High-$T_c$ superconductivity with quadratic electron-phonon coupling

Abstract

High-temperature superconductivity in cuprates remains a central challenge in condensed matter physics due to the complex interplay of lattice dynamics and electronic correlations. Traditional BCS theory fails to capture these effects, while competing models emphasize either phonons or correlations but rarely both. We propose a novel Hamiltonian that integrates linear and quadratic phonon-mediated interactions with strong electronic correlations in a two-dimensional cuprate lattice, representing the first unified framework incorporating quadratic electron-phonon coupling (QEPC) for enhanced pairing. This innovation addresses limitations in prior models by including multiple phonon modes, momentum-dependent coupling, and QEPC, which enables quantum bipolaron formation and significantly boosts $T_c$. Through rigorous derivations using extended Eliashberg equations, we obtain analytical and numerical expressions for the critical temperature ($T_c$) and superconducting gap ($\Delta$), incorporating a frequency- and momentum-dependent pairing potential enhanced by QEPC, absent in prior models. Our proofs elucidate the synergy between linear/quadratic electron-phonon coupling (g, $\gamma$) and on-site repulsion (U), predicting enhanced $T_c$ up to $\approx 100$ K in cuprates, surpassing previous limits, along with a dome-shaped phase diagram peaking at optimal doping. Numerical simulations, with improved self-consistent solutions and larger grids, validate these results against experimental data, revealing non-monotonic trends in $T_c$ and $\Delta$ with respect to g, $\gamma$, and doping x, as well as a doping-dependent isotope coefficient that minimizes at optimal doping and increases in the underdoped and overdoped regimes. This unified framework bridges phonon- and correlation-driven mechanisms, offering novel insights for material design, such as engineered superlattices for QEPC, and resolving longstanding controversies in high-$T_c$ superconductivity by predicting higher $T_c$ through quantum effects.

Introduction

High-temperature superconductors, notably cuprates discovered by Bednorz and Müller in 1986¹, exhibit critical temperatures ($T_c$) exceeding 100 K, far beyond the predictions of BCS theory^2,3. These materials, such as $\text {La}_{2-x}\text {Sr}_{x}\text {CuO}_4$ and $\text {YBa}_{2}\text {Cu}_{3}\text {O}_7$, challenge conventional understanding due to their complex interplay of lattice dynamics, strong electronic correlations, and unconventional d-wave pairing⁴. The microscopic process is still unclear after decades of research, with arguments focusing on the relative importance of electronic correlations and phonons^5,6,7.

Phonon-mediated pairing, central to conventional superconductors^8,9, was initially downplayed in cuprates due to strong Coulomb repulsion (U) in the $\text {CuO}_2$ planes^10,11. However, recent studies highlight considerable electron-phonon coupling, especially with oxygen-related modes, according to investigations such as neutron scattering and angle-resolved photoemission spectroscopy (ARPES)^12,13. Recent advances emphasize quadratic electron-phonon coupling (QEPC), where pairing arises from phonon zero-point fluctuations, enabling quantum bipolarons and potentially higher $T_c$^14,15,16, as demonstrated in bipolaronic models for high-temperature superconductivity¹⁷, incorporating quantum polaron effects¹⁸. Conversely, correlation-driven mechanisms, such as the resonating valence bond (RVB) state^5,19, emphasize the Mott insulator physics of undoped cuprates. This Mott physics can lead to insulator-superconductor transitions via bipolaronic mechanisms²⁰. Competing theories, including spin fluctuations²¹ and stripe phases^22,23, further complicate the picture. No single model adequately incorporates linear and quadratic phonons with correlations to forecast $T_c$ and the superconducting gap ($\Delta$).

To overcome these challenges, we propose a unified Hamiltonian that combines linear and quadratic phonon-mediated interactions with electronic correlations in a two-dimensional lattice, marking the first integration of QEPC in cuprate models for enhanced superconductivity. We overcome simplifications in previous methods by including momentum-dependent coupling, multiple phonon modes (acoustic and optical), and QEPC to improve accuracy and enable quantum bipolaron formation. We derive novel analytical expressions for $T_c$ and $\Delta$, incorporating a dynamic pairing potential that captures frequency- and momentum-dependent interactions enhanced by QEPC, absent in static BCS or Hubbard models^24,25. We derive a doping-dependent modulation in the effective coupling from carrier density considerations, resulting in a dome-shaped phase diagram. Our predictions are confirmed by numerical simulations over experimental data, which reveal enhanced $T_c \approx 100$ K, non-monotonic behaviors with respect to coupling strength and doping, and unusual isotope effects consistent with cuprate findings. Our model bridges phonon- and correlation-driven mechanisms, offering insights for designing new high-$T_c$ materials through QEPC engineering and resolving key controversies.

The coexistence of strong electron-phonon coupling and electronic correlations in cuprates and related materials like nickelates has been convincingly demonstrated by recent developments in experimental techniques, such as high-resolution ARPES and resonant inelastic X-ray scattering (RIXS)^26,27,28. These results indicate that pairing is enhanced by mode-selective phonon interactions, particularly in the presence of charge density waves (CDW) and stripe ordering, with QEPC playing a key role in quantum fluctuations¹⁶. This implies that a unified model incorporating both lattice dynamics, QEPC, and correlation effects is necessary to correctly predict superconducting properties^29,30. Quantitative predictions in layered superconductors are made possible by theoretical advancements in ab initio calculations and Feshbach resonance techniques, which highlight the orbital-selective character of these interactions^{31,32,33,34,35}.

To better elucidate the relevance of these recent findings, we note that the mode-selective electron-phonon coupling observed in RIXS experiments²⁸ supports our choice of multiple phonon modes in the model, as it captures the dominant contributions from oxygen vibrations that are critical for pairing enhancement. Furthermore, our paradigm may be more broadly applicable to other layered superconductors since the synergy we describe for cuprates is similar to the cooperation between electron-phonon coupling and correlations in nickelates²⁷. Recent Eliashberg calculations incorporating correlations and QEPC predict $T_c$ values around 50-100 K for electron-doped systems, aligning with our refined predictions.

Section Theoretical model presents the theoretical model, including the Hamiltonian and pairing potential. Section Derivations of key formulas details the derivations of $T_c$ and $\Delta$. Section Results and analysis discusses numerical results, including comparisons to experiments. Section Discussion analyzes implications, limitations, and material design strategies, while Section Conclusion summarizes findings and future directions.

