In this thesis, multi-component quantum systems were studied, such as multiband superconductors, nano and atomically thin films, and multi-component Bose-Einstein condensates. New phenomena, which appear exclusively in these multi-component systems, are inherently connected with the complexity of interacting many-particle system stemming from the multi-component nature and the competing effects arising between constituent components of the system. Due to those exciting properties, multi-component systems provide numerous prospects and challenges for future studies. In this thesis, the solid platform for such studies was provided by deriving the consistent Ginzburg-Landau theory for multiband superconductors and superconducting nanofilms, bearing in mind that the GL theory is to date the most convenient theoretical tool to study superconducting properties and related phenomena in the proximity of the critical temperature Tc. Rotating and harmonically trapped multi-component Bose-Einstein condensates were also studied in this thesis as example of a multicomponent system that exhibits rich and pronounced quantum phenomena, which can then be experimentally realized and further manipulated in a broad range of parameters.

In what follows, the main results of the thesis are summarized in concreto.

In Chapter 2, the Ginzburg-Landau theory for the multiband superconductors was derived from the multiband BCS Hamiltonian. The derivation is based on the Gor'kov truncation of the matrix gap equation. The procedure invokes the expansion of the band gaps in powers of τ (where τ=1-T/Tc). In the expansion, we removed incomplete contributions into the band gaps which are of orders higher than τ1/2, i. e., higher than the precision of the solution of the original Gor'kov truncation. After performing this procedure, the accuracy of the gap matches the accuracy of the Gor'kov truncation. Further, we considered two scenarios for the solution for the critical temperature Tc. When the solution for Tc is not degenerate, we found that the Ginzburg-Landau theory of a multiband superconductor maps onto a single-component GL formalism in which only a unique order parameter exists, and, as a consequence, the spatial profiles of all band gaps are equivalent. If the solution for Tc is degenerate, which appears due to a symmetry of the system, the Ginzburg-Landau theory acquires multiple order parameters. The detailed analysis was performed for the three-band system, treated as a prototype of a multiband superconductor. For the simple three-band model of pnictides with dominant interband couplings, it was shown that the solution for Tc is twofold degenerate, resulting in existence of two order parameters in the theory, which in turn leads to appearance of non trivial phase difference between the gaps or, so-called, chiral solutions. It was demonstrated that chiral state can be the ground state of such a three-band superconductor, thus can be experimentally realized. The explicit microscopic expressions for the otherwise phenomenological coefficients of the Ginzburg-Landau theory were found.

In Chapter 3, it was demonstrated that due to the size quantization of the electron motion limited in one of the dimensions, single-crystalline metallic nanofilms exhibit multiband structure. The Ginzburg-Landau theory appropriate for single-crystalline metallic nanofilms was derived. In the derivation, the suitable BCS Hamiltonian was constructed for such a 2D system by integrating out the coordinate in which the electron motion is limited. The matrix gap equations were obtained for this multiband system from the newly constructed Hamiltonian. Subsequently, the same procedure of the Gor'kov truncation on removing incomplete contributions in the band gaps, similar to the one used in Chapter 2, was employed. The explicit expressions for the coefficients of the Ginzburg-Landau formalism for the single-crystalline metallic nanofilms were also found. The obtained formalism is computationally convenient and efficient, and will serve as a powerful theoretical tool for further investigations of the effects of the multiple-subband structure on, e.g., vortex configurations, critical phenomena and superconducting fluctuations in nano-thin single-crystalline samples.

In the second half of the thesis, the multicomponent Bose-Einstein condensates were studied. In Chapter 4, rotating and harmonically trapped Rabi-coupled three-component Bose-Einstein condensates were studied. This system was shown to host unconventional vortex lattices in the rotating ground state of the system. It was demonstrated that the found states can be topologically characterized as a two-dimensional lattice of skyrmions. To classify the states with different skyrmionic lattices, the average topological index was calculated as the average of pairwise CP1 topological indices of each pair of components. Different skyrmionic vortex lattices were then grouped into two phase diagrams in the parameter space of the intercomponent interaction strength and Rabi coupling frequency of the two components, for other parameters fixed. In this study, equal contributions of all three components were assumed, and the Rabi couplings were the only varied parameters which is possible to achieve in current experiments. Besides the emerging plethora of possible vortex lattice topologies, it was also shown that at certain combinations of signs and values of the Rabi frequencies relative phase frustration arises in the system resulting in some of the pairwise Rabi energies to become heavily suppressed. Such Rabi suppression leads to an effective reduction of the three-component BEC to a density-density-coupled two-component BEC. These exciting features are not only of fundamental importance, but their observation in an experiment can be used as a proof of frustration in the system under consideration.

In Chapter 5, two-component Bose-Einstein condensates in rotating harmonic traps were investigated in three different regimes. The first regime is miscible with intercomponent attractive interaction, the second represents the miscible regime with intercomponent repulsive interaction, while the third is the immiscible regime with intercomponent repulsive interaction, with the consequently separated phases. Stable multiquantum vortices in all three regimes were found, and it was shown that their stability is an inherent property of the harmonically confined, mass-imbalanced two-component system. The discovered states were classified into spin-skyrmion (coreless) and spin-vortex (cored) variants, both of which can be realized in a 87Rb-41K BEC with the current experimental techniques, and both of which are novel phases in the field.