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

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 renormalized phonon dispersion, the electron dispersion and electron spectral functions. I demonstrate how vertex corrections stabilize the solution, stopping a catastrophic softening of the (π, π) phonon mode. A kink in the electron dispersion is found in the normal state along the (ζ, ζ) symmetry direction in both the vertex- and non-vertex-corrected theories for low phonon frequencies, corresponding directly to the renormalized phonon frequency at the (π, 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.