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Possible Manifestation of Q-Ball Mechanism of High-T_{c} Superconductivity in X-ray Diffraction

## Abstract

**:**

## 1. Introduction

_{4}[7] below ${T}_{c}\approx 30$ K, followed by the steep rise of the measured ${T}_{c}$ up to ${T}_{c}\approx 135$ K, and under pressure up to ${T}_{c}\approx 164$ K, in another perovskite HgBa

_{2}Ca

_{2}Cu

_{3}O${}_{8+\delta}$ [8]. In parallel to the discoveries in cuprates, a superconductivity below ${T}_{c}=39$ K was found in magnesium diboride MgB

_{2}[9], as well as in iron-based superconductors, where the ${T}_{c}$ values approach 60 K [10,11]. The angle-resolved photoemission experiments (ARPES) from Bi2201 [12] revealed that at temperatures below ${T}^{*}$, and well above the superconducting ${T}_{c}$, a pseudogap appears in the remnant fermionic Fermi-surface antinodal regions. The scanning tunnelling microscopy (STM) reveals both the particle–hole symmetry breaking and pronounced spectral broadening indicative of a spatial state distinct from homogeneous superconductivity [13]. Similarly, neutron scattering experiments [14] reveal superconducting correlations coexisting with stripe fluctuations (‘dynamic stripes’) that correspond to coupled spin- and charge-density-waves fluctuations. Importantly, pairing correlations without global phase coherence persist up to temperatures of the order of ${T}^{*}$ and provide simultaneously diamagnetism, which is observed up to about 150 K in the optimally doped YBa

_{2}Cu

_{3}O${}_{7-\delta}$ with ${T}_{c}=92$ K [5]. Hence, the discovered HTS was called unconventional with respect to the weak coupling Bardeen–Cooper–Schrieffer (BCS) theory [15], because it occurs in strongly correlated electron systems, where the Cooper-pair formation is dominated by repulsive electron–electron interactions, that are expected to cause competing and intertwined orders of stripe phases and pair-density waves, electronic liquid-crystal phases, etc. [16]. The phase diagram of the temperature versus the hole doping level for the copper oxides, which contains insulating antiferromagnetic, pseudogap, ‘strange metal’, superconducting, and Fermi liquid phases in the different doping intervals, is discussed in the reviews, see, e.g., [17]. Thus, turning back to the Q-ball picture, we mention that an essential prerequisite for the Q-balls emergence is the attraction between condensed elementary bosonic spin-/charge-density-wave excitations. It is self-consistently triggered by the formation of Cooper-pairs condensates inside Euclidean Q-balls [1]. Hence, the binding of the fermions into Cooper/local pairs inside the Q-balls occurs via an exchange with semiclassical density fluctuations of a finite amplitude below a high enough temperature ${T}^{*}$. The latter is of the order of the excitation ‘mass’, i.e., proportional to the inverse of the correlation length of the short-range spin-/charge-density-wave fluctuations. The Q-ball charge Q counts the number of condensed elementary bosonic excitations forming the finite amplitude spin-/charge-density wave inside the Q-ball volume. The amplitude of the Q-ball fluctuation lies in the vicinity of the local minimum of the free energy of the Q-ball, thus making it stable. Euclidean Q-balls arise due to the global invariance of the effective theory under the $U\left(1\right)$ phase rotation of the Fourier amplitudes of the spin-/charge-density fluctuations, leading to the conservation of the ‘Noether charge’ Q in Matsubara time. This is reminiscent of the Q-balls formation in the supersymmetric standard model, where the Noether charge responsible for the baryon number conservation in real time is associated with the $U\left(1\right)$ symmetry of the squarks field [18,19,20]. Contrary to the squark Q-balls, the Euclidean Q-balls arise at finite temperature ${T}^{*}$ and the phase of the dominating Fourier component of the spin-/charge-density wave fluctuation rotates with bosonic Matsubara frequency $\Omega =2\pi T$ in the Euclidean space time. Simultaneously, the local minimum of the Q-ball potential energy located at the finite value of the modulus of the Fourier amplitude arises due to the local/Cooper pairing [2]. A ‘bootstrap’ condition is an exchange with fluctuations of a finite amplitude that causes the local/Cooper pairing of fermions inside Q-balls already at high temperatures. An idea of a semiclassical ‘pairing glue’ between fermions in cuprates, but for an itinerant case, was proposed earlier in [21]. Hence, the proposed superconducting pairing mechanism inside Q-balls is distinct from the usual phonon- [22] or spin-fermion coupling models [23] considered previously for high-${T}_{c}$ cuprates, based upon the exchange with infinitesimal spin- and charge-density fluctuations [24] or polarons [25] in the usual Fröhlich picture.

## 2. Quintessence of Euclidean Q-Balls Picture

## 3. Summary of Theoretical Predictions for Q-Balls

## 4. Theoretical Predictions for Some Measurable Q-Balls Manifestations

#### 4.1. Charge Q Conservation

#### 4.2. Manifestations of Superconducting Condensates Inside Q-Balls

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Q-ball volume ${V}_{Q}\sim {\xi}^{3}$ as function of the CDW/SDW amplitude squared ${M}^{2}\sim A$ (normalised) is plotted in the arbitrary units using Equation (18). Here, coherence volume ${\xi}^{3}$ and amplitude A are characteristics of the X-ray scattering peak from the Q-balls in the pseudogap phase of high-${T}_{c}$ cuprates, see text.

**Figure 3.**(

**a**) Theoretical temperature dependences of the SDW/CDW amplitude ${M}^{2}=A$ (blue curve) and Q-ball size $R=\xi $ (red curve), temperature expressed in units of ${T}^{*}={\mu}_{0}/2\pi $; (

**b**) temperature dependence of the smallest Q-ball charge ${Q}_{min}$ from Equation (26), see text. Corresponding experimental data points are presented in G. Campi et al. paper [3].

**Figure 4.**Theoretical prediction for the temperature dependence of the weighted scattering amplitude ${A}_{w}$, see text, in the units of ${T}^{*}={\mu}_{0}/2\pi $.

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**MDPI and ACS Style**

Mukhin, S.
Possible Manifestation of Q-Ball Mechanism of High-*T*_{c} Superconductivity in X-ray Diffraction. *Condens. Matter* **2023**, *8*, 16.
https://doi.org/10.3390/condmat8010016

**AMA Style**

Mukhin S.
Possible Manifestation of Q-Ball Mechanism of High-*T*_{c} Superconductivity in X-ray Diffraction. *Condensed Matter*. 2023; 8(1):16.
https://doi.org/10.3390/condmat8010016

**Chicago/Turabian Style**

Mukhin, Sergei.
2023. "Possible Manifestation of Q-Ball Mechanism of High-*T*_{c} Superconductivity in X-ray Diffraction" *Condensed Matter* 8, no. 1: 16.
https://doi.org/10.3390/condmat8010016