Superconductivity-induced shift of phonon frequency in the system with itinerant and strongly correlated electrons

We investigate a system consisting of strongly correlated localized and itinerant electron states mixed by optical vibration. The linear vibronic interaction causes the softening of active optical mode and it may induce a structural instability. The modification of renormalized phonon frequency by the gap in the energy of itinerant electrons in the superconducting phase has been established. The disposition of the electron spectrum and chemical potential influences substantially the corresponding effect. It is demonstrated that the presence of van Hove singularity in the center of itinerant electron band introduces qualitative changes into the superconductivity-induced shifts of phonon frequency.


Introduction
The mutual influence between superconductivity and phonon dynamics, including structural instability, has been an object of research for many years, see review [1] for earlier publications in this field. A number of effects related to the phonon dynamics, structural instabilities and symmetry breaking present in high temperature superconductors [2,3] have been investigated intensively both experimentally and theoretically. In particular, various aspects of phonon selfenergy effects caused by superconductivity were analyzed in Refs. [4,5,6,7].
In the present contribution we study the changes in phonon frequency, renormalized by the vibronic mixing of itinerant and strongly correlated localized electron states, due to the electron pairing gap in the spectrum of itinerant electrons in the superconducting phase. In the absence of superconductivity, the model was examined in Ref. [8] and the special case of such a scheme was considered as a possible reason for the tetragonal-orthorhombic transition in La 2−x M x CuO 4 [9,10]. The vibronic hybridization of electron states has been accepted as quite general mechanism for structural (ferroelectric) phase transitions [11,12].

Basic equations
We use the following model Hamiltonian for the electron-phonon system: Here H el is the Hamiltonian of electron subsystem containing localized strongly correlated electrons (Hamiltonian H l ) and itinerant superconducting band electrons (Hamiltonian H i ), H ph is the Hamiltonian of phonon subsystem and H el−ph describes the vibronic mixing of localized and itinerant electron states. In the case of localized electrons the one-electron annihilation and creation operators have been replaced by the Hubbard operators: where j is the index of lattice site, s is the spin index and η(↑, ↓) = ±1. In the Hamiltonian (2), ε 1 = ε d − µ and ε 2 = 2ε d + U − 2µ where ε d is the energy of a localized electron without Coulomb correlation, U/2 is the energy of Coulomb interaction between two electrons per one spin direction and µ is the chemical potential. In the Hamiltonian (3),

Renormalized phonon frequency
One can find the renormalized squared phonon frequency by applying the Shrieffer-Wolff transformation [13], generalized for the case of phonon-mediated dynamic hybridization between itinerant and localized electron states, to the Hamiltonian (1).
We will consider the disposition of electron states where the band of itinerant electrons is located between the lower Hubbard (LH) and upper Hubbard (UH) levels, i.e. ε d < ε min and ε d + U > ε max , see figure 2. Note also that such a set of electron states is roughly similar to the effective electron spectrum in copper-oxide systems [14,15] stemming from the Emery model [16,17]. Chemical potential intersects the conduction band of itinerant electrons. In what follows, we neglect the contributions containing AA, Then, by introducing the average numbers of elementary excitations and electrons, one obtains for the squared phonon frequency renormalized by electron-phonon interaction where It has been assumed that ε min − ε d ≫hω q and ε d + U − ε max ≫hω q in Eq. (6).
In the case T = 0 the average numbers ν(k) = n 0 = n 2 = 0, n 1 = 1 and we have for the squared frequency (6) the expression Further, we will analyze the superconductivity-induced shift of squared phonon frequency at zero temperature where ω ≡ ω q , Ω ≡ Ω q and Ω n is the value of Ω if ∆ sc = 0. where and K(m) is the complete elliptic integral of the first kind. The superconductivity-induced shifts ∆Ω 2 /ω 2 as the functions of chemical potential position are shown in figures 3, 4 for various dispositions of Hubbard levels. In the figures the energy of lower Hubbard leval ε d is fixed and the energy of the higher Hubbard level ε d + U is tuned by the variation of U .
In figure 3 the value U = 3 eV corresponds to the symmetric location of the Hubbard levels with regard to conduction band. In this case the curve ∆Ω 2 /ω 2 vs µ is also symmetric, see thick broken line in figure 3. The increase of U in figure 3 introduces moderate asymmetry for the location of the Hubbard levels. As a result the strong dependence ∆Ω 2 /ω 2 vs µ appears in the center of conduction band (µ = 0) related to the presence of van Hove singularity accompanied by the asymmetry of the function ∆Ω 2 (µ) in the domains µ < 0 and µ > 0. One can observe in figure 3 that ∆Ω 2 (µ) > 0, i.e. superconductivity induces here the hardening of phonon dynamics which is, however, suppressed in the region µ > 0 by the increase of U . If the chemical potential approaches the edges of conduction band, the effect of hardening increases.

Constant density of electron states
To illustrate more precisely the role of van Hove singularity in the superconductivity-induced shift of phonon frequency, we will consider for the comparison a model where the density of with κ = 2|V | 2 ρ/N 0 ω 2 .  For the constant density of itinerant electron states, the superconductivity-induced shifts ∆Ω 2 /ω 2 vs chemical potential position calculated on the basis of Eqs. (12) and (10)

Summary
• The impact of electron energy gap formation in the superconducting phase on phonon frequency renormalized by the vibronic mixing of itinerant and localized electron states has been studied.
• If the location of the lower and upper Hubbard levels with regard to the band of itinerant electrons is symmetric or weakly asymmetric, superconductivity induces the hardening of phonon dynamics. The dependence of the effect on µ is stronger near the edges and near the center of conduction band.
• If the location of the Hubbard levels is sufficiently asymmetric, the softening of phonon dynamics appears in certain regions of the chemical potential position.
• For the constant density of electron states the strong dependence of superconductivity-induced shift of phonon frequency on the chemical potential position near the center of conduction band disappears because in this case the van Hove singularity is absent in the center of conduction band.