Hierarchy of Self-Consistent Approximate Methods and the Parquet ...

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Motivation For the strongly-correlated electron systems, the Hubbard model is a minimum model H = −t ∑ 〈i,j〉,σ (c† i,σ c j,σ + h.c.) + U ∑ i n i,↑n i,↓ Diagrammatic method is a popular approach to analyze this model Central question: How to sum up as many as possible physically relevant Feynman diagrams =⇒ Self-consistent diagrammatic approximate methods Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 4
Green function and self-energy (x 2 , t 2 ) (x 1 , t 1 ) • 1-particle Green function G ψ(x 2 , t 2 ) = ∫ G(x 2 , t 2 ; x 1 , t 1 )ψ(x 1 , t 1 ) dx 1 dt 1 ⊲ amplitude (phase) accumulated when a particle moving around some path ⊲ long-ranged • self-energy Σ ⊲ renormalization due to the interaction with other particles ⊲ short-ranged • Similarly, we can define 2-partilce Green function and 2-particle “self-energy” (vertex). Hierarchy of Self-Consistent Approximate Methods and the Parquet Formalism 5
Page 1 and 2: Hierarchy of Self-Consistent Approx
Page 3: Motivation For the strongly-correla
Page 7 and 8: Schwinger-Dyson equation Schwinger-
Page 9 and 10: Vertex v + initial 3 1 + + 4 H 2 P
Page 11 and 12: scattering channels p-h horizontal
Page 13 and 14: 1-particle reducibility reducible +
Page 21 and 22: Bethe-Salpeter equation and Parquet
Page 23 and 24: Hierarchy of self-consistent approx
Page 31 and 32: Parquet approximation: Λ ≈ v •
Page 33 and 34: DCA/QMC + parquet: Λ II ≈ Λ I,D
Page 35 and 36: application I: toy model The Hamilt
Page 37 and 38: application II: Hubbard model (N c
Page 39: Conclusion • Parquet formalism

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