Journal of Low Temperature Physics

, Volume 151, Issue 5–6, pp 1149–1163 | Cite as

Breakdown of Migdal–Eliashberg Theory via Catastrophic Vertex Divergence at Low Phonon Frequency

Article

Abstract

We investigate the applicability of Migdal–Eliashberg (ME) theory by revisiting Migdal’s analysis within the dynamical mean-field theory framework. First, we compute spectral functions, the quasi-particle weight, the self energy, renormalised phonon frequency and resistivity curves of the half-filled Holstein model. We demonstrate how ME theory has a phase-transition-like instability at intermediate coupling, and how the Engelsberg–Schrieffer (ES) picture is complicated by low-energy excitations from higher order diagrams (demonstrating that ES theory is a very weak coupling approach). Through consideration of the lowest-order vertex correction, we analyse the applicability of ME theory close to this transition. We find a breakdown of the theory in the intermediate coupling adiabatic limit due to a divergence in the vertex function. The region of applicability is mapped out, and it is found that ME theory is only reliable in the weak coupling adiabatic limit, raising questions about the accuracy of recent analyses of cuprate superconductors which do not include vertex corrections.

Keywords

Interacting electron systems Electron-phonon interactions Migdal–Eliashberg theory 

PACS

71.10.Fd 71.30.+h 71.38.Ht 71.38.Mx 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of WarwickWarwickUK
  2. 2.Department of PhysicsLoughborough UniversityLoughboroughUK

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