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 ${\bf n}$, 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.