Theoretical model

We model a two-dimensional cuprate lattice (Fig. 1) with a Hamiltonian that integrates electronic, phononic, and interaction terms, now enhanced with quadratic electron-phonon coupling:

$$\begin{aligned} H = H_{\text {el}} + H_{\text {ph}} + H_{\text {el-ph}} + H_{\text {q-el-ph}} + H_{\text {corr}}, \end{aligned}$$

(1)

To ensure the interactions–linear EPC, QEPC, and correlations–are treated consistently without full decoupling, we solve the coupled Eliashberg equations self-consistently (Section Derivations of key formulas), where the electron and phonon self-energies incorporate mutual feedback via the dressed propagators. This approach is justified in the intermediate coupling regime ($\lambda \approx 2$), where the Migdal approximation holds, and vertex corrections are small, as verified in recent studies on cuprate superconductivity^14,36. For stronger couplings, suppression effects may arise, but our parameters align with the enhancement observed in experiments, as in breakdowns of Migdal-Eliashberg in strong-coupling regimes³⁷.

where the terms are defined as follows:

• The electronic term, describing electron hopping with $t=0.3$ eV and chemical potential $\mu$ to control doping x, typical for $\text {CuO}_2$ planes³⁸, is given by:

  $$\begin{aligned} H_{\text {el}} = -t \sum _{\langle i,j \rangle , \sigma } (c_{i\sigma }^\dagger c_{j\sigma } + \text {h.c.}) - \mu \sum _{i,\sigma } n_{i\sigma }. \end{aligned}$$

  (2)

• The phononic term, extended to include multiple modes–an optical mode with $\omega _{\textbf{q}} = \omega _D = 0.07$ eV and an acoustic mode with $\omega '_{\textbf{q}} = \omega '_D |\sin (q_x/2) + \sin (q_y/2)|$ where $\omega '_D = 0.006$ eV, representing diverse oxygen and lattice vibrations¹²–is expressed as:

  $$\begin{aligned} H_{\text {ph}} = \sum _{\textbf{q}} \omega _{\textbf{q}} b_{\textbf{q}}^\dagger b_{\textbf{q}} + \sum _{\textbf{q}} \omega '_{\textbf{q}} a_{\textbf{q}}^\dagger a_{\textbf{q}}. \end{aligned}$$

  (3)

  The linear summation of mode contributions in $V_{\text {dyn}}$ is an approximation valid for non-resonant, weakly interacting phonons with distinct symmetries, as observed in cuprates²⁸. QEPC is applied to the optical mode, which dominates quantum bipolaron effects due to its higher energy¹⁶. This choice enhances generality while maintaining computational tractability.

• The linear electron-phonon coupling term, with coupling strengths $g=0.35$ eV and $g'=0.08$ eV, momentum-dependent to capture realistic interactions (e.g., stronger coupling near Brillouin zone edges), is given by:

  $$\begin{aligned} H_{\text {el-ph}} = \sum _{\textbf{k},\textbf{q},\sigma } \bigg ( g(\textbf{q}) n_{\textbf{k}+\textbf{q},\sigma } (b_{\textbf{q}} + b_{-\textbf{q}}^\dagger ) + g'(\textbf{q}) n_{\textbf{k}+\textbf{q},\sigma } (a_{\textbf{q}} + a_{-\textbf{q}}^\dagger ) \bigg ), \end{aligned}$$

  (4)

  where $g(\textbf{q}) = g (1 + 0.2 \cos q_x + 0.2 \cos q_y)$ and similarly for $g'(\textbf{q})$, justified by neutron scattering data¹³; this extension refines prior local coupling approximations³⁹.

• The novel quadratic electron-phonon coupling term, with strength $\gamma =0.1$ eV, introducing quantum fluctuations for bipolaron formation¹⁶, is:

  $$\begin{aligned} H_{\text {q-el-ph}} = \sum _{\textbf{k},\textbf{q},\sigma } \gamma (\textbf{q}) n_{\textbf{k}+\textbf{q},\sigma } (b_{\textbf{q}} + b_{-\textbf{q}}^\dagger )^2, \end{aligned}$$

  (5)

  where $\gamma (\textbf{q}) = \gamma (1 + 0.1 \cos q_x + 0.1 \cos q_y)$, motivated by recent theoretical advances¹⁴ and soliton states in quadratic coupling⁴⁰.

• The correlation term, introducing on-site Coulomb repulsion with $U=2$ eV, reflecting strong correlations⁵, is defined as:

  $$\begin{aligned} H_{\text {corr}} = U \sum _i n_{i\uparrow } n_{i\downarrow }. \end{aligned}$$

  (6)

The lattice in Fig. 1 illustrates the square planar structure with Cu (blue) and O (red) atoms, nearest-neighbor hopping paths and phonon modes localized on O sites. This structure supports small bipolaron formation in 2D Holstein-Hubbard models under adiabatic limits⁴¹. This structure is essential for understanding the d-wave symmetry of the pairing, as the square lattice symmetry facilitates the $\cos k_x - \cos k_y$ form factor, which is consistent with experimental observations in cuprates⁴.

The parameters defining the Hamiltonian in Eq. (1), as summarized in Table 1, are carefully selected based on a combination of experimental data and theoretical estimates to accurately model the physical properties of cuprate superconductors, now including QEPC. This selection accounts for single-polaron properties in double-well electron-phonon couplings⁴². ARPES measurements and band structure computations revealed a bandwidth of around 1-2 eV in the $\text {CuO}_2$ planes³⁸, including the electron hopping integral $t = 0.3 \, \text {eV}$; Debye frequencies $\omega _D = 0.07 \, \text {eV}$ and $\omega '_D = 0.006$ eV, corresponding to apical oxygen vibrations and acoustic modes identified as key for electron-phonon coupling in neutron scattering and RIXS experiments^12,28; electron-phonon coupling strengths $g = 0.35 \, \text {eV}$ and $g' = 0.08 \, \text {eV}$, consistent with kink features in ARPES spectra⁴⁶; quadratic coupling $\gamma = 0.1 \, \text {eV}$, drawn from recent models¹⁶; and the on-site Coulomb repulsion $U = 2 \, \text {eV}$, capturing the strong correlation regime without immediate Mott insulation at half-filling⁵. The damping factor $\eta = 10^{-4} \, \text {eV}$ guarantees numerical stability in Eliashberg calculations³⁹, while the density of states $N(0) = 1 \, \text {eV}^{-1}$ approximates the 2D logarithmic divergence near the van Hove singularity, which is essential for improving pairing in cuprates⁴⁷. For many cuprate families, the empirically observed peak in $T_c$ is reflected in the optimal doping level $x_{\text {opt}} = 0.16$⁴⁸. Table 1 provides a full explanation of these parameters, which are divided into electronic, phononic, electron-phonon coupling, and correlation terms.

Table 1 Model parameters, their physical significance, and experimental/theoretical justifications.

The coupling $g=0.35$ eV is tuned to yield $T_c \approx 100$ K, consistent with strong EPC from ARPES kinks^12,46, where effective $\lambda \sim 1-2$. This is larger than anisotropic estimates⁴⁹ due to QEPC inclusion. $g'=0.08$ eV and $\gamma =0.1$ eV follow from multi-mode and bipolaron models^16,30. This adjustment aligns with phonon dispersions in Bi-2212, where acoustic modes reach  6-10 meV at BZ boundary⁴⁵.

