Analytic treatment of a polaron in a nonparabolic conduction band

TL;DR

This paper develops and compares analytical methods for studying polarons in non-parabolic, finite-width conduction bands, extending Feynman's variational approach to lattice systems and benchmarking against exact numerical results.

Contribution

It introduces an extended Feynman variational method for lattice polarons in non-parabolic bands, applicable across coupling regimes, and benchmarks it against numerical techniques.

Findings

Modified Feynman method achieves high accuracy in energy and dispersion.

Analytical approaches successfully describe polarons with spin-orbit coupling.

Methods outperform traditional continuum models in lattice systems.

Abstract

We develop and compare several analytical approximations for the polaron problem in finite-width, non-parabolic conduction bands. The main focus of the work is an extension of the Feynman variational method to a tight-binding lattice, where the effective-mass approximation is no longer applicable. The resulting variational formulation is not restricted to a specific phonon dispersion or electron-phonon interaction and provides a uniform description across weak-, intermediate-, and strong-coupling regimes. We revisit and generalize other analytical approaches traditionally formulated for continuum polarons, including canonical transformations and self-consistent Wigner-Brillouin-type approximations. For lattice polarons, these methods exhibit qualitative features absent in the continuum case, such as a nontrivial connection between weak- and strong-coupling limits. We show that an improved Wigner-Brillouin scheme yields a momentum-dependent polaron self-energy free of resonances and in good agreement with numerically exact results over the whole range of momenta within the Brillouin zone. All methods are applied to the Holstein model and are benchmarked against numerically exact calculations, including Diagrammatic Monte Carlo (both our calculations and preceding works), exact diagonalization, and density-matrix renormalization-group results. The analytical approaches are extended to polarons with Rashba-type spin-orbit coupling, providing a stringent test of their applicability in systems with nontrivial band structure. Our results demonstrate that the modified Feynman variational method yields ground-state energies and dispersions with accuracy comparable to, and in many cases exceeding, that of other established analytical approaches. The developed framework offers a versatile and reliable analytical description of lattice polarons beyond the continuum approximation.