On singularities of a superconductor order parameter due to an electronic topological phase transition

It is shown that if the difference between the Fermi energy εF and the critical energy εK at which an electronic topological transition occurs takes a value in the energy range 0 ≤ | εF − εK | ≤ ωr under the effect of elastic lattice deformations, then two kinds of root singularities occur in the dependence of the superconductor order parameter Δ (p,ω) on the momentum p and the frequency ω is the characteristic frequency of the order of the boundary frequency of the phonon spectrum). The first kind of singularity in Δ (p,ω) is related to the fact that according to the Éliashberg equation, a layer of isoenergetic surfaces ε(p) = εF ± ω′ takes part in the formation of the function Δ (p,ω). The quantity ω′ varies over the range from zero to ωr If the surface ε(p) = εK is in this layer, then for a virtual change in the Fermi energy εF ± ω′ at the frequency ω′ = | εF − εK | an electronic topological transition occurs. The singularity in Δ(p,ω) indeed corresponds to this “virtual” topological transition, which exerts no influence on the properties of metals in the normal state. These singularities will be observed in a broad range of elastic stresses (pressures) satisfying the condition | εF − εK | ≲ ωr. The second kind of singularity is associated with the fact that the topology of the Fermi surface changes under the effect of elastic stresses. Then two new points of tangency of sections of the Fermi surface form ε(p + q) = εF (or old ones vanish) in the neighborhood of the point q + p = pk with the isoenergetic phonon surfaces ωλ (q) = ω (pk is the singular point on the surface ε(p) = εK at which V(pK) = ∇pε(p) = 0). In this case the singularities in Δ(p,ω) will be observed at pressures P ≳ P0. The pressure P0 corresponds to the condition β = | εF − εK | ≃ (Δ0m1/3S)2/3 (Δ0 is the magnitude of the energy gap in the superconductor, m is the mass of an electron, and S is the speed of sound).

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