The dynamic pairing potential, generalizing static BCS pairing and now momentum-dependent with QEPC contributions, is:

$$\begin{aligned} V_{\text {dyn}}(\textbf{k}, \textbf{q}, \omega ) = -\frac{g(\textbf{q})^{2} \omega _{\textbf{q}} + g^{\prime }(\textbf{q})^{2} \omega ^{\prime }_{\textbf{q}} + 2\gamma (\textbf{q})^2 \omega _{\textbf{q}} }{\omega ^{2} - \left( \omega _{\textbf{q}}^{2} + \left( \omega ^{\prime }_{\textbf{q}}\right) ^{2}\right) /2 + i \eta }, \end{aligned}$$

(7)

where $\eta = 10^{-4}$ eV ensures numerical stability³⁹, and the QEPC term enhances attraction via zero-point fluctuations¹⁶. This form captures retardation effects, multi-mode contributions, and quantum bipolaronic pairing, crucial for high-$T_c$ systems⁸. For small $\omega$, we approximate:

$$\begin{aligned} V_{\text {dyn}}(\textbf{q}, \omega ) \approx [g(\textbf{q})^2 + g'(\textbf{q})^2 + 2\gamma (\textbf{q})^2] \left( \frac{1}{\bar{\omega }_{\textbf{q}}} + \frac{\omega ^2}{\bar{\omega }_{\textbf{q}}^3} \right) , \end{aligned}$$

(8)

where $\bar{\omega }_{\textbf{q}} = (\omega _{\textbf{q}} + \omega '_{\textbf{q}})/\sqrt{2}$, derived by expanding the denominator for $\omega \ll \bar{\omega }_{\textbf{q}}$. The effective $\bar{\omega }_{\textbf{q}} = \sqrt{ (\omega _{\textbf{q}}^2 + (\omega '_{\textbf{q}})^2)/2 }$ is the root-mean-square average, derived from the quadrature sum in the denominator of $V_{\text {dyn}}$, providing a physically motivated scale for multi-mode dynamics⁵⁰. The expansion starts from:

$$\begin{aligned} \omega ^2 - \bar{\omega }_{\textbf{q}}^2 \approx -\bar{\omega }_{\textbf{q}}^2 \left( 1 - \frac{\omega ^2}{\bar{\omega }_{\textbf{q}}^2}\right) ^{-1} \approx -\bar{\omega }_{\textbf{q}}^2 \left( 1 + \frac{\omega ^2}{\bar{\omega }_{\textbf{q}}^2}\right) , \end{aligned}$$

(9)

leading to the leading attractive term and the dynamic correction, consistent with simplified Eliashberg spectral functions⁵⁰. To clarify this approximation, we start from the exact form:

$$\begin{aligned} V_{\text {dyn}} \approx - \frac{[g(\textbf{q})^2 \omega _{\textbf{q}} + g'(\textbf{q})^2 \omega '_{\textbf{q}} + 2\gamma (\textbf{q})^2 \omega _{\textbf{q}}]}{\omega ^2 - \bar{\omega }_{\textbf{q}}^2}. \end{aligned}$$

(10)

Neglecting the small imaginary part for analytical purposes (as $\eta$ is infinitesimal), we factor out $-\bar{\omega }_{\textbf{q}}^2$ from the denominator:

$$\begin{aligned} \omega ^2 - \bar{\omega }_{\textbf{q}}^2 = -\bar{\omega }_{\textbf{q}}^2 \left( 1 - \frac{\omega ^2}{\bar{\omega }_{\textbf{q}}^2}\right) . \end{aligned}$$

(11)

Then, using the binomial expansion for small $\epsilon = \omega ^2 / \bar{\omega }_{\textbf{q}}^2$:

$$\begin{aligned} (1 - \epsilon )^{-1} \approx 1 + \epsilon , \end{aligned}$$

(12)

we get:

$$\begin{aligned} (\omega ^2 - \bar{\omega }_{\textbf{q}}^2)^{-1} \approx -\bar{\omega }_{\textbf{q}}^{-2} \left( 1 + \frac{\omega ^2}{\bar{\omega }_{\textbf{q}}^2}\right) . \end{aligned}$$

(13)

Multiplying by $-[g(\textbf{q})^2 \omega _{\textbf{q}} + g'(\textbf{q})^2 \omega '_{\textbf{q}} + 2\gamma (\textbf{q})^2 \omega _{\textbf{q}}]$ yields:

$$\begin{aligned} [g(\textbf{q})^2 + g'(\textbf{q})^2 + 2\gamma (\textbf{q})^2] \left( \frac{1}{\bar{\omega }_{\textbf{q}}} + \frac{\omega ^2}{\bar{\omega }_{\textbf{q}}^3} \right) , \end{aligned}$$

(14)

approximately, which is the dynamic correction that becomes significant at limited frequency plus the attractive static term. In the vicinity of the Fermi surface where pairing takes place, this expansion is true as long as $\omega / \bar{\omega }_{\textbf{q}} < 1$. By including QEPC, multiple modes, and momentum dependency, the potential is refined, resulting in more precise predictions that are in line with recent Eliashberg studies incorporating quantum effects^16,36,51.

Fig. 1

Schematic of the two-dimensional $\text {CuO}_2$ lattice, with Cu (blue) and O (red) atoms. Electron hopping (t) occurs between nearest neighbors, and phonons represent oxygen vibrations. The lattice illustrates the square planar structure with nearest-neighbor hopping paths and phonon modes localized on O sites.

The model now includes QEPC, multiple phonon modes, and momentum-dependent pairing, motivated by experimental evidence and recent theoretical advances^4,16,28. Anisotropic U or multi-band effects could be included in future extensions. The doping level x is incorporated through $\mu$, but for analytical derivations, we derive a doping-dependent modulation in the effective coupling from carrier density and QEPC effects, as detailed in Section Derivations of key formulas. In contrast to correlation-induced suppression at high doping levels brought on by decreased coherence or competing instabilities like charge order, this modulation is physically motivated by the increase in accessible charge carriers with doping, which improves pairing, enhanced by quantum bipolarons.

Derivations of key formulas

We employ the extended Eliashberg framework to account for strong linear and quadratic electron-phonon coupling and correlations^8,16. The self-energy is now solved self-consistently:

$$\begin{aligned} \Sigma (\textbf{k}, i\omega _n) = T \sum _{\omega _m} \int \frac{d^2\textbf{q}}{(2\pi )^2} \bigg ( V_{\text {dyn}}(\textbf{k}, \textbf{q}, i\omega _n - i\omega _m) G(\textbf{k} - \textbf{q}, i\omega _m)\bigg ), \end{aligned}$$

(15)

where $G(\textbf{k}, i\omega _m) = [i Z(\omega _m) \omega _m - \xi _{\textbf{k}} - \Sigma (\textbf{k}, i\omega _m)]^{-1}$ is the dressed Green’s function with renormalization factor Z, $\xi _{\textbf{k}} = \epsilon _{\textbf{k}} - \mu$, and $\omega _n = (2n+1)\pi T$ are Matsubara frequencies. Renormalization of the quasiparticle dispersion and the development of the superconducting gap are made possible by this self-energy, which takes into account the effects of the dynamic pairing potential enhanced by QEPC. Coupled equations result from the electrical feedback that also dresses the bosonic self-energy (phonon propagator).

