Beyond-quasiparticle transport with vertex correction: self-consistent ladder formalism for electron-phonon interactions

TL;DR

This work develops a self-consistent ladder-sc$GD_0$ framework to compute phonon-limited electronic transport from first principles, incorporating beyond-quasiparticle spectral features and vertex corrections within equilibrium Keldysh formalism. Building on the sc$GD_0$ electron self-energy, the authors derive a self-consistent ladder equation for the current vertex, unifying the Boltzmann transport equation and bubble approximations while enforcing charge conservation via the Ward identity. They validate the method on model Hamiltonians and apply it to Si, ZnO, and SrVO$_3$, achieving quantitative agreement with dc conductivities and THz optical/dielectric measurements, and they show the phonon-assisted current is essential for nonlocal electron–phonon coupling. This approach provides a practical, ab initio route to transport in materials with strong electron–phonon interactions and establishes a framework for extending many-body transport to additional response functions.

Abstract

We present a self-consistent many-body framework for computing phonon-limited electronic transport from first principles, incorporating both beyond-quasiparticle effects and vertex corrections. Using the recently developed first-principles scGD0 method, we calculate spectral functions with nonperturbative effects such as broadening, satellites, and energy-dependent renormalization. We show that the scGD0 spectral functions outperform one-shot G0D0 and cumulant approximations in model Hamiltonians and real materials, eliminating unphysical spectral kinks and correctly predicting the phonon emission continuum. Building on this, we introduce the self-consistent ladder formalism for transport, which captures vertex corrections due to electron-phonon interactions. This approach unifies and improves upon the two state-of-the-art approaches for first-principles phonon-limited transport: the bubble approximation and the Boltzmann transport equation. Moreover, as a charge-conserving approximation, it enables consistent calculations of the optical conductivity and dielectric function. We validate the developed method against numerically exact results for model Hamiltonians in the dilute polaronic limit and apply it to real materials. Our results show quantitative agreement with the experimental dc conductivities in intrinsic semiconductors Si and ZnO and the SrVO3 metal, as well as excellent agreement with the experimental THz optical and dielectric properties of Si and ZnO. This work unifies first-principles and many-body approaches for studying transport, opening new directions for applying many-body theory to materials with strong electron-phonon interactions.

Figures (37)

• Figure 1: Overview of the ladder-sc$GD_0$ transport formalism. (a) Feynman diagrams for the self-consistent $GD_0$ (sc$GD_0$) self-energy and the self-consistent ladder equation. Single solid lines show bare electron Green's functions; double solid lines show dressed ones. Blue wavy lines indicate the bare phonon propagator, black dots represent the e-ph coupling, gray squares indicate Debye--Waller coupling, and white circles represent coupling to external electric fields. Green shaded regions indicate the renormalized current vertex function. (b) Hierarchy of linear-response e-ph transport methods. The Boltzmann transport equation (BTE) Ponce2020Review, self-energy relaxation time approximation (SERTA) Zhou2016Ponce2018, and the bubble approximation Basov2011RMPZhou2019STO can be derived as approximations to ladder-sc$GD_0$. (c) Real part of the normalized ac conductivity of $n$-doped ZnO at $T=90~\mathrm{K}$ with calculations compared to experimental data Baxter2009 (see also Fig. \ref{['fig:ZnO_ac']}). Ladder-sc$GD_0$ accurately captures the frequency dependence, whereas other methods yield a decay that is too slow with frequency.
• Figure 2: Time ordering on the Schwinger--Keldysh contour.
• Figure 3: Feynman diagrams of the (a) Dyson equation for the Green's function, (b) self-consistent $GD_0$ (sc$GD_0$) self-energy including the Debye--Waller (DW) term, (c) one-shot $G_0 D_0$ self-energy, and (d) low-order diagrams not included in the sc$GD_0$ approximation.
• Figure 4: Flowchart for the self-consistent calculation of the self-energy.
• Figure 5: (a)--(d) Spectral functions of the undoped 1D Holstein model with parameters $t = \omega_0 = 1$, $\lambda = 0.5$, and $T=0$ computed using the Rayleigh--Schrödinger perturbation theory (RS), cumulant approximation, one-shot $G_0 D_0$, and self-consistent $GD_0$ (sc$GD_0$) methods. Dashed black lines show the bare electron band. Red arrows mark nonphysical features in the RS dispersion and the cumulant spectral function. Solid horizontal line in (d) indicates the quasiparticle energy at $k_x=0$, and the dashed line indicates the energy above it by the phonon frequency $\omega_0$. (e) One-dimensional cuts of the spectral functions at $k_x=0$, shown on a logarithmic $x$ scale. We also show the exact result from the hierarchical equations of motion (HEOM) method Mitric2022 as a thin black line. Note that the splitting of the satellite peak in the HEOM result around $\varepsilon = -1.5 \omega_0$ is a finite-size artifact Mitric2022. (f)--(j) The same for the 1D Peierls model with $t = \omega_0 = 1$, $\lambda = 0.5$, and $T=0$. (k)--(o) The same for the 3D Fröhlich model with $m_0 = \omega_0 = 1$, $\alpha = 2$, and $T=0$, where the wavevector $\abs{{{\bm{\mathrm{k}}}}}$ is given in units of $k_0 = \sqrt{2m_0 \omega_0}$. We used an artificial broadening of $\eta = 0.01$ for all calculations, which determines the width of the low-energy quasiparticle peaks.