#
Euclidean Q-Balls of Fluctuating SDW/CDW in the ‘Nested’ Hubbard Model of High-T_{c} Superconductors as the Origin of Pseudogap and Superconducting Behaviors

## Abstract

**:**

## 1. Introduction

## 2. Effective Model

## 3. Free Energy of the Cooper-Pairing Fluctuations inside the Q-Balls

## 4. Eliashberg Equations and Bound States along the Axis of Matsubara Time

## 5. The Q-Balls’ Sizes

## 6. The ${\mathbf{T}}_{\mathbf{c}}$ vs. Superconducting Density ${\mathbf{N}}_{\mathbf{s}}$: The Uemura Plot

#### 6.1. The Size of the Cooper-Pair Function

## 7. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Anomalous self-energy ${\Sigma}_{2p,\sigma}$ (short-dashed line) and fermionic dispersion ${E}_{p}$ (solid line) as a function of bare fermionic dispersion ${\epsilon}_{p}$ in the vicinity of the Fermi-level ${\mu}_{f}=0$ near the ‘nested’/anti-nodal points of the bare Fermi surface inside a Q-ball with local superconducting (pseudo)gap ${g}_{0}$, see Equations (22) and (23).

**Figure 2.**Effective potential energy ${U}_{eff}$ of the SDW/CDW fluctuation as a function of the amplitude M weighted by Matsubara frequency $\mathsf{\Omega}=2\pi nT$ at two different temperatures ${T}_{1}={T}^{*}$—curve 1, and ${T}_{2}={T}_{c}$—curve 2, see text.

**Figure 3.**The contour plots of self-consistency Equation (40) in the plane $\{M/\mathsf{\Omega},\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}\}$ are presented: (

**a**) the numerated 1–3-...curves are plotted for different values of parameter $\kappa \equiv 4g\nu {\epsilon}_{0}$ in the ‘weak coupling’ interval of values $\kappa <{\kappa}^{*}$, see text after Equation (42); (

**b**) same as (

**a**), but with added curves numerated 1–3 in the interval of ‘strong’ couplings $\kappa \ge {\kappa}^{*}$, and with curves numerated 4–7 in the ‘weak coupling’ interval $\kappa <{\kappa}^{*}$ kept for convenience of comparison.

**Figure 4.**The phase diagram that follows from Equation (41), where $\kappa \equiv c{\displaystyle \frac{4g\nu {\epsilon}_{0}}{3}}$, see text.

**Figure 5.**The plots of the potential energy $-{\tilde{U}}_{eff}$ for different values of Matsubara frequency $\mathsf{\Omega}=2\pi T$ at fixed coupling strength value $\kappa $ corresponding to contour curves 4 in Figure 3b. (

**a**) The set of curves in the superconducting domain $T\le {T}_{c}$ in Figure 3; (

**b**) the set of curves in the $T\ge {T}^{*}$ domain in Figure 3.

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**MDPI and ACS Style**

Mukhin, S.
Euclidean *Q*-Balls of Fluctuating SDW/CDW in the ‘Nested’ Hubbard Model of High-*T _{c}* Superconductors as the Origin of Pseudogap and Superconducting Behaviors.

*Condens. Matter*

**2022**,

*7*, 31. https://doi.org/10.3390/condmat7020031

**AMA Style**

Mukhin S.
Euclidean *Q*-Balls of Fluctuating SDW/CDW in the ‘Nested’ Hubbard Model of High-*T _{c}* Superconductors as the Origin of Pseudogap and Superconducting Behaviors.

*Condensed Matter*. 2022; 7(2):31. https://doi.org/10.3390/condmat7020031

**Chicago/Turabian Style**

Mukhin, Sergei.
2022. "Euclidean *Q*-Balls of Fluctuating SDW/CDW in the ‘Nested’ Hubbard Model of High-*T _{c}* Superconductors as the Origin of Pseudogap and Superconducting Behaviors"

*Condensed Matter*7, no. 2: 31. https://doi.org/10.3390/condmat7020031