Role of the ward identity and relevance of the G⁰W⁰ approximation in normal and superconducting states

ABSTRACT

On the basis of the self-consistent calculation scheme for the electron self-energy Σ with the use of the three-point vertex function Γ always satisfying the Ward identity, we find that the obtained quasi-particle dispersion in the normal state in gapped systems, such as semiconductors, insulators, and molecules, is well reproduced by that in the one-shot GW (or G₀W₀) approximation. In calculating the superconducting transition temperature T_c, we also find a similar situation; the result for T_c in the gauge-invariant self-consistent framework including the effect of Γ satisfying the Ward identity is different from that in the conventional Eliashberg theory (which amounts to the GW approximation for superconductivity) but is close to that in the G₀W₀ approximation. Those facts indicate that the G₀W₀ approximation actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner and thus we can understand that the G₀W₀ approximation is better than the fully self-consistent GW one in obtaining some of physical quantities.

GRAPHICAL ABSTRACT

Keywords:

• Superconductivity
• vertex correction
• ward identity
• GW approximation
• Eliashberg theory
• gauge-invariant self-consistent (GISC) theory

Acknowledgements

Y. Takada thanks H. Maebashi and S. Ishii for valuable discussions for the result in the normal state.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan: Innovative Area ‘Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics [grant number 22104011]