Critical temperature ($T_c$)

The linearized gap equation for the superconducting order parameter is: The d-wave form $\Delta (\textbf{k}, \omega ) = \Delta _0(\omega ) (\cos k_x - \cos k_y)$ is assumed based on experimental evidence in cuprates⁴, where correlations from U suppress s-wave and favor d-wave symmetry⁵. The momentum-dependent kernel $V_{\text {dyn}}(\textbf{k}, \textbf{q}, \omega )$ (Eq. 7) supports anisotropy, and the self-consistent solution is projected onto this form for alignment with observations, as validated in hybrid phonon-correlation models^14,29,30.

$$\begin{aligned} \Delta (\textbf{k}, i\omega _n) = T \sum _{\omega _m} \int \frac{d^2\textbf{q}}{(2\pi )^2} \bigg \{ V_{\text {dyn}}(\textbf{k}, \textbf{q}, i\omega _n - i\omega _m) \frac{\Delta (\textbf{k} - \textbf{q}, i\omega _m)}{Z(\omega _m)^2 |\omega _m|^2 + \xi _{\textbf{k} - \textbf{q}}^2 + \Delta (\textbf{k} - \textbf{q}, i\omega _m)^2}\bigg \}. \end{aligned}$$

(16)

For d-wave pairing, common in cuprates⁴, we assume $\Delta (\textbf{k}, \omega ) = \Delta _0(\omega ) (\cos k_x - \cos k_y)$.

This form captures the characteristic d-wave symmetry prevalent in cuprates, leading to gap nodes along the Brillouin zone diagonals ($k_x = \pm k_y$), where the superconducting order parameter vanishes. According to ARPES observations of gapless quasiparticles and the lack of a complete gap^4,38, this nodal structure promotes greater $T_c$ by enhancing pairing near the antinodes ($(\pi ,0)$ and equivalents) due to van Hove singularities in the density of states, further boosted by QEPC. The visualization in Fig. 2 plots $|\Delta (\textbf{k})|$ across the two-dimensional Brillouin zone using the predicted $\Delta _0 \approx 25$ meV, highlighting the fourfold symmetry and zero-gap lines that distinguish unconventional superconductivity from isotropic s-wave pairing. As demonstrated by experimental data³⁸, this structure emphasizes the synergy between phonon-mediated interactions, QEPC, and electronic correlations. We solve the coupled Eliashberg equations for $\Sigma$, Z, and the phonon self-energy $\Pi (\textbf{q}, i\nu _l) = -T \sum _m \int d^2\textbf{k} G(\textbf{k}, i\omega _m) G(\textbf{k} + \textbf{q}, i\omega _m + i\nu _l)$, updating the phonon propagator $D(\textbf{q}, i\nu _l) = [\nu _l^2 + \omega _{\textbf{q}}^2 + \Pi (\textbf{q}, i\nu _l)]^{-1}$ (similarly for acoustic mode). This full self-consistency, absent in prior approximations, reduces $T_c$ estimates by 10-20% compared to decoupled cases but is offset by QEPC enhancement³⁶, consistent with recent Eliashberg studies on validity in intermediate coupling regimes¹⁴.

For small $\omega$, we use the approximated $V_{\text {dyn}}$ but iterate numerically for precision. Integrating over the Brillouin zone with density of states N(0), assuming a logarithmic contribution from the 2D DOS near the van Hove singularity or bandwidth cutoff, now with momentum dependence and QEPC:

$$\begin{aligned} \int \frac{d^2\textbf{q}}{(2\pi )^2} V_{\text {dyn}}(\textbf{k}, \textbf{q}) \approx N(0) \int d\xi \, V_{\text {eff}}(\xi ) \ln \left( \frac{\bar{\omega }_D}{\omega }\right) \left( 1 + \frac{\omega ^2}{\bar{\omega }_D^2} \right) , \end{aligned}$$

(17)

where $V_{\text {eff}}(\xi )$ averages over q, improved by QEPC. To derive this, we note that in 2D systems, the momentum integral over the pairing potential can be converted to an energy integral using the DOS: $\int d^2\textbf{q}/(2\pi )^2 \approx N(0) \int d\xi$, where the logarithmic form arises from the BCS-like cutoff, $\int _{-\bar{\omega }_D}^{\bar{\omega }_D} d\xi / |\xi | \approx 2 \ln (\bar{\omega }_D / \omega )$, but adjusted for the frequency, momentum dependence, and QEPC. Specifically, for the static term, it is $[g^2 + g'^2 + 2\gamma ^2] N(0) / \bar{\omega }_D \cdot \ln (\bar{\omega }_D / \omega )$, but since $\bar{\omega }_q$ varies, it simplifies to the form above. The dynamic term follows similarly, with an extra $\omega ^2 / \bar{\omega }_D^2$ factor from the expansion. The Matsubara sum is approximated as:

$$\begin{aligned} \sum _m \frac{1}{Z_m^2 |\omega _m|^2 + \xi ^2 + \Delta _m^2} \approx \int _0^{\bar{\omega }_D} \frac{d\omega }{2\pi T} \frac{1}{\omega ^2 + \xi ^2 + \Delta ^2} \approx \frac{1}{2T} \ln \left( \frac{\bar{\omega }_D}{T}\right) , \end{aligned}$$

(18)

where we use the low-temperature limit near $T_c$ (where $T_c \ll \bar{\omega }_D$), and cutoff at $\bar{\omega }_D$, consistent with strong-coupling approximations²⁴. As long as T is not too low in relation to $\bar{\omega }_D$, this approximation substitutes the discrete sum with a continuous integral, and the logarithmic divergence captures the crucial pairing instability close to $T_c$. For accuracy, simulations employ full numerical summation. This leads to an eigenvalue problem, yielding the effective coupling:

$$\begin{aligned} \lambda _{\text {eff}} = \frac{[g^2 + g'^2 + 2\gamma ^2] N(0)}{\bar{\omega }_D}, \end{aligned}$$

(19)

where the denominator accounts for correlation suppression, with the argument of the logarithm reversed from initial forms to ensure physical suppression for large U. To derive this, we incorporate the Coulomb repulsion U as a pseudopotential that renormalizes the phonon attraction, following McMillan’s approach²⁴, where the effective $\lambda = \lambda _{ph} / (1 + \mu ^* + \lambda _{ph})$, but here $\mu ^* \approx 0.1$ approximates the correlation-induced repulsion, typical for cuprates²⁵. The critical temperature is:

$$\begin{aligned} T_c = \frac{\bar{\omega }_D}{1.2} \exp \left( -\frac{1.04 (1 + \lambda _{\text {eff}})}{\lambda _{\text {eff}} - \mu ^* (1 + 0.62 \lambda _{\text {eff}})}\right) , \end{aligned}$$

(20)

The correctness of this approximation is confirmed by numerical solutions in Section Results and analysis, where complete Eliashberg yields $T_c \approx 100$ K. Although it can fluctuate in strong coupling, this form offers a fair analytical approximation that is refined by self-consistency and QEPC. The factor 1.2 originates from the strong-coupling Eliashberg limit where $2\Delta / T_c \approx 4-5$, but corrected for the gap-to-$T_c$ ratio.

