Electron and phonon dispersions of the two dimensional Holstein model: Effects of vertex and non-local corrections

TL;DR

This study uses the dynamical cluster approximation to analyze electron and phonon dispersions in the 2D Holstein model, revealing how vertex and non-local corrections influence phonon stability and electron spectral features.

Contribution

It introduces the application of DCA to include spatial fluctuations and vertex corrections in the Holstein model, providing new insights into phonon and electron dispersion behaviors.

Findings

Vertex corrections stabilize phonon modes, preventing softening.

A kink in electron dispersion correlates with renormalized phonon frequency.

Non-local corrections amplify effects at high phonon frequencies.

Abstract

I apply the newly developed dynamical cluster approximation (DCA) to the calculation of the electron and phonon dispersions in the two dimensional Holstein model. In contrast to previous work, the DCA enables the effects of spatial fluctuations (non-local corrections) to be examined. Approximations neglecting and incorporating lowest-order vertex corrections are investigated. I calculate the phonon density of states, the renormalised phonon dispersion, the electron dispersion and electron spectral functions. I demonstrate how vertex corrections stabilise the solution, stopping a catastrophic softening of the $(\pi,\pi)$ phonon mode. A kink in the electron dispersion is found in the normal state along the $(\zeta,\zeta)$ symmetry direction in both the vertex- and non-vertex-corrected theories for low phonon frequencies, corresponding directly to the renormalised phonon frequency at the $(\pi,0)$ point. This kink is accompanied by a sudden drop in the quasi-particle lifetime. Vertex and non-local corrections enhance the effects at large bare phonon frequencies.