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A correlation is established between the theories of superconductivity based on the concept of charge density waves (CDWs) and the translation invariant (TI) bipolaron theory. It is shown that CDWs are originated from TI-bipolaron states in the pseudogap phase due to the Kohn anomaly and form a pair density wave (PDW) for wave vectors corresponding to nesting. Emerging in the pseudogap phase, CDWs coexist with superconductivity at temperatures below those of superconducting transition, while their wave amplitudes decrease as a Bose condensate is formed from TI bipolarons, vanishing at zero temperature.

Presently, there is no agreement about the microscopic nature of the high-temperature superconductivity (HTSC). At the same time, there are phenomenological models such as a Ginzburg–Landau model [

TI bipolarons are formed at a temperature

In describing the Kohn anomaly, one usually proceeds from Frӧhlich Hamiltonian of the following form [

Renormalization of phonon frequencies corresponding to (2) is determined by the expression [

The Kohn anomaly describes vanishing of renormalized phonon modes

In the TI-bipolaron theory of SC [

It should be noted that spectral

The experimental evidence of the occurrence of renormalized phonons with zero energy in layered HTSC cuprates is the absence of a gap in the nodal direction in these materials which, by definition, is the phonon frequency in the TI bipolaron theory of SC.

The wave function of a TI-bipolaron with the wave vector

It follows from (6) that in the case of a weak and intermediate coupling (when TI-bipolaron states are metastable for

Subject to availability of a Fermi surface with a sharp boundary, the TI-bipolaron gas under consideration will have some peculiarities. Thus, if there are rather large fragments on this surface which can be superimposed by transferring one of them onto vector

The gain in the energy is caused by the above considered Kohn anomaly [

The general expression for

The suggested TI-bipolaron spectrum for a charge density wave. It looks as the roton spectrum, but is sharper at

For this spectrum, TI bipolarons will pass on to the state with the energy minimum

Hence, for

Energetical advantageousness of а condensate phase follows from expressions (7), (8), which suggest that a homogeneous Bose condensate has a lower energy on condition:

It should be noted that the scenario considered is close in many respects to the Fröhlich superconductivity model [

To be more specific, let us consider the case of a HTSC such as YBCO. The vector of a CDW in YBCO lies in

Still greater softening of phonon modes can be expected for

It should be noted that in the approach suggested, a difference between CDWs and PDWs disappears and

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

VL contributed to conception and design of the study, wrote the manuscript, read, and approved the submitted version.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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