Fig. 2

Momentum dependence of the superconducting gap $|\Delta (\textbf{k})| = \Delta _0 |\cos k_x - \cos k_y|$ in the Brillouin zone, with $\Delta _0 \approx 25$ meV. The color scale shows gap magnitude in eV, with d-wave nodes (dark blue, $\Delta =0$) along $k_x = \pm k_y$ and fourfold symmetry.

Superconducting gap ($\Delta _0$)

At $T=0$, the gap equation is nonlinear and coupled:

$$\begin{aligned} \Delta _0(\textbf{k}) = N(0) \int _0^{\bar{\omega }_D} d\omega \left( \frac{[g^2 + g'^2 + 2\gamma ^2] \omega }{\omega ^2 + \Delta _0(\textbf{k})^2} \tanh \left( \frac{\sqrt{\omega ^2 + \Delta _0(\textbf{k})^2}}{2T}\right) \right) , \end{aligned}$$

(21)

solved self-consistently with Z and $\Pi$. In the weak-coupling limit ($T \rightarrow 0$, $\tanh \rightarrow 1$):

$$\begin{aligned} \Delta _0 \approx N(0) \int _0^{\bar{\omega }_D} d\omega \frac{[g^2 + g'^2 + 2\gamma ^2] \omega }{\omega ^2 + \Delta _0^2}. \end{aligned}$$

(22)

For small $\Delta _0$, we approximate $\omega ^2 + \Delta _0^2 \approx \omega ^2$, yielding:

$$\begin{aligned} \Delta _0 \approx [g^2 + g'^2 + 2\gamma ^2] N(0) \ln \left( \frac{\bar{\omega }_D}{\Delta _0}\right) . \end{aligned}$$

(23)

To complete this approximation, note that the integral $\int _{\Delta _0}^{\bar{\omega }_D} d\omega / \omega \approx \ln (\bar{\omega }_D / \Delta _0)$, where the lower limit is shifted to $\Delta _0$ to avoid divergence, as for $\omega < \Delta _0$, the denominator is dominated by $\Delta _0^2$. This is a standard BCS-like logarithmic approximation. Solving self-consistently with $\lambda _{\text {eff}}$, we obtain:

$$\begin{aligned} \Delta _0 \approx 2 \bar{\omega }_D \exp \left( -\frac{1}{\lambda _{\text {eff}}}\right) , \end{aligned}$$

(24)

verified by numerical solutions in Section Results and analysis and consistent with residue theorem calculations³⁹. The factor 2 comes from the BCS relation $\Delta _0 \approx 2 \bar{\omega }_D \exp (-1/\lambda )$ in the weak-coupling limit, adjusted for our effective $\lambda _{\text {eff}}$, with multi-mode and QEPC effects enhancing $\Delta _0$ slightly, incorporating insights from mobile bipolaron models in strong-coupling regimes⁶².

Doping dependence of $T_c$

To capture the essential dome-shaped phase diagram observed in cuprates, we derive the doping-dependent modulation to the effective coupling from carrier density and QEPC considerations. The pairing strength is improved by carrier concentration but decreased at high doping because of conflicting phases or diminished coherence, naturally emerging from the model via QEPC-enhanced carrier interactions. This leads to a functional form for the doping-modulated effective coupling:

$$\begin{aligned} \lambda _{\text {eff}}(x) = \lambda _{\text {eff}} \cdot \frac{x}{x_{\text {opt}}} \exp \left( 1 - \frac{x}{x_{\text {opt}}}\right) , \end{aligned}$$

(25)

where $\lambda _{\text {eff}}$ is the base coupling from the previous subsection (evaluated at optimal conditions), and $x_{\text {opt}} = 0.16$ is the optimal doping level. The phonon-mediated attraction is derived to scale linearly with hole density x (increasing available pairs) in order to derive this form. Additionally, an exponential decay factor describes overdoping consequences, such as increased scattering or phase separation, enhanced by QEPC quantum effects. A maximum at $x = x_{\text {opt}}$ is ensured by the exponential, and the peak value is normalized by the prefactor. To prove the maximality, we consider the function:

$$\begin{aligned} f(x) = \frac{x}{a} \exp \left( 1 - \frac{x}{a}\right) , \end{aligned}$$

(26)

where $a = x_{\text {opt}}$. Differentiating with respect to x yields:

$$\begin{aligned} \frac{df}{dx} = \frac{1}{a} \exp \left( 1 - \frac{x}{a}\right) \left( 1 - \frac{x}{a}\right) , \end{aligned}$$

(27)

which vanishes at $x = a$, confirming the peak. To further verify, the second derivative at $x = a$ is:

$$\begin{aligned} \frac{d^2 f}{dx^2} \bigg |_{x=a} = -\frac{1}{a^2} \exp (0) < 0, \end{aligned}$$

(28)

proving it is a maximum. This modulation is phenomenologically derived from carrier density and QEPC effects, leading to a dome shape that aligns with experiments, though not purely self-consistent without the ansatz. The doping-dependent critical temperature is then:

$$\begin{aligned} T_c(x) = \frac{\bar{\omega }_D}{1.2} \exp \left( -\frac{1.04 (1 + \lambda _{\text {eff}}(x))}{\lambda _{\text {eff}}(x) - \mu ^* (1 + 0.62 \lambda _{\text {eff}}(x))}\right) . \end{aligned}$$

(29)

This expression predicts a non-monotonic dome shape, aligning with experimental phase diagrams in cuprates⁴⁸, and represents a key novelty of our unified model with QEPC enhancement. The multi-mode and QEPC extensions make the dome slightly more asymmetric, with a steeper underdoped side.

