Effective enhancement of the electron-phonon coupling driven by nonperturbative electronic density fluctuations

TL;DR

This work demonstrates that near the Mott metal–insulator transition, nonperturbative electronic density fluctuations can dramatically renormalize the electron-phonon coupling in a correlated 2D Hubbard model. Using DMFT and a two-particle Bethe-Salpeter framework, the authors express the static charge susceptibility as $\chi^c_{\mathbf{q}}(0) \simeq \sum_{\alpha} (\lambda_{\alpha}^{-1} + \beta \, {\cal T}_{\mathbf{q}})^{-1} w_{\alpha}$ with ${\cal T}_{\mathbf{q}} \approx t^2[\cos(q_x)+\cos(q_y)]$, linking momentum structure to eigenvalues of the local susceptibility. They show that approaching the MIT causes a low-$\mathbf{q}$ peak and an $\xi$-controlled, Ornstein–Zernike–like behavior for both the charge response and the renormalized el-ph vertex $\gamma_{\mathbf{q}}^{\nu}$, which in turn amplifies the phonon-mediated pairing interaction, yielding a static $\Gamma_{pair}$ that scales as $\sim \xi^2$ in 2D. The analysis further reveals that the renormalized el-ph coupling can be dramatically enhanced at small $\mathbf{q}$ while the compressibility shows a more nuanced evolution depending on spectral weights $w_{\alpha}$ and the lowest eigenvalue $\lambda_I$, a distinction sharpened by including next-nearest-neighbor hopping $t'$. Finally, a DMFT-based D$\Gamma$A–like approach highlights potential divergences and the limits of perturbative treatments near phase transitions, motivating future self-consistent extensions. Together, these results reveal a robust, nonperturbative mechanism for lattice–electron interplay in strongly correlated systems and point to strong forward-scattering tendencies that could influence superconductivity in Hund’s metals and related oxides.

Abstract

We present a dynamical mean-field study of the nonperturbative electronic mechanisms, which may lead to significant enhancements of the electron-phonon coupling in correlated electron systems. Analyzing the effects of electronic correlations on the lowest-order electron-phonon processes, we show that in the proximity of the Mott metal-to-insulator transition of the doped square lattice Hubbard model, where the isothermal charge response becomes particularly large at small momenta, the coupling of electrons to the lattice is strongly increased. This, in turn, induces significant corrections to both the electronic self-energy and phonon-mediated pairing interaction, indicating the possible onset of a strong interplay between lattice and electronic degrees of freedom even for small values of the bare electron-phonon coupling.

Figures (13)

• Figure 1: Momentum dependence of the static charge susceptibility $\chi^c_{\mathbf{q}}$ computed in DMFT for the 2D-Hubbard model on the square lattice for $\beta=55.75$ and $n=0.9966$ at the bosonic frequency $\omega=0$ and evaluated for varying $U$ values.
• Figure 2: Renormalized el-ph vertex $\gamma_{\mathbf{q}} \!= \! \tilde{g}_{\mathbf{q}}/g_0$ extracted from $\chi^c_{\mathbf{q}}$ as a function of $U$ at the lowest bosonic/fermionic frequencies at $\beta=55.75$. Inset: Contribution of the lowest eigenvalue to $\gamma_{\mathbf{q}}$ ($\gamma_I$), in comparison to the contribution of the remaining eigenvalues ($\gamma_{R}$).
• Figure 3: Left panel: Lowest order contributions in $g_0$ to the phonon-mediated pairing interaction and to the electronic self-energy. Central panel: $U$-dependence of the pairing diagram (violet) in units of $\lambda$ for $\beta=55.75$ and $\omega_0 =0.1$, split in its contributions arising from the zero (red) and finite frequency (blue) components. Inset: Dependence of the zero-frequency contribution to the pairing diagram on the correlation length and inverse temperature. Right panel: Same plots as for the central panel, but for the self-energy corrections (s. text).
• Figure S1: Diagrammatic representation of the Bethe-Salpeter equation. Here, $\chi_c$ denotes the full charge susceptibility and the first term, $\chi_0$, corresponds to the bubble diagram. $\Gamma_c$ contains all two-particle irreducible diagrams (here in the charge channel within the particle-hole convention).
• Figure S2: Real and imaginary part of $\beta\mathcal{T}_{\mathbf{q}=0}(\nu)$ as a function of fermionic Matsubara frequencies $\nu$ for increasing values of $U$.