Abstract:
A topological crossover, associated with the collapse of the Fermi surface in strongly correlated Fermi systems, is examined. It is demonstrated that in these systems, the temperature domain where standard Fermi liquid results hold dramatically narrows, because the Landau regime is replaced by a classical one. The impact of the collapse of the Fermi surface on pairing correlations is analyzed. In the domain of the Lifshitz phase diagram where the Fermi surface collapses, splitting of the BCS superconducting phase transition into two different ones of the same symmetry is shown to occur.

Abstract:
Two different scenarios of the quantum critical point (QCP), a zero-temperature instability of the Landau state, related to the divergence of the effective mass, are investigated. Flaws of the standard scenario of the QCP, where this divergence is attributed to the occurrence of some second-order phase transition, are demonstrated. Salient features of a different {\it topological} scenario of the QCP, associated with the emergence of bifurcation points in equation $\epsilon(p)=\mu$ that ordinarily determines the Fermi momentum, are analyzed. The topological scenario of the QCP is applied to three-dimensional (3D) Fermi liquids with an attractive current-current interaction.

Abstract:
Employing the duality between the momentum distribution $n(p)$ and density distribution $\rho(r)$, problems of theory of systems with flat bands, pinned to the Fermi surface, are discussed. We propose that the Lifshitz topological phase transition associated with the formation of additional pockets of the Fermi surface is the precursor of fermion condensation.

Abstract:
The fundamental structure of the full set of solutions of the BCS $^3 P_2$ pairing problem in neutron matter is established. The relations between different spin-angle components in these solutions are shown to be practically independent of density, temperature, and the specific form of the pairing interaction. The spectrum of pairing energies is found to be highly degenerate.

Abstract:
The separation method developed earlier by us [Nucl. Phys. {\bf A598} 390 (1996)] to calculate and analyze solutions of the BCS gap equation for $^1$S$_0$ pairing is extended and applied to $^3$P$_2$--$^3$F$_2$ pairing in pure neutron matter. The pairing matrix elements are written as a separable part plus a remainder that vanishes when either momentum variable is on the Fermi surface. This decomposition effects a separation of the problem of determining the dependence of the gap components in a spin-angle representation on the magnitude of the momentum (described by a set of functions independent of magnetic quantum number) from the problem of determining the dependence of the gap on angle or magnetic projection. The former problem is solved through a set of nonsingular, quasilinear integral equations, providing inputs for solution of the latter problem through a coupled system of algebraic equations for a set of numerical coefficients. An incisive criterion is given for finding the upper critical density for closure of the triplet gap. The separation method and its development for triplet pairing exploit the existence of a small parameter, given by a gap-amplitude measure divided by the Fermi energy. The revised BCS equations admit analysis revealing universal properties of the full set of solutions for $^3$P$_2$ pairing in the absence of tensor coupling, referring especially to the energy degeneracy and energetic order of these solutions. The angle-average approximation introduced by Baldo et al. is illuminated in terms of the separation-transformed BCS problem and the small parameter expansion...

Abstract:
We study the renormalization of the Fermi-liquid parameters in the vicinity of a density wave quantum phase transition, which should occur in MOSFET systems at low densities. First, using a perturbative RPA treatment of fluctuations, we calculate the electronic self-energy and show that the effective mass diverges at the density wave transition point. Second, we go beyond perturbation theory, making use of the exact Pitaevskii identities. Within this exact analysis, we also find a divergence of the effective mass, which occurs at higher densities in the fluctuation region, as compared to the perturbation theory. This result signals the break-down of conventional Fermi-liquid description in the vicinity of the transition point. The divergence of the effective mass gives rise to a singular behavior of the electronic compressibility. We suggest that the experimentally observed enhancement of the effective mass is a precursor to a second order thermodynamic phase transition into a glassy density wave state.

Abstract:
Properties of superfluid states of two-dimensional electron systems with critical antiferromagnetic fluctuations are investigated. These correlations are found to result in the emergence of rapidly varying in the momentum space terms in all components of the mass operator, including the gap function $\Delta({\bf p})$. It is shown that a domain, where these terms reside, shrinks with the temperature, leading to a significant difference between the temperature $T_c$, at which superconductivity is terminated, and the temperature $T^*$, where the gap in the single-particle spectrum vanishes.

Abstract:
The self-consistent theory of the fermion condensation, a specific phase transition which results in a rearrangement of the single particle degrees of freedom in strongly correlated Fermi systems is developed. Beyond the phase transition point, the single particle spectra are shown to be flat. The interplay between the flattening and the damping of the single particle spectra at $T\to 0$ is investigated. The width $\gamma(\epsilon)$ of the single particle states is found to grow up linearly with $\epsilon$ over a wide range of energy as in a marginal Fermi liquid. Our results gain insight into the success of the phenomenological theory of the normal states of high-temperature superconductors by Varma et al.

Abstract:
Properties of strongly correlated two-dimensional (2D) electron systems in solids are studied on the assumption that these systems undergo a phase transition, called fermion condensation, whose characteristic feature is flattening of the electron spectrum $\epsilon({\pf p})$. Unlike the previous models in the present study, the decay of single-particle states is properly taken into account. Remarkably, the value of the topological charge ($N=1/2$) remains unchanged, supporting the view that systems with a fermion condensate form a separate class of Fermi liquids. Results of calculations are found to be in qualitative agreement with ARPES data.

Abstract:
The problem of pairing in anisotropic electron systems possessing patches of fermion condensate in the vicinity of the van Hove points is analyzed. Attention is directed to opportunities for the occurrence of non-BCS pairing correlations between the states belonging to the fermion condensate. It is shown that the physical emergence of such pairing correlations would drastically alter the behavior of the single-particle Green function, the canonical pole of Fermi-liquid theory being replaced by a branch point.