Isotope effect coefficient ($\alpha$)

To further elucidate the role of phonons in our unified model, we derive the oxygen isotope effect coefficient. This coefficient is defined as:

$$\begin{aligned} \alpha = -\frac{d \ln T_c}{d \ln M}, \end{aligned}$$

(30)

where M is the oxygen mass. Given the dependence of the Debye frequencies on mass:

$$\begin{aligned} \bar{\omega }_D \propto M^{-1/2}, \end{aligned}$$

(31)

we obtain:

$$\begin{aligned} \frac{d \ln \bar{\omega }_D}{d \ln M} = -\frac{1}{2}, \end{aligned}$$

(32)

leading to:

$$\begin{aligned} \alpha = \frac{1}{2} \frac{d \ln T_c}{d \ln \bar{\omega }_D}. \end{aligned}$$

(33)

The critical temperature is expressed as:

$$\begin{aligned} T_c = \frac{\bar{\omega }_D}{1.2} \exp \left( -\frac{1.04 (1 + \lambda _{\text {eff}}(x))}{\lambda _{\text {eff}}(x) - \mu ^* (1 + 0.62 \lambda _{\text {eff}}(x))}\right) , \end{aligned}$$

(34)

from which we compute:

$$\begin{aligned} \frac{d \ln T_c}{d \ln \bar{\omega }_D} = 1 + \frac{1}{\lambda _{\text {eff}}(x)^2} \frac{d \lambda _{\text {eff}}(x)}{d \ln \bar{\omega }_D}. \end{aligned}$$

(35)

The effective coupling is defined as:

$$\begin{aligned} \lambda _{\text {eff}}(x) = \lambda _{\text {base}} f(x), \end{aligned}$$

(36)

where the doping-dependent function is:

$$\begin{aligned} f(x) = \frac{x}{x_{\text {opt}}} \exp \left( 1 - \frac{x}{x_{\text {opt}}}\right) , \end{aligned}$$

(37)

and the base coupling strength is:

$$\begin{aligned} \lambda _{\text {base}} = \frac{\kappa }{\bar{\omega }_D}, \end{aligned}$$

(38)

with:

$$\begin{aligned} \kappa = [g^2 + g'^2 + 2\gamma ^2] N(0). \end{aligned}$$

(39)

To compute the derivative, we express:

$$\begin{aligned} \lambda _{\text {base}} = \frac{\kappa }{D}, \end{aligned}$$

(40)

where:

$$\begin{aligned} D = \bar{\omega }_D. \end{aligned}$$

(41)

The derivative of D with respect to $\bar{\omega }_D$ is:

$$\begin{aligned} \frac{dD}{d\bar{\omega }_D} = 1, \end{aligned}$$

(42)

leading to:

$$\begin{aligned} \frac{d \lambda _{\text {base}}}{d \bar{\omega }_D} = -\frac{\kappa }{D^2} = -\frac{\kappa }{\bar{\omega }_D^2}. \end{aligned}$$

(43)

Multiplying by $\bar{\omega }_D$, we obtain:

$$\begin{aligned} \frac{d \lambda _{\text {base}}}{d \ln \bar{\omega }_D} = \bar{\omega }_D \frac{d \lambda _{\text {base}}}{d \bar{\omega }_D} = -\frac{\kappa }{\bar{\omega }_D}. \end{aligned}$$

(44)

Substituting into the expression for the derivative of $\ln T_c$, we get:

$$\begin{aligned} \frac{d \ln T_c}{d \ln \bar{\omega }_D} = 1 - \frac{\bar{\omega }_D - \kappa }{\kappa f(x)}. \end{aligned}$$

(45)

Thus, the isotope effect coefficient is:

$$\begin{aligned} \alpha (x) = \frac{1}{2} \left[ 1 - \frac{\bar{\omega }_D - \kappa }{\kappa f(x)} \right] . \end{aligned}$$

(46)

This novel expression captures the interplay between phonon frequency, linear/quadratic electron-phonon coupling, and doping modulation, considering phonon dispersion effects on the competition between pairing and charge order⁴⁴. For parameters where $\kappa > \bar{\omega }_D$ (e.g., stronger $g \approx 0.28$ eV, achievable in certain cuprates or under external tuning like pressure), $\alpha > 0.5$ in underdoped regimes, decreasing to a minimum near optimal doping–a behavior absent in conventional BCS theory and reflective of the unconventional isotope effects in cuprates. The capacity of the model to anticipate isotope effects without the need for extra parameters is demonstrated in this derivation, which offers a cohesive perspective on phonon-correlation synergy amplified by quadratic coupling. Multi-mode and QEPC effects somewhat increase $\alpha$ variations.

Results and analysis

Numerical calculations use a mean-field approach extended to full Eliashberg self-consistency with parameters from Table 1. The gap equation is solved iteratively on a refined $200 \times 200$ k-space grid using a Newton-Raphson solver coupled with equations for Z and $\Pi$, with convergence criteria of $10^{-6}$ eV for $\Delta$, Z, and $\Pi$. The temperature is incremented in steps of 0.1 K to determine $T_c$, defined as the point where $\Delta < 10^{-8}$ eV. Monte Carlo methods estimate error bars, assuming 5% uncertainty in g, $g'$, $\gamma$, and U, with 1000 iterations to ensure statistical reliability. Calculations were performed on a standard workstation, taking approximately 20 hours for convergence due to the added complexity. The Gaussian-like shape arises from the analytical form of $\lambda _{\text {eff}}$, where the denominator grows logarithmically with $g + g' + 2\gamma$, leading to an optimal coupling. Simulations verify that the analytical version of $\lambda _{\text {eff}}(x)$ properly reproduces the dome, even while $\mu$ is adjusted to reach x for doping dependency. To elaborate on the numerical procedure, the Newton-Raphson iteration for the gap involves initializing $\Delta ^{(0)} = 0$, $Z^{(0)}=1$, $\Pi ^{(0)}=0$, then updating $\Delta ^{(n+1)} = \Delta ^{(n)} - f(\Delta ^{(n)}) / f'(\Delta ^{(n)})$, where $f(\Delta ) = \Delta - RHS$ of the gap equation, and the derivative is computed numerically or analytically where possible, with similar updates for Z and $\Pi$. To precisely capture the d-wave nodes, the k-space grid is centered on the Fermi surface. For increased precision, integration is carried out using Simpson’s method. More in line with current Eliashberg forecasts incorporating QEPC, this refinement lessens overestimation from earlier single-mode approximations^15,16,36.

Sensitivity analysis

To assess the robustness of our results against parameter uncertainties, we varied each key parameter in Table 1 by $\pm 10\%$ and recomputed $T_c$, $\Delta _0$, and $\alpha$ self-consistently. The maximum variation in $T_c$ is 15% (e.g., +10% in g yields +12% in $T_c$), in $\Delta _0$ is 10%, and in $\alpha$ is 8%, with the phase diagram and d-wave symmetry unchanged. This stability underscores the reliability of our predictions, as the effective coupling $\lambda _{\text {eff}}$ balances enhancements from EPC/QEPC with suppression from correlations.

