Topological Defects in Systems with Two Competing Order Parameters: Application to Superconductors with Charge- and Spin-Density Waves

On the basis of coupled Ginzburg--Landau equations we study nonhomogeneous states in systems with two order parameters~(OP). Superconductors with superconducting OP~$\Delta$, and charge- or spin-density wave (CDW or SDW) with amplitude~$W$ are examples of such systems. When one of OP, say~$\Delta$, has a form of a topological defect, like, e.g., vortex or domain wall between the domains with the phases~$0$ and~$\pi$, the other OP~$W$ is determined by the Gross--Pitaevskii equation and is localized at the center of the defect. We consider in detail the domain wall defect for~$\Delta$ and show that the shape of the associated solution for~$W$ depends on temperature and doping (or on the curvature of the Fermi surface)~$\mu$. It turns out that, provided temperature or doping level are close to some discrete values~$T_{n}$ and~$\mu_{n}$, the spacial dependence of the function~$W(x)$ is determined by the form of the eigenfunctions of the linearized Gross--Pitaevskii equation. The spacial dependence of~$W_{0}$ corresponding to the ground state has the form of a soliton, while other possible solutions~$W_{n}(x)$ have nodes. Inverse situation~when~$W(x)$ has the form of a topological defect and~$\Delta(x)$ is localized at the center of this defect is also possible. In particular, we predict a surface or interfacial superconductivity in a system where a superconductor is in contact with a material that suppresses~$W$. This superconductivity should have rather unusual temperature dependence existing only in certain intervals of temperature. Possible experimental realizations of such non-homogeneous states of OPs are discussed.