On the applicability of the cumulant expansion method for the calculation of transport properties in electron-phonon systems

TL;DR

The paper critically evaluates the cumulant expansion (CE) method within the independent-particle approximation (IPA) for calculating charge mobility in electron–phonon systems, using the Peierls and Fröhlich models as benchmarks against Boltzmann transport, Migdal (MA), and self-consistent Migdal (SCMA) approaches. It shows that CE, despite being perturbative in nature, can yield accurate mobility predictions at weak-to-moderate coupling and not-too-low temperatures, particularly when the interaction is momentum-dependent, where vertex corrections become significant. A central theoretical contribution is the argument that CE reproduces spectral sum rules up to order four, offering a justification for its improved performance over MA in several regimes; a practical criterion based on the convergence of n_e from sum rules guides where CE is reliable. The study also highlights limitations, including spectral tails at low temperatures and nonconvergence in Fröhlich-like cases with strong coupling, which can be mitigated by SCMA or alternative methods. Overall, CE emerges as a computationally efficient and reasonably accurate tool for mobility calculations within IPA, with caveats linked to convergence and tail effects that warrant careful numerical and analytical checks.

Abstract

We assess the accuracy of the cumulant expansion (CE) method, combined with the independent-particle approximation (IPA), for calculating charge mobility in electron-phonon systems. As representative testbeds, we consider the Peierls and Fröhlich models, which serve as simplified frameworks where accurate or numerically exact benchmarks are available. These are used to compare the CE results with those obtained using the Boltzmann formalism, the Migdal approximation, and its self consistent extension-approaches that are presently the most commonly employed alternatives for transport calculations. Supported by analytical arguments based on spectral sum rules and by our previous results for the Holstein model, we argue that, for weak to moderate coupling strengths and not-too-low temperatures, the CE within the IPA framework yields accurate results. In the case of the Peierls model, the role of vertex corrections is also discussed.

Figures (25)

• Figure 1: Feynman diagrams for (a) $\mu_e$, (b) $\mu_{ph}$ within the independent-particle approximation, and (c) corrections beyond the independent-particle approximation.
• Figure 2: Self-energy in the Migdal approximation.
• Figure 3: Comparison of the band-mass to renormalized-mass ratio $m_0/m^*$ and the ground-state energy $E_p$ obtained from CE, GGCE, SCMA, and MA for $\omega_0 = 0.5$ at $T = 0$. The red point corresponds to the CE result for $\lambda=0.25$.
• Figure 4: Comparison of different methods for calculating the inverse of (a) the total mobility $1/\mu$ (see Eq. \ref{['eq:total_mobility_4342']}), (b) the electronic contribution to the mobility $1/\mu_e$ (see Eq. \ref{['muedc']}), and (c) the phononic contribution to the mobility $1/\mu_{ph}$ (see Eq. \ref{['eq:muph']}), for $\omega_0 = 0.5$ and $\lambda = 0.25$.
• Figure 5: Comparison of HEOM, CE, and SCMA spectral functions for $\omega_0=0.5$, $\lambda=0.25$, $T=1.0$.