The kinetic characteristics of the doped Mott-Hubbard material are considered within the realistic spin-fermion model which takes into account the strong spin-charge coupling. The kinetic equation constructed on the basis of the mechanism of carrier scattering on the spin fluctuations is solved using the multi-moment method, which allows one to analyze the temperature behavior of nonequilibrium distribution function in the problems of electrical resistivity rho and the Hall coefficient R H . The calculated dependences rho(T) and R H (T) for the underdoped and optimally doped regimes demonstrate good qualitative agreement with the experimental data. In particular, the Hall coefficient calculated for the underdoped regime reproduces the experimentally observed sharp drop and even a change in sign at low temperatures.

The review discusses the emergence of the spin-fermion model of cuprates and the formation of the spin-polaron concept of the electronic structure of hole-doped cuprate superconductors. This concept has allowed describing the properties of cuprates in the normal phase as well as the features of superconducting pairing in the unified approach. The derivation of the spin-fermion model from the Emery model in the regime of strong electronic correlations is described, demonstrating the appearance of strong coupling between the spins of copper ions and holes on oxygen ions. Such a strong interaction against the background of the singlet state of the spin subsystem of copper ions (quantum spin liquid) leads to the formation of special Fermi quasiparticles — nonlocal spin polarons. Under doping, the spin-polaron ensemble exhibits instability with respect to superconducting d-wave pairing, whereas superconducting s-wave pairing is not implemented. At the optimal doping, the transition to the superconducting phase occurs at temperatures corresponding to experimental data. It is shown that the superconducting d-wave pairing of spin-polaron quasiparticles is not suppressed by the Coulomb repulsion of holes located on neighboring oxygen ions. It is emphasized that, when the spec-tral characteristics of spin-polaron quasiparticles are taken into account, the calculated temperature and doping dependences of the London penetration depth are in good agreement with experimental data.

The spectral properties of an ensemble of spin-polaron quasiparticles have been studied within the spin–fermion model of cuprate superconductors using the method combining the Feynman diagram technique and the diagram technique for spin operators. It has been shown that strong spin–charge coupling results in the formation of the lower spin-polaron band separated by a wide energy gap from the band of bare holes. It has been shown that the spin-polaron band has a local minimum near the (π/2, π/2) point of the Brillouin zone. A class of diagrams for the self-energy part that have a fundamental significance for the description of the main features of the spin-polaron spectrum has been determined.

The Fermi excitation spectrum, the problem of Cooper instability, and the Londons magnetic field penetration depth in cuprate superconductors are considered using the unified conception based on accounting for the strong coupling between the spin of copper ions and holes at oxygen ions. This coupling leads to strong renormalization of the primary spectrum of oxygen holes with the formation of spin-polaron quasiparticles. Analysis of Cooper instability performed using the spin-polaron concept for different channels has shown that only the superconducting d -wave pairing occurs in the ensemble of spin-polaron quasiparticles, and there are no solutions corresponding to the s -wave pairing. It has been demonstrated that the superconducting d -wave pairing is not suppressed by the Coulomb repulsion of holes located at neighboring oxygen ions. This effect is due to peculiarities in the crystallographic structure of the CuO 2 plane and the aforementioned strong spin–fermion coupling. As a result, such interaction of holes is omitted in the kernel of the integral equation for the superconducting order parameter with the d -wave symmetry. It has been shown the Hubbard repulsion of holes and their interaction for the second coordination sphere of the oxygen sublattice for actual intensities of the interaction do not suppress the d -wave type of superconductivity. For the spin-polaron ensemble, we have analyzed the dependence of the Londons magnetic field penetration depth on the temperature and hole concentration. It has been established that the peculiarities of this dependence are closely related to specific features of the spin-polaron spectrum.

Taking into account the real crystalline structure of the CuO_2 plane within the spin-fermion model and using the diagram technique, the spin-polaron concept of the fermionic excitations in cuprate superconductors is implemented. It is shown that an account of the on-site scattering processes leads to considerable binding energy of the spin-polaron quasiparticles. An account of the two-site spin-fermion scattering processes results in the energy spectrum and spectral properties of the spin-polaron quasiparticles which agree well with experimental data on cuprate superconductors.

Taking into account the real crystalline structure of the \(\hbox {CuO}_2\) plane and the strong spin-fermion coupling, we study the influence of the intersite Coulomb repulsion between holes on the Cooper instability of the spin-polaron quasiparticles in cuprate superconductors. The analysis shows that only the superconducting d-wave pairing is implemented in the whole region of doping, whereas the solutions of the self-consistent equations for the s-wave pairing are absent. It is shown that intersite Coulomb interaction \(V_1\) between the holes located at the nearest oxygen ions does not affect the d-wave pairing, because its Fourier transform \(V_q\) vanishes in the kernel of the corresponding integral equation. The intersite Coulomb interaction \(V_2\) of quasiparticles located at the next-nearest oxygen ions does not vanish in the integral equations, however, but it is also shown that the d-wave pairing is robust toward this interaction for physically reasonable values of \(V_2\).

Taking into account the real crystalline structure of the CuO2 plane and the strong spin-fermion coupling, the influence of the on-site Coulomb repulsion of holes Up and the intersite Coulomb repulsion V2 between holes located at the next-nearest-neighbor oxygen ions on the formation of the superconducting gap with the d-wave symmetry of the order parameter of the spin-polaron quasiparticles is studied. It is shown that the formation of the resulting superconducting gap within the spin-fermion model is caused by three components. The dependence of the narrowing of the superconducting gap on the values Up and V2 is analyzed.

