Quantum Interactions: Electrons, Phonons, and Hubbard Interaction

Exploring the complexities of electron-electron and electron-phonon interactions, nonequilibrium Green's functions, Hartree-Fock method, Coulomb's law, quantum operator forms, Hubbard interaction, and electron-phonon interactions from first principles. The interactions delve into the behavior of charges, quantum operators, and modes, highlighting key principles in quantum mechanics.

• Quantum Interactions
• Electrons
• Phonons
• Hubbard Interaction
• Greens Functions

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1. Week 4 Electron-electron, electron- phonon interactions, nonequilibrium Green s functions, Hartree-Fock method

2. Coulombs law between charges q1, r1 q2, r2 1 1 2 q q r = = = F m 12 1 ( , ) r r , 4 1 (atomic u nits), 8.854187813 10 V 1 2 0 0 r 4 0 1 2 r ( ) = = 1 = 2 ( ) r r r r , , ( ) ( ) V q q 1 1 1 2 1 2 0 = = = = E E F E , , q V 1 0

3. Coulomb interaction in quantum operator form = = = j j q n , , (for electron) q c c c c q e j j j j j j 1 1 ? = = 1 2 q q , 1 2 q q 1 2 1 1 2 2 q q c c c c v ij 4 r r 0 i j N N N N ( ) = = + j j i ij i v c cc c ij i v c c c c ij i v c c ij i v c c c c i j ij j j ij j j i = = = = , 1 , 1 , 1 , 1 i j i j i j i j c c 1 2 N e ( ) 2 = + = = 1 2 N , , , , H c Hc ij i v c c c c c c c c c j j i 2 = , 1 i j c N

4. Hubbard interaction Onsite interaction: putting two electrons of the same spin on the same location is not possible due to Pauli exclusion principle. Putting two electrons on the same site must have opposite spins. = 0 c c j j j j Uc c c c j j = j Un n j

5. Electron phonon interaction, a simple consideration (SSH type) tc c + + + i i i h.c. ( ) h.c. tc c gc c u u j j j i j hopping t = + + 2 ( ) ( ) t a '( ) t a ( ) ( ) t x x M u u O u j i j i = = + , x M u x a M u i i j j distance

6. Electron-phonon interaction, first principles 1 2 ( ) Problem: work out the explicit relationship between real space and mode space representations of the Hamiltonian. = + + + T j 2 H c Hc p u Ku c M c u j j 1 2 k q = + + n v c c k a a q k k q q n n v v n v 1 N ( ) + + q v mn m v ( ) k g c c a a + k q k q q n v , , , , m n k q n,k ,q R ( ) R 1 M H = k ij g | | e M i j k k m,k+q

7. Electron Greens function of a single mode 1 i dc dt i i t = = = = , [ , c H ] , ( ) c t (0) H c c c c e i i i i t i t = = = ( ) ( ) (0) c t c (1 ) g t e cc e f 1 1 i i i t = + = = = = ( ) (0) ( ) c t , , g t c e f f c c + ( ) 1 e k T B i i t = = , ( ) 1 if = r ( ) ( ) t ( ) ( ) ( ) t e 0 and 0 if 0 g t g t g t t t t i i t = = t e a ( ) ( ) ( ) ( ) ( ) g t t g t g t = = + + t g t = = + + = = + = t r r a ( ) ( ) ( ) ( ) t g t ( ) ( ) ( ( ) ) t g t ( ) ( ) g g g g t g t t g t g g g g g g g g iA t r t tg

8. Electron Greens function of a single mode, go into E space + + E E dE i t i t = = = ( ) g E ( ) ( ) ( ) g E e , / g t e dt g t E 2 = = = = 2 ( ) E 2 ( )( 1) 1 = g i E f i ( ) E 2 ( ) ( ), ( ) A E 2 ( ) g i E f i f A E E + E 1 i i i t i t t + = ( ) t e = = * r a ( ) ( ) E 0 g E e dt g + E = = + + = = + = t r r a g g g g g g g g g g g g g g iA t r t t