Numerical results

Figure 3 shows $T_c$ versus g (with $g'$, $\gamma$ fixed at 0.08 eV, 0.1 eV), peaking at $T_c \approx 100$ K for $g \approx 0.35$ eV. This non-monotonic behavior is consistent with $\text {YBa}_{2}\text {Cu}_{3}\text {O}_7$ data ($T_c \approx 92$ K) but strengthened through quadratic phonon interactions, indicating an ideal balance between correlation suppression and electron-phonon enhancement³⁸. However, it is adjusted higher due to QEPC. In accordance with cuprate data⁴⁶, the curve displays a slightly asymmetric bell shape with a peak at $g \approx 0.35$ eV and $T_c \approx 100$ K. This is because of the competition between phonon-mediated attraction (which increases with $g + g' + 2\gamma$) and correlation-induced repulsion (which is dominated by U at higher g). Error bars indicate 5% parameter uncertainty, derived from Monte Carlo sampling of parameter variations, and are larger near the peak due to the proportional scaling with $T_c$. This bell shape can be understood qualitatively as follows: for small g, $T_c$ increases exponentially with $\lambda \propto g^2 + g'^2 + 2\gamma ^2$, but as g grows, the correlation term in the denominator of $\lambda _{\text {eff}}$ becomes significant, leading to saturation and eventual decrease. The shift in peak position and increased maximum compared to prior versions arise from QEPC and momentum-dependent coupling.

Fig. 3

Critical temperature ($T_c$) versus electron-phonon coupling (g). The curve shows a slightly asymmetric bell shape peaking at $T_c \approx 100$ K for $g \approx 0.35$ eV, with error bars larger near the peak.

Figure 4 plots $\Delta _0$ versus g, mirroring $T_c$ trends with a maximum $\Delta _0 \approx 25$ meV, larger than BCS predictions due to dynamic pairing and QEPC but adjusted by self-consistency⁵⁰. The results agree with ARPES measurements in $\text {Bi}_{2}\text {Sr}_{2}\text {CaCu}_{2}\text {O}_8$³⁸. The trend follows $T_c$, exhibiting a slightly asymmetric bell shape with a peak at $g \approx 0.35$ eV and $\Delta _0 \approx 25$ meV, with enhancement over BCS due to dynamic effects and QEPC⁵⁰. Error bars reflect numerical uncertainties, derived from Monte Carlo sampling, and are larger near the peak due to proportional scaling with $\Delta _0$. The enhancement beyond BCS (where $2\Delta / T_c \approx 3.52$) arises from the retardation in $V_{\text {dyn}}$, which effectively strengthens the pairing for energies within $\bar{\omega }_D$, but multi-mode and QEPC introduce slight broadening.

Fig. 4

Superconducting gap ($\Delta _0$) at $T=0$ versus g. The curve mirrors $T_c$ with peak at $\Delta _0 \approx 25$ meV for $g \approx 0.35$ eV, error bars larger near peak.

Figure 5 illustrates $T_c$ versus doping x, exhibiting a characteristic dome shape with a peak at $x \approx 0.16$ and $T_c \approx 100$ K. This non-monotonic trend arises from the derived doping modulation in $\lambda _{\text {eff}}(x)$, where superconductivity is enhanced near optimal doping due to balanced carrier density and minimal competing orders, but suppressed in under- and overdoped regimes, further boosted by QEPC. The form captures the synergy of phonon pairing with correlation-driven doping effects, aligning with phase diagrams in $\text {La}_{2-x}\text {Sr}_{x}\text {CuO}_4$ and other cuprates⁴⁸. The asymmetry in the dome (steeper decay in underdoped side due to multi-mode and QEPC effects) reflects the exponential suppression term, providing a quantitative match to experiments. The curve shows a dome shape with a peak at $x \approx 0.16$ and $T_c \approx 100$ K, due to the novel modulation in the effective coupling that balances phonon-mediated pairing with doping-induced correlations and competing phases. Error bars indicate 5% uncertainty in parameters like $x_{\text {opt}}$ and $g + g' + 2\gamma$, derived from Monte Carlo sampling, and are larger near the peak due to proportional scaling with $T_c$. This prediction reproduces the universal cuprate phase diagram⁴⁸. The steeper underdoped decay can be attributed to enhanced phonon scattering in low-doping regimes, mitigated by QEPC.

Fig. 5

Critical temperature ($T_c$) versus doping level (x). The dome peaks at $T_c \approx 100$ K for $x \approx 0.16$, with steeper underdoped decay and error bars larger near peak.

Figure 6 depicts the isotope coefficient $\alpha$ versus doping x for $g = 0.28$ eV (where $\kappa > \bar{\omega }_D$), showing a minimum near optimal doping ($x \approx 0.16$, $\alpha \approx 0.55$) and increasing values in both underdoped and overdoped regions, with a stronger rise in the underdoped side due to the asymmetric doping modulation f(x) and multi-mode/QEPC contributions. The model’s innovative integration of dynamic phonon effects with correlations and QEPC is shown in this behavior, which results from $\lambda _{\text {eff}}(x)$’s increased sensitivity to $\bar{\omega }_D$ changes at low f(x). The prediction aligns with experimental trends in cuprates, where $\alpha$ exceeds the BCS value of 0.5 in underdoped samples and minimizes at optimal doping^12,48. Error bars indicate 5% parameter uncertainty in g, $g'$, $\gamma$, and $\bar{\omega }_D$ from Monte Carlo sampling, larger away from optimal doping due to the inverse dependence on f(x). The curve exhibits a minimum at $x \approx 0.16$ with $\alpha \approx 0.55$, increasing strongly (slightly) in underdoped (overdoped) regimes due to the novel doping modulation enhancing phonon sensitivity away from optimality. This unconventional trend, absent in standard BCS theory, underscores the phonon-correlation synergy enhanced by QEPC and matches cuprate experiments⁴⁸. The minimum at optimal doping reflects the point where the pairing is least sensitive to phonon frequency changes, as the effective coupling is maximized, reducing the relative impact of $\delta \bar{\omega }_D$, with multi-mode and QEPC effects slightly lowering the minimum.

Fig. 6

Isotope coefficient ($\alpha$) versus doping level (x) for $g = 0.28$ eV. The curve minimizes at $\alpha \approx 0.55$ for $x \approx 0.16$, with stronger underdoped increase and error bars larger away from peak.

The predicted $T_c \approx 100$ K and $\Delta _0 \approx 25$ meV align with cuprate experiments within 10-15% but enhanced by QEPC^38,46, refined from prior overestimations. Discrepancies may arise from simplified phonon spectra or doping variations. For example, in $\text {YBa}_{2}\text {Cu}_{3}\text {O}_7$, experimental $T_c = 92$ K and $\Delta \approx 20-25$ meV, showing close agreement, with our model’s enhancement due to QEPC and full self-consistency.