Within the spin-fermion model for cuprate superconductors, the influence of the intersite Coulomb interactions V-2 and V'(2) between holes located at the next-nearest-neighbor oxygen ions of CuO2 plane on the implementation of the d(x)(2)-y(2)-wave pairing is studied. It is shown that d-wave pairing can be suppressed only for unphysically large values of V-2 and V'(2).

The effect of the Coulomb repulsion of holes on the Cooper instability in an ensemble of spin–polaron quasiparticles has been analyzed, taking into account the peculiarities of the crystallographic structure of the CuO2 plane, which are associated with the presence of two oxygen ions and one copper ion in the unit cell, as well as the strong spin–fermion coupling. The investigation of the possibility of implementation of superconducting phases with d-wave and s-wave of the order parameter symmetry has shown that in the entire doping region only the d-wave pairing satisfies the self-consistency equations, while there is no solution for the s-wave pairing. This result completely corresponds to the experimental data on cuprate HTSC. It has been demonstrated analytically that the intersite Coulomb interaction does not affect the superconducting d-wave pairing, because its Fourier transform V q does not appear in the kernel of the corresponding integral equation.

It has been shown that, because of the two-orbital character of the subsystem of holes located at oxygen sites and the spatial separation of this subsystem from that of spins at copper ions, the superconducting phase in high-T (c) superconductors is stable with respect to the strong Coulomb repulsion of holes located at nearestneighbor oxygen sites if the order parameter has the symmetry. This effect is due to the symmetry characteristics of the Coulomb potential, owing to which the equation determining the Cooper pairing in the channel does not include this potential.

We demonstrate the instability of the normal state of purely repulsive fermionic systems towards the transition to the Kohn-Luttinger superconducting state. We construct the superconducting phase diagrams of these systems in the framework of the Hubbard and Shubin-Vonsovsky models on the square and hexagonal lattices. We show that an account for the long-range Coulomb interactions, as well as the Kohn- Luttinger renormalizations, lead to an increase in the critical superconducting temperatures in various materials, such as high-temperature superconductors, idealized monolayer, and bilayer of doped graphene. Additionally, we discuss the role of the structural disorder and the nonmagnetic impurities in superconducting properties of real graphene systems.

It is shown that an account for the space separatedness of the two-orbital subsystem of the oxygen holes and the subsystem of the localized spins of copper ions in high-T$_c$ cuprate superconductors leads to the stability of the superconducting $d_{x^2-y^2}$-wave pairing towards the strong Coulomb repulsion between holes located at the nearest oxygen ions. This effect is due to the fact that the Coulomb potential slips out of the equation for the Cooper pairing in the $d_{x^2-y^2}$-wave channel owing to the properties of symmetry.

The effect of the long-range Coulomb interaction on the formation of the Kohn–Luttinger superconductivity in monolayer doped graphene is studied disregarding the Van der Waals potential of the substrate and both magnetic and non-magnetic impurities. It is shown that the allowance for the Kohn–Luttinger renormalizations up to the second order in perturbation theory in the on-site Hubbard interaction inclusively, as well as in the intersite Coulomb interaction, significantly affects the interplay between the superconducting phases with the f -wave, p+ip -wave, and d + id -wave symmetries of the order parameter. It is demonstrated that taking Coulomb repulsion of electrons located at the next-nearest neighboring atoms in such a system into account changes qualitatively the phase diagram and enhances the critical temperature of the transition to the superconducting phase.

The effect of Coulomb interaction between Dirac fermions on the formation of the Kohn-Luttinger superconducting state in bilayer doped graphene is studied disregarding of the effect of the van der Waals potential of the substrate and impurities. The phase diagram determining the boundaries of superconductive domains with different types of symmetry of the order parameter is built using the extended Hubbard model in the Born weak-coupling approximation with allowance for the intratomic, interatomic, and interlayer Coulomb interactions between electrons. It is shown that the Kohn-Luttinger polarization contributions up to the second order of perturbation theory in the Coulomb interaction inclusively and an account for the long-range intraplane Coulomb interactions significantly affect the competition between the superconducting phases with the f -, p + ip -, and d + id -wave symmetries of the order parameter. It is demonstrated that the account for the interlayer Coulomb interaction enhances the critical temperature of the transition to the superconducting phase.

We discuss the mechanisms of unconventional superconductivity and superfluidity in 3D and 2D fermionic systems with purely repulsive interaction at low densities. We construct phase diagrams of these systems and find the areas of the superconducting state in free space, as well as on the lattice in the framework of the Fermi-gas model with hard-core repulsion, the Hubbard model, the Shubin-Vonsovsky model, and the $t-J$ model. We demonstrate that the critical superconducting temperature can be greatly increased in the spin-polarized case or in a two-band situation already at low densities. The proposed theory is based on the Kohn-Luttinger mechanism or its generalizations and explains or predicts anomalous $p$-, $d$-, and $f$-wave pairing in various materials, such as high-temperature superconductors, the idealized monolayer and bilayer of doped graphene, heavy-fermion systems, layered organic superconductors, superfluid $^3$He, spin-polarized $^3$He mixtures in $^4$He, ultracold quantum gases in magnetic traps, and optical lattices.