Fermi liquid theory of electronic topological transitions and screening anomalies in metals

General expressions for the contributions of the Van Hove singularity (VHS) in the electron density of states to the thermodynamic potential Ω are obtained in the framework of a microscopic Fermi liquid theory. The renormalization of the singularities in Ω connected with the Lifshitz electronic topological transition (ETT) is found. Screening anomalies due to virtual transitions between VHS and the Fermi level are considered. It is shown that, in contrast with the one-particle picture of ETT, the singularity in Ω turns out to be two-sided for interacting electrons. 71.10.Ay Typeset using REVTEX 1 Electronic topological transitions (ETT) can take place in metals and alloys at the motion of the Van Hove singularities (VHS) of the electron density of states across the Fermi level at the variation of external parameters. Their investigation is now a welldeveloped branch of solid state physics (see, e.g., the reviews). Recently the interest in the problem has been revived by the observations of anomalies in the pressure dependence of lattice properties of Cd and Zn, which were explained in terms of ETT. To describe qualitatively the behavior of thermodynamic and transport properties of metals near ETT, correlation effects are to be taken into account since they are by no means small at typical electron densities. Up to now the influence of the interelectron interactions on ETT has been considered only for simplified models. An exact expression for the many-electron renormalization of the most singular contribution to the thermodynamic potential Ω near ETT has been obtained in the model of nearly free electrons. In comparison with the oneparticle expression for singular contribution to the thermodynamic potential Ω, correlation effects result in appearance of numerical factors which are expressed in terms of the effective mass of the quasiparticles and three-leg vertex γ for the wave vectors connecting VHS points in the Brillouin zone. Dzyaloshinskii demonstrated that in the two-dimensional (2D) case, in contrast with 3D one, γ diverges and, moreover, the ground state of many-electron system can be of a non-Fermi-liquid type provided that VHS are close enough to the Fermi level (see also the recent papers). In the 3D case the Fermi-liquid description is valid near ETT. However, the Fermi-liquid renormalization factors contain singularities themselves, and some anomalous contributions to the physical properties appear due to virtual transitions between the VHS point in the electron energy spectrum and the Fermi level (screening anomalies). It was found in that these anomalies make the singularity in Ω at ETT to be two-sided, i.e., of order of (±z) θ (±z)−const (∓z) θ (∓z) where z = EF −Ec; EF and Ec are the Fermi energy and the energy of VHS, θ (x > 0) = 1, θ (x < 0) = 0. However, these considerations were based on the calculations of particular diagrams in perturbation theory. In this work we present a general consideration of the singularities in the thermodynamic properties of metals near ETT in the framework of the rigorous microscopic Fermi liquid theory. We 2 restrict ourselves only to the consideration of the 3D case and do not discuss the much more complicated 2D case where the applicability of the Fermi liquid theory is doubtful. Let λ = nkσ be the set of electron quantum numbers in a metal: band index, quasimomentum and spin projection, respectively. For the normal Fermi liquid we have the following general expression for the Green function near the chemical potential level μ Gλ (E) = 1 E − ελ + μ− Σλ (E, μ) (1) = Zλ E + μ− ελ +G λ (E) , where ελ is the bare electron energy, ελ is the renormalized one that satisfies the equation ελ = ε 0 λ + Σλ (ελ − μ, μ) , (2) Σλ (E, μ) is the self-energy, ImΣλ (E = 0, μ) = 0, Zλ is the residue of the Green function in the pole and G λ (E) is the regular (incoherent or nonquasiparticle) part of the Green function. To find the singular contribution to the thermodynamic potential connected with the closeness of VHS in the renormalized spectrum to the Fermi energy EF = μ it is suitable to start from the Luttinger theorem