Discussion

Our model successfully integrates linear and quadratic phonon-mediated pairing^12,16 with electronic correlations⁵, predicting enhanced $T_c$ and $\Delta$ consistent with experiments³⁸. The dynamic pairing potential $V_{\text {dyn}}$ provides a unifying framework for cuprates by capturing retardation effects and quantum bipolarons not present in static models²⁴. For material design, the non-monotonic dependency of $T_c$ on $g + 2\gamma$ indicates an ideal coupling regime, which includes doping with Sr or Ba to tune $g + g' + 2\gamma$⁵². Furthermore, the derived doping-dependent $\lambda _{\text {eff}}(x)$ explains the superconducting dome, bridging undoped Mott insulator and overdoped metallic phases. This dome arises from the interplay where low doping limits pair formation due to insufficient carriers, optimal doping maximizes coherence, and high doping introduces scattering or Fermi liquid damping that weakens pairing, with QEPC providing additional stability. Predictions are in accordance with previous Feshbach and Eliashberg studies^14,15 while the QEPC, multi-mode, and momentum-dependent extensions improve the model’s realism and enable higher $T_c$. The model features several prescribed aspects: the d-wave pairing form is assumed to align with experiments, as correlations suppress s-wave but do not derive symmetry ab initio; the dome-shaped doping dependence uses a phenomenological modulation; momentum-dependent couplings are empirical from data; multi-mode summation approximates non-interacting phonons. Derived naturally are the enhanced $T_c$ and $\Delta$ from QEPC-linear EPC-correlation synergy, non-monotonic trends (e.g., $T_c$ vs. g), and doping-dependent $\alpha$, all via self-consistent Eliashberg solutions. Future work could derive these via ab initio or DMFT extensions.

A partial gap above $T_c$ is reflected in the pseudogap phase, which is seen in underdoped cuprates and may be caused by conflicting orders or preformed pairs⁷. Our model does not explicitly include this phase but could be extended by introducing a pseudogap order parameter, $\Delta _{\text {PG}}$, coupled to $\Delta$ via a Ginzburg-Landau framework⁵³, potentially modulated by the doping factor for consistency, as suggested by studies on pseudogap crossover in electron-phonon systems⁵⁴. For instance, one could add a term $\Delta _{\text {PG}} \propto \sqrt{1 - x/x_{\text {opt}}}$ for $x < x_{\text {opt}}$, reflecting the pseudogap’s strength in the underdoped regime, which would interact with the superconducting order through a free energy functional. Future work could incorporate this via DMFT extensions.

We explain ARPES kinks by combining both processes, in contrast to spin-fluctuation models²¹ that ignore phonons¹². Our model bridges the Mott insulator and superconducting phases by quantifying phonon contributions enhanced by QEPC, in contrast to RVB theory⁵. The d-wave symmetry aligns with cuprate experiments⁴, but the model could extend to s-wave systems like iron pnictides⁵⁵. For example, in pnictides, the multi-orbital nature might require adjusting the form factor, but the dynamic potential with QEPC remains applicable. Recent Eliashberg works predict similar $T_c$ ranges³⁶.

Limitations include the assumption of two phonon modes, which, while an improvement, still simplifies cuprate phonon spectra¹². Our intermediate coupling regime aligns with heuristic bounds on superconductivity and strategies to exceed them⁵⁶. Anisotropic U or multi-band effects (e.g., Cu d-orbitals) could refine predictions⁵⁷. Numerical results assume a uniform doping level, whereas cuprates exhibit doping-dependent phases⁴⁸. Additionally, the model neglects quantum fluctuations beyond mean-field, which could be incorporated via dynamical mean-field theory (DMFT) extensions to better capture strong correlation effects at low doping. The momentum dependence, while added, is phenomenological; ab initio derivations could further improve accuracy.

The model suggests that tuning $g + g' + 2\gamma$ via chemical doping (e.g., Sr, Ba) or lattice strain could optimize $T_c$, as seen in Hg-based cuprates ($T_c \approx 133$ K)⁴⁸. This tuning extends to strain-modulated electron-phonon coupling in topological edge states of 2D materials, incorporating nonlinear and anharmonic effects⁵⁸. The non-monotonic $T_c$ behavior indicates an optimal $g \approx 0.35$ eV, achievable through oxygen isotope substitution or pressure⁵⁹, with QEPC enabling higher values. Given their comparable lattice dynamics, extensions to nickelates or pnictides⁶⁰ show promise. Moreover, the doping dome in Fig. 5 suggests that exact control of x at 0.16 by substitution might optimize $T_c$, providing methods for improving superconductivity in layered materials. For nickelates, recent experiments²⁷ suggest similar electron-phonon synergy, where our model could predict $T_c$ by adjusting $\bar{\omega }_D$ to match Ni-O modes.

Figures 3, 4, 5, and 6 demonstrate the non-monotonic trends predicted by our model, with peaks aligning within 10-15% of experimental values from ARPES and neutron scattering, highlighting the role of dynamic pairing enhanced by QEPC in exceeding BCS limits. The model’s prediction of unusual pairing symmetry congruent with cuprate tests is further supported by the visualization of the d-wave gap structure in Fig. 2³⁸. Self-consistent solutions of the gap equation confirm the approximate Gaussian form employed in simulations, which is reflected in the modest asymmetry around $g = 0.35$ eV. The shape of the doping dome highlights the correctness of the derived modulation. The isotope effect in Fig. 6 further validates the phonon role, predicting unconventional doping dependence consistent with observations.

Our framework extends the quantum bipolaron superconductivity from QEPC in¹⁶, where pairing arises from phonon zero-point fluctuations without classical lattice distortion, by embedding it in a cuprate-specific model with correlations, doping, and multi-modes. This enables novel predictions like the dome-shaped phase diagram and unconventional isotope effects, not addressed in the general treatment of¹⁶. In intermediate coupling, superconductivity prevails over competing orders¹⁴, aligning with optimal-doped cuprates⁴⁸, though stronger regimes may favor CDW.

Conclusion

We present a unified phonon-electron model with quadratic coupling for high-temperature superconductivity, deriving novel expressions for $T_c$ and $\Delta$ that incorporate dynamic pairing, QEPC, and correlations. Numerical results ($T_c \approx 100$ K, $\Delta _0 \approx 25$ meV) align with cuprate experiments, validating the model’s predictive power. The framework resolves key controversies by bridging phonon- and correlation-driven mechanisms through QEPC, offering a versatile tool for studying unconventional superconductors, including the characteristic doping dome and isotope effects. Future research should explore multi-phonon effects, doping dependence, and pseudogap physics. Experimental tests, such as isotopic substitution to vary $\bar{\omega }_D$, could further validate the model. Applications to emerging materials like nickelates or twisted bilayer graphene⁴⁸ could expand its impact. For instance, strain-tunable approaches in 2D materials could further optimize pairing interactions, as explored in related frameworks⁶¹. For twisted bilayer graphene, the model’s dynamic potential with QEPC might explain phonon-assisted pairing in moiré superlattices, with adjustments for the flat band DOS.