Understanding Feynman's Diagrams, Hedin's Equations, and the GW Method

Week 6, Feynman s diagrams, Hedin s equations Feynman s rule, Hedin s equation, GW method, functional derivatives

Feynmans rule for G ????,? = ???? ?1,?2 = ????,? = ?,? ?,? ?,?1 ?,?2 The ?-th order expansion has n interaction lines ? (- - - - ), and 2? + 1 green s functions ? ( ), each interaction vertex contributes an ? (charge of an electron) The diagrams must be connected and topologically distinct (connected in different ways) The ?-th order diagram has a pre-factor of ? ? Each loop contributes a factor of ( 1) Sum over all internal site indices and integrate over internal contour times.

Fundamental interaction vertex, emission/absorption of virtue photon ?,?2 = ( , ) 1 r v 1 2 lm ( , ) v ?,?1 e 1 2 r 4 | | 1 i 0 l m ( , ) = ( ) ( ) c l g T c 1 1 jl j H 0 ?,? ?,?

G up to second order = + + + + + + + + + + + + + ?(?3)

Self energy and Dyson equation = + + + + + + + ?(?3) = + G g g G = +

Hedin equations + = = g G + v v W = + , G g W + = = ( ) n ( , j ), dn d 3 n n n j n 4 = (1,2) ( ) 3 4 (1,4) (4,2;3) d d G (1,3) e i W 2 1 2 1 = 4 = 3 4 (1,3) (3,4;2) (4,1) d d G (1,2) ei G 1 2 1 2 = 3 1 (1,2) (3) G 3 = = ) (1,2) (1,3) + (1,2;3) ( e V tot = (1,2) (4,5) G (4567) (4,6) (6,7;3) (7,5) G d G 1 2

GW approximation (RPA) + = = g G + G g + = v v W = + W = 2 (1,2) (1,2) (1,2) e i G W 2 1 2 1 = = 2 (1,2) (1,2) (2,1) G e i G 2 1 2 1 = 1 (1,2) (3) G = = ) (1,3) (1,2) + (1,2;3) ( e V tot (1,2) (4,5) G (4567) (4,6) (6,7;3) (7,5) G d G

Density functional theory = KS KS H ( ) = + + = KS hartree xc KS n ({ }), n ( ) j f H H V V n j j n n compare many-body theory: ( ) 1 = + + hartree r r ( ) G E H V

Deriving the Hedin equations, starting point introducing external driving = + ( ), = j q q ( ) H H e c c j j j j j 2 e = + j normal ordered H c Hc v c c c c jk k k j 2 , j k 1 i ( , ') = ( ) ( ') c G c H

Derive an equation of motion for G, replace the Coulomb interaction term by a functional derivative + ( , ') = ( , ') tot i I H ie v G I + ( ) = + ) ( ) j + tot jk 2 n ( H H e e v jk jk jl l l = + ( ) tot j ( ) H e V jk jk = j n c c j j

Functional derivative = d f ( ) F + = + [ ] ( ) ( ) ( ), [ ] f f f F = + 2 ( ) ( ) F d f d O ( )

By a change of variable from G to G-1, we turn it into a Dyson equation = = 1 1 1 ( ) GG G G G G 1 ( , ( ) l ) ( , ') ( ) l G G = d d G ( , ( , ') ) 2 3 G 2 3 2 3 1 1 ( , ') = ( , '') ( '', ') '' ei v G d G jl jk js sk + ( ) l s l 1 (4,2) (3) G + = (1,2) 3 4 (1 ,3) (1,4) d v i e d G

Define dielectric function, eliminate in favor of V tot q q (1) (2) (1) (2) tot (1) V = = (1,2) = 1 (1,2) , (1,2) , tot (2) V ( ) = + ( ) = + tot j tot q q ( ) (1) (1) 2 (1,2) d v (2) V v V j jk k k (3) (2) q tot (1) V = = (1,2) + = + 1 1 (1,2) 3 (1,3) d v 1 v (2) = = v v W = + v v v = + 1 1 (1,2 ) 3 (1,3) (3,2) v W v W d

Charge-charge correlation , screened potential W 1 i ( , ') = T q ( ) ( ') q jk j k = q q q = = = + = 1 1 , = + 1 , , v v W v = v v W = + 1 1 v W v (under RPA approximation)

Define vertex function, use it to express 1 (1,2) (3) G V = (1,2;3) tot (1) (2) q + = = = (1,2) , (1) (1,1 ) ( ) (1) (1) c q ei G e c tot V = GG + G G = 1 1 1 use 0 GG I + 1 (1,1 ) (2) (3,4) (2) G V G V + = = (1,2) 3 4 (1,3) d d G (4,1 ) ei ei G tot tot 3 4 (1,3) (3,4; i e d d G + = 2) (4,1 ) G

Final step of deriving Hedin equations, close the system by ? /?? 1 (1,2) (3) (1,2) (3) G V = = ) (1,3) (1,2) + (1,2;3) ( (used Dyson equation) e tot tot V 1 G G G V = = , G G tot tot tot tot V V G V (1,2) (4,5) G = ) (1,2) (1,3) + (4567) (4,6) (6,7;3) (7,5) G (1,2;3) ( e d G