9. Greens functions of many degrees, diagonalize H H H H H H c c 11 12 1 1 N ( ) 21 22 2 = = 1 2 N , , , H c Hc c c c H H c 1 N NN N N = = = = i c H c H H S S SS I ij j = , 1 i j 0 0 1 0 ( ) 2 = = = , , , , , S HS H S 1 2 n n n N 0 0 0 N d d 1 N 2 = = = = n , H c Hc d d d S c n n = 1 n d N

10. Many degree Greens function i i as N N matrix = = k ( ) t ( ) (0) or c t c ( ) ( ) (0) c t c G G t jk j = = = = m , , , (1 ) c Sd c d S S HS d d f n n nm m , , , , , a r t t , , , , , a r t t = ( or ) t ( or ) t G E S g E S , , r a t ( ) t dG ( ) 1 = + i = , ,( ) r r a t ( ) ( ) , ( ) t I G E E I H i HG t dt

11. Fluctuation-dissipation theorem in thermal equilibrium = = r a ( ) ( ) f E ( ) G E G G if A E Prove this important result using the Lehmann representation. 1 ( ) = = a r ( ) f E , G G + ( ) E 1 e ( ) = = ( ) E G G e G G

12. Greens function, complex z i = ( ) t e = / / r iHt iHt ( ) compare: ( ) (0) G t t e N ( ) 1 = = n n ( ) z (resolvent of ) G z zI H H z = 1 n n + i r Retarded Advanced ( ) is obtainef if ( ) is obtained if G E G E z E i a z E + (2 1) n + = = 0, 1, 2, M Matsubara ( ) is obtained if , , G z i n n n n

13. Equation of motion of (free) Green s functions , , r a t ( ) t dG = , , r a t ( ) t ( ) t I i HG Problem: Prove this from the definitions of Green s functions, with the Heisenberg equation of motion for c, ? ?? dt , ( ) t dG = , ( ) t 0 i HG dt ??= ??. t ( ) dG t dt = t ( ) ( ) t I i HG t

14. Contour ordered Greens function + - ( ) ( ') if c ' c i i ( , ') = ( ) ( ') c = G T c ( ') ( ) if ' c c ( , ), t = + + ++ + t G G G G G G G G ( , ') = = = ' ' ( , ') t t ( ') G G G t t + t

15. Equation of motion on contour ? + ? + ( , ') G ( , ') = ( , ') i HG I ( , ') ' = ( '), 1 t t ( , ) t

16. Handle Coulomb interaction, the simplest way, mean-field theory 1 2 = + 2 j k H c Hc e v c c c c jk k j jk j k c c c c j k j k c c k c c c c c c k j k j j + = HF 2 k j 2 j k c c k H c Hc e v c c c c e v c c c H c jk k j jk j jk jk Hartree tern Fock term

17. Hartree and Fock self energies m j k = F jk 2 k e v c c jk j j = + + HF H F H H = H jk 2 m m c c e v jk jm m HF H c H c

18. Greens function in Hartree- Fock approximation + = HF 2 k j 2 j k c c k H c Hc e v c c c c e v c c c H c jk k j jk j jk jk 1 = + i + + r 2 k ( ) ( ) G E E H e e v c c jk jk j jk jk j jk 1 1 = = = k ( ) , e v c c f E j jk k + ( ) E 1 e k T k B + i ( ) = = * iEt r jk r kj k / ( ) E (0) ( ) c t ( ) f E G ( ) ( ) E G c e dt E G jk j + dE = = = HF n k * kn ( ) E ( ) , ( ) j c c i G S f S S j jk jn jn n 2 n

19. A computational procedure for Hartree-Fock self-consistency 1) Compute ???matrix, set the Coulomb interaction terms (Hartree and Fock terms) to 0 in the first step. 2) Solve the eigenvalue problem ??? ?= ?? ?. Normalize ? to 1. 3) Compute ?? 4) Go back to 1). Repeat until energy and eigen functions are converged. ?? = ? ?? ?(??) ? (?).

20. Solve the benzene ring with PPP model using Hartree-Fock? Need spin! 1 2 6 = + + i h.c. ( 1)( 1) H t c c V n n + 1, ' i ij i j = = , , , ' i j 1, , i U = V ij + 2 r r 1 | | i j