Variational Polaron Equations Applied to the Anisotropic Fr\"ohlich Model

TL;DR

This paper develops a variational polaron equation framework in Bloch space for anisotropic Fr"ohlich models, enabling detailed energy decomposition and improved convergence, validated against known solutions and Gaussian ansatz results.

Contribution

It introduces a novel variational approach in Bloch space for anisotropic polaron models, with an efficient optimization algorithm and analytical treatment of divergence issues.

Findings

Validated against isotropic Fr"ohlich model solutions

Demonstrated energy decomposition consistent with Pekar's theorem

Improved convergence through analytical divergence treatment

Abstract

Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr\"ohlich model with anisotropic non-degenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr\"ohlich electron-phonon matrix elements at $\Gamma$-point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr\"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr\"ohlich model.