Superconductivity in Electron Liquids: Precision Many-Body Treatment of Coulomb Interaction

TL;DR

The paper tackles the long-standing challenge of a controlled first-principles treatment of Coulomb repulsion in conventional superconductivity. It develops an effective field theory (EFT) framework synergized with variational Diagrammatic Monte Carlo (DiagMC) to compute the Coulomb pseudopotential $\mu^*$ and the electron-phonon coupling $\lambda$ from the microscopic Hamiltonian, using a downfolded Bethe-Salpeter equation on the Fermi surface. By applying this to the Uniform Electron Gas (UEG), it derives a precise, scale-separated workflow that bridges high-energy renormalizations to a low-energy, frequency-only ME equation, and establishes a robust method for predicting Tc from normal-state data, including near quantum critical points. The work also demonstrates that DFPT can reliably reproduce the effective electron–phonon coupling in simple metals and provides concrete Tc predictions for Li, Na, Mg, Al, and Zn under ambient and high-pressure conditions, offering explanations for long-standing discrepancies and predicting a pressure-induced transition in Al. Overall, the framework advances reliable Tc calculations and material design beyond the weak-correlation limit by anchoring Coulomb and phonon contributions to microscopic vertex functions.

Abstract

More than a century after discovery, the theory of conventional superconductivity remains incomplete. While the importance of electron-phonon coupling is understood, a controlled first-principles treatment of Coulomb interaction is lacking. Current ab initio calculations of superconductivity rely on a phenomenological downfolding approximation, replacing Coulomb interaction with a repulsive pseudopotential μ*, and leaving ambiguities in electron-phonon coupling with dynamical Coulomb interactions unresolved. We address this via an effective field theory approach, integrating out high-energy electronic degrees of freedom using variational Diagrammatic Monte Carlo. Applied to the uniform electron gas, this establishes a microscopic procedure to implement downfolding, define the pseudopotential, and express dynamical Coulomb effects on electron-phonon coupling via the electron vertex function. We find the bare pseudopotential significantly larger than conventional values. This yields improved pseudopotential estimates in simple metals and tests density functional perturbation theory accuracy for effective electron-phonon coupling. We present an ab initio workflow computing superconducting Tc from the anomalous vertex's precursory Cooper flow. This infers Tc from normal state calculations, enabling reliable estimates of very low Tc (including near quantum phase transitions) beyond conventional reach. Validating our approach on simple metals without empirical tuning, we resolve long-standing discrepancies and predict a pressure-induced transition in Al from superconducting to non-superconducting above ~60GPa. We propose ambient-pressure Mg and Na are proximal to a similar critical point. Our work establishes a controlled ab initio framework for electron-phonon superconductivity beyond the weak-correlation limit, paving the way for reliable Tc calculations and novel material design.

Figures (15)

• Figure 1: Normal component of the electron self-energy approximated by the self-consistent Fock diagram with the phonon-mediated e-e interaction $W^{ph}$. According to Migdal's theorem, higher-order vertex corrections based on $W^{ph}$ are suppressed by $O(\omega_\text{D}/E_\text{F})$.
• Figure 2: Diagrammatic representation of the phonon-mediated e-e interaction, $W^{\rm ph}$, composed of the phonon propagator, $D$, bare coupling $g^{(0)}$, vertex function $\Gamma_3^e$, and the dielectric function $\epsilon_{\mathbf{q}\nu}$. The last two are combined to form the screened electron-phonon coupling.
• Figure 3: Self-consistent Bethe-Salpeter equation for the anomalous vertex $\Lambda(\mathbf{k},-\mathbf{k};\mathbf{q}=0)$ in momentum space, where $\mathbf{k}$ and $-\mathbf{k}$ are the momenta of the outgoing electrons. The total momentum $\mathbf{q}$ is set to zero, as this corresponds to the leading Cooper instability in our case. The kernel consists of the particle-particle irreducible 4-point vertex $\tilde{\Gamma}^{e}$, which is a purely electronic contribution, and the phonon-mediated interaction $W^{ph}$ described in Fig. \ref{['fig:wph_def']}; higher-order vertex corrections are small according to Migdal's theorem.
• Figure 4: Dimensionless "bare" Coulomb pseudopotential, $\mu_{E_F}$, as a function of $r_\text{s}$ for the 3D UEG extracted from VDiagMC data for $\mu_{\omega_\text{c}}$ by inverting relation (\ref{['eq:mu_E_F']}); the solid line represents a linear fit to the VDiagMC data. The exact VDiagMC values corresponding to these integer $r_\text{s}$ points are listed in Table \ref{['tab:vdiagmc_data']}. At $r_\text{s}> 0.5$, the VDiagMC results demonstrate a dramatic deviation from predictions of three standard approximations: static random phase approximation (RPA) $\mu_{\text{RPA-static}}$ (red dashed curve), $\mu_{\text{MA}}$ (green dashed curve) based on the Yukawa interaction with Thomas-Fermi screening momentum, and the dynamic RPA (see, e.g., Ref. pengcheng), $\mu_{\text{RPA-dynamic}}$. Note perfect agreement between all curves at $r_\text{s} \ll 1$.
• Figure 5: Comparison between the precursory Cooper flow solutions of the full and downfolded Bethe-Salpeter equations for a toy model with the two-particle-irreducible electron vertex function approximated by the Coulomb interaction screened by RPA polarization and a typical phonon-mediated interaction. The calculation is performed at moderate $r_\text{s}=1.91916$ case (representative of Al) with $T_\text{c}^{(\text{full})}/T_\text{F}=10^{-5.668}$ and $T_\text{c}^{(\text{approx})}/T_F=10^{-5.667}$ (difference $\sim0.2\%$), confirming the validity of the downfolding approximation.