# Phase Separation and Pairing Fluctuations in Oxide Materials

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## Abstract

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## 1. Introduction

## 2. Phase Separation in the Mean-Field Approximation

## 3. Phase Separation and Fluctuations

**l**with the spin projection $\sigma =\pm 1$, ${a}_{\mathbf{l}\sigma}^{\u2020}$ and ${a}_{\mathbf{l}\sigma}$ are electron creation and annihilation operators. As seen from Equation (8), the Hamiltonian has four eigenvectors $|\lambda \rangle $: the empty state $|0\rangle $ with the eigenenergy ${E}_{0}=0$, two singly occupied degenerate states $|\sigma \rangle $ with the energy ${E}_{1}=-\mu $, and the doubly occupied state $|2\rangle $ with the energy ${E}_{2}=U-2\mu $. As follows from the energy expressions, with the change of $\mu $, these states become alternately the ground states of the atom: for $\mu <0$ it is $|0\rangle $, for $0<\mu <U$ the degenerate singly occupied states are the lowest ones, and for $U<\mu $ the ground state is $|2\rangle $.

**k**is the wave vector and ${t}_{\mathbf{k}}$ the Fourier transform of hopping constants. The drastic variation of electron bands near $\mu \approx 0$ and $\mu \approx U$ can be characterized as their pronounced non-rigidity—a strong dependence of the electron dispersion on the chemical potential/electron concentration. This non-rigidity is the origin of the NEC observed near these values of the chemical potential.

## 4. Pair Correlations In Cuprates

#### 4.1. Pairing from Long Range Electron-Phonon Interaction

#### 4.2. Pair Fluctuations in the Symmetry-Broken Ground State

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Gram-susceptibility of an ’as-prepared’ La${}_{2}$CuO${}_{4+\delta}$ ($\delta \sim $ 0.01) sample as a function of temperature. The sample was rapidly quenched from room temperature to the indicated temperatures and subsequently the magnetization (${B}_{\mathrm{ext}}\sim $ 9 mT) was collected by slowly cooling the sample. Beginning from the lowest data set each curve was shifted upwards by a value of 5 × 10${}^{-7}$ cm${}^{3}$/g compared to the preceding one. (Adapted from Figure 1a, Ref. [5,6] by permission from Springer/Nature/Palgrave).

**Figure 2.**Full energy of the crystal E vs hole concentration c in the hole-rich (stripe) region. Initial (mean) hole concentration is 0.05. The dotted line corresponds to the rigid AF lattice.

**Figure 4.**The dependence $x(\mu )$ in the Hubbard-Kanamori model with two orbitals. $U=6t$, $J=1.5t$, and $T=0.13t$.

**Figure 5.**Spin (

**a**) and charge (

**b**) structure of $d=4$ bond centered stripes at $\delta =1/8$. Also shown are the interaction parameters used in the spin (${J}_{AF}$, ${J}_{F}$) and charge (V) channel which determine the pairing fluctuations in Equation (16).

**Figure 6.**GA band structure for $d=4$ bond-centered stripes oriented along the y-direction. Shown are the dispersions for $q=0$ (black, solid) and $q=\pi /4$ (dashed). The energy is measured with respect to the chemical potential (horizontal grey line) and the lattice constant is set to $a\equiv 1$. LHB/UHB: Lower and upper Hubbard band.

**Figure 7.**Instantaneous order parameter fluctuations for $d-$wave (

**a**,

**b**) and extended $s-$wave symmetry (

**c**) obtained from the removal part ($\omega <0$). Panel (

**a**) corresponds to the TDGA ladder result for $\langle {\mathsf{\Delta}}_{q}^{d}{\mathsf{\Delta}}_{q}^{d}\rangle =\int d\omega {P}_{q}(\omega )$ whereas panels (

**b**) and (

**c**) show the vertex contributions $\int d\omega \mathsf{\Delta}{P}_{q}(\omega )$. Coupling parameter $g=1$.

**Figure 8.**Order parameter fluctuations at $\mathbf{q}=0$ for $d-$wave (

**a**) and extended $s-$wave symmetry (

**b**). Red: TDGA; Black: GA. The inset covers the energy range around the bound states of the removal spectra. Also shown (blue dashed) is the TDGA result without spin and charge attractive interactions ($g=0$) and the strong coupling case (green dashed dotted) for $g=2$.

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**MDPI and ACS Style**

Bill, A.; Hizhnyakov, V.; Kremer, R.K.; Seibold, G.; Shelkan, A.; Sherman, A.
Phase Separation and Pairing Fluctuations in Oxide Materials. *Condens. Matter* **2020**, *5*, 65.
https://doi.org/10.3390/condmat5040065

**AMA Style**

Bill A, Hizhnyakov V, Kremer RK, Seibold G, Shelkan A, Sherman A.
Phase Separation and Pairing Fluctuations in Oxide Materials. *Condensed Matter*. 2020; 5(4):65.
https://doi.org/10.3390/condmat5040065

**Chicago/Turabian Style**

Bill, Andreas, Vladimir Hizhnyakov, Reinhard K. Kremer, Götz Seibold, Aleksander Shelkan, and Alexei Sherman.
2020. "Phase Separation and Pairing Fluctuations in Oxide Materials" *Condensed Matter* 5, no. 4: 65.
https://doi.org/10.3390/condmat5040065