# Translation-Invariant Bipolarons and Charge Density Waves in High-Temperature Superconductors

- Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow, Russia

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.

## Introduction

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 [1], a model of charge density waves (CDWs) or pair density waves (PDWs) (the recent review on the theory and experiment with CDW/PDW in high-temperature superconductors, ultracold atomic gases, and mesoscopic devices was published by Agterberg et al. [2]) and a model of spin density waves which enable one to describe numerous HTSC experiments [3]. These models are silent on the nature of paired states taking part in SС. In author’s works [4–7], paired states mean translation invariant (TI) bipolaron states formed by a strong electron–phonon interaction similar to Cooper pairs (for the review of earlier pioneering works devoted to superconductivity based on the theory of small-radius bipolarons, see [8]). TI bipolarons are plane waves with a small correlation length capable of forming a Bose–Einstein condensate with high transition temperature which possesses SC properties. A correlation between the Bardeen–Cooper–Schrieffer (BCS) theory [9] and the Ginzburg–Landau theory was established by Gor’kov [10]. The aim of this work was to establish a correlation between the TI-bipolaron theory of SC and CDW (PDW).

## Results

### General Relations for the Spectrum of a Moving TI Bipolaron

TI bipolarons are formed at a temperature

In describing the Kohn anomaly, one usually proceeds from Frӧhlich Hamiltonian of the following form [11]:

where the first term corresponds to a free electron gas;

Renormalization of phonon frequencies corresponding to (2) is determined by the expression [11]:

where

The Kohn anomaly describes vanishing of renormalized phonon modes

In the TI-bipolaron theory of SC [4–7], bipolarons are believed to be immersed into an electron gas. The properties of such bipolarons are also described by Fröhlich Hamiltonian of the form (2), but with a field of already renormalized phonons with energies

It should be noted that spectral equation 1 is independent of the form of

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

where

where

With the use of (6) expression for

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

where *α* is a constant of the electron–phonon interaction; that is, mass *α*, good approximations for

### TI Bipolarons and Charge Density Waves

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

that is,

The gain in the energy is caused by the above considered Kohn anomaly [14] which implies that for

The general expression for *δ*-shaped, in which the *δ*-shaped minimum will correspond to

**FIGURE 1**. 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 [15]. In the Frӧhlich model, it was assumed that two electrons with opposite momenta (as in BCS) on the Fermi surface are connected by a phonon with the wave vector

## Discussion—Comparison With Experiment

To be more specific, let us consider the case of a HTSC such as YBCO. The vector of a CDW in YBCO lies in *ab*-plane and has two equally likely directions: along the *a*-axis *b*-axis

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

## Data Availability Statement

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

## Author Contributions

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

## Conflict of Interest

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.

## Publisher’s Note

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Keywords: paired states, TI-bipolaron mass, Peierls transition, Kohn anomaly, charge density waves

Citation: Lakhno VD (2021) Translation-Invariant Bipolarons and Charge Density Waves in High-Temperature Superconductors. *Front. Phys.* 9:662926. doi: 10.3389/fphy.2021.662926

Received: 01 February 2021; Accepted: 19 August 2021;

Published: 04 October 2021.

Edited by:

Morten Ring Eskildsen, University of Notre Dame, United StatesReviewed by:

Konrad Jerzy Kapcia, Adam Mickiewicz University, PolandDaniel Agterberg, University of Wisconsin–Milwaukee, United States

Copyright © 2021 Lakhno. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Victor D. Lakhno, lak@impb.ru