The study of strongly correlated systems remains highly relevant due to recent theoretical and experimental advances. One example is quantum dots—zero-dimensional electronic systems in which electron motion is confined in all spatial directions. When a quantum dot is coupled to metallic leads, it forms a nanoscale electronic device. Of particular interest is the description of nonequilibrium electron transport through such systems, where strong Coulomb correlations and the influence of external magnetic or electric fields play a crucial role. These structures provide the basis for modern nanoscale transistors, qubits, and other quantum devices.
The nonequilibrium electron transport is investigated within the Keldysh nonequilibrium Green's function formalism. To account for strong correlations, both analytical approaches (equations of motion, the Hubbard approximation, and the non-crossing approximation (NCA)) and numerical methods (quantum Monte Carlo (QMC), inchworm QMC, numerical renormalization group (NRG), and density matrix renormalization group (DMRG)) are employed.
Another important class of strongly correlated systems is represented by interacting relativistic bosons (pions, kaons, and gluons), which are of considerable interest in the physics of neutron stars, heavy nuclei, and heavy-ion collisions. Their theoretical description requires accounting for both interparticle interactions and the possibility of Bose–Einstein condensation. Commonly used approaches include the polarization operator, chiral perturbation theory, perturbative QCD, and lattice QCD.
The first part of the dissertation is devoted to the investigation of nonequilibrium transport through a single-level quantum dot described by the Anderson model in the presence of an external magnetic field. The Green's functions are calculated within an improved approximation similar to the Hubbard-I approximation, which becomes exact both in the atomic limit and in the absence of electron–electron interactions. Analytical expressions for the electric current are derived for both the stationary regime and the time-dependent bias voltage within the wide-band limit approximation. Numerical calculations of the transport characteristics are performed and compared with the results obtained using the NCA and nonequilibrium QMC methods.
The second part addresses a double quantum dot system with both intra-dot and inter-dot Coulomb correlations, where the upper quantum dot acts as a gate. The transport properties are studied within the NCA, which allows the description of the Kondo effect. It is shown that at low temperatures the system enters a narrow Kondo regime, resulting in highly sensitive switching between three distinct states: an insulating state, a normal conducting state, and a resonant tunneling state.
The final part of the dissertation investigates the thermodynamic properties of a system of interacting bosons and antibosons at finite temperatures within a Skyrme-like mean-field model. Two cases are considered: a system with zero chemical potential and a system with a conserved isospin density.
It is shown that for sufficiently strong attractive interactions and zero chemical potential, the system undergoes a first-order phase transition to a Bose-condensed phase. The condensed phase is characterized by a constant particle density and a finite critical temperature, T_c>0. The discontinuity of the energy density at T_c confirms that the phase transition is a phase transition of the first order.
For the system with a conserved isospin density, it is demonstrated that, regardless of the strength of the attractive interaction, a second-order phase transition to the Bose–Einstein condensed phase occurs at the critical temperature T_c. This phase exists in the temperature interval 0≤T≤T_c. It is shown that the heat capacity exhibits a discontinuity in its first derivative at T_c, while Bose–Einstein condensation occurs only for the particle species with the higher number density, namely the π− mesons.
