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ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First Principles Trieste, 19-23 March 2018 Lecture Wed.1 Many-body theory of electron-phonon interactions Feliciano Giustino Department of Materials, University of Oxford Department of

  1. Time evolution of field operators ˆ Ground state of N -electron system H | N � = E N | N � ˆ s -th excited state of N +1 -electron system H | N + 1 , s � = E N +1 ,s | N + 1 , s � Excitation energy ε s = E N +1 ,s − E N Heisenberg time evolution i � ∂ � � Ht/ � ˆ ψ ( x , t ) = e i ˆ ψ ( x ) e − i ˆ ˆ Ht/ � ˆ ψ ( x , t ) , ˆ ˆ ψ ( x , t ) = H ∂t Exercise Ht/ � ˆ � N | e i ˆ ψ ( x ) e − i ˆ Ht/ � | N + 1 , s � � N | ψ ( x , t ) | N + 1 , s � = � N | e iE N t/ � ˆ ψ ( x ) e − iE N +1 ,s t/ � | N + 1 , s � = Giustino, Lecture Wed.1 12/35

  2. Time evolution of field operators ˆ Ground state of N -electron system H | N � = E N | N � ˆ s -th excited state of N +1 -electron system H | N + 1 , s � = E N +1 ,s | N + 1 , s � Excitation energy ε s = E N +1 ,s − E N Heisenberg time evolution i � ∂ � � Ht/ � ˆ ψ ( x , t ) = e i ˆ ψ ( x ) e − i ˆ ˆ Ht/ � ˆ ψ ( x , t ) , ˆ ˆ ψ ( x , t ) = H ∂t Exercise Ht/ � ˆ � N | e i ˆ ψ ( x ) e − i ˆ Ht/ � | N + 1 , s � � N | ψ ( x , t ) | N + 1 , s � = � N | e iE N t/ � ˆ ψ ( x ) e − iE N +1 ,s t/ � | N + 1 , s � = � N | ˆ ψ ( x ) | N + 1 , s � e − iε s t/ � = Giustino, Lecture Wed.1 12/35

  3. Time evolution of field operators ˆ Ground state of N -electron system H | N � = E N | N � ˆ s -th excited state of N +1 -electron system H | N + 1 , s � = E N +1 ,s | N + 1 , s � Excitation energy ε s = E N +1 ,s − E N Heisenberg time evolution i � ∂ � � Ht/ � ˆ ψ ( x , t ) = e i ˆ ψ ( x ) e − i ˆ ˆ Ht/ � ˆ ψ ( x , t ) , ˆ ˆ ψ ( x , t ) = H ∂t Exercise Ht/ � ˆ � N | e i ˆ ψ ( x ) e − i ˆ Ht/ � | N + 1 , s � � N | ψ ( x , t ) | N + 1 , s � = � N | e iE N t/ � ˆ ψ ( x ) e − iE N +1 ,s t/ � | N + 1 , s � = � N | ˆ e − iε s t/ � = ψ ( x ) | N + 1 , s � � �� � f s ( x ) Dyson orbital Giustino, Lecture Wed.1 12/35

  4. The Green’s function at zero temperature Time-ordered Wick’s time-ordering operator Green’s function G ( x t, x ′ t ′ ) = − i � � N | ˆ T ˆ ψ ( x t ) ˆ ψ † ( x ′ t ′ ) | N � Giustino, Lecture Wed.1 13/35

  5. The Green’s function at zero temperature Time-ordered Wick’s time-ordering operator Green’s function G ( x t, x ′ t ′ ) = − i � � N | ˆ T ˆ ψ ( x t ) ˆ ψ † ( x ′ t ′ ) | N � � � electron in x ′ at time t ′ � � Giustino, Lecture Wed.1 13/35

  6. The Green’s function at zero temperature Time-ordered Wick’s time-ordering operator Green’s function G ( x t, x ′ t ′ ) = − i � � N | ˆ T ˆ ψ ( x t ) ˆ ψ † ( x ′ t ′ ) | N � � � � electron in x ′ at time t ′ � � electron in x at time t Giustino, Lecture Wed.1 13/35

  7. The Green’s function at zero temperature Time-ordered Wick’s time-ordering operator Green’s function G ( x t, x ′ t ′ ) = − i � � N | ˆ T ˆ ψ ( x t ) ˆ ψ † ( x ′ t ′ ) | N � � � � electron in x ′ at time t ′ � � electron in x at time t x t • • x ′ t ′ Giustino, Lecture Wed.1 13/35

  8. The Green’s function at zero temperature Consider t > t ′ (electron injection) − i � � N | ˆ ψ ( x t ) ˆ G ( x t, x ′ t ′ ) ψ † ( x ′ t ′ ) | N � = Giustino, Lecture Wed.1 14/35

  9. The Green’s function at zero temperature Consider t > t ′ (electron injection) − i � � N | ˆ ψ ( x t ) ˆ G ( x t, x ′ t ′ ) ψ † ( x ′ t ′ ) | N � = − i � � N | e i ˆ Ht/ � ˆ ψ ( x ) e − i ˆ Ht/ � e i ˆ Ht ′ / � ˆ ψ † ( x ′ ) e − i ˆ Ht ′ / � | N � = Giustino, Lecture Wed.1 14/35

  10. The Green’s function at zero temperature Consider t > t ′ (electron injection) − i � � N | ˆ ψ ( x t ) ˆ G ( x t, x ′ t ′ ) ψ † ( x ′ t ′ ) | N � = − i � � N | e i ˆ Ht/ � ˆ ψ ( x ) e − i ˆ Ht/ � e i ˆ Ht ′ / � ˆ ψ † ( x ′ ) e − i ˆ Ht ′ / � | N � = − i ψ ( x ) e − i ( ˆ H − E N )( t − t ′ ) / � ˆ � � N | ˆ ψ † ( x ′ ) | N � = Giustino, Lecture Wed.1 14/35

  11. The Green’s function at zero temperature Consider t > t ′ (electron injection) − i � � N | ˆ ψ ( x t ) ˆ G ( x t, x ′ t ′ ) ψ † ( x ′ t ′ ) | N � = − i � � N | e i ˆ Ht/ � ˆ ψ ( x ) e − i ˆ Ht/ � e i ˆ Ht ′ / � ˆ ψ † ( x ′ ) e − i ˆ Ht ′ / � | N � = − i ψ ( x ) e − i ( ˆ H − E N )( t − t ′ ) / � ˆ � � N | ˆ ψ † ( x ′ ) | N � = � s | N + 1 , s �� N + 1 , s | Giustino, Lecture Wed.1 14/35

  12. The Green’s function at zero temperature Consider t > t ′ (electron injection) − i � � N | ˆ ψ ( x t ) ˆ G ( x t, x ′ t ′ ) ψ † ( x ′ t ′ ) | N � = − i � � N | e i ˆ Ht/ � ˆ ψ ( x ) e − i ˆ Ht/ � e i ˆ Ht ′ / � ˆ ψ † ( x ′ ) e − i ˆ Ht ′ / � | N � = − i ψ ( x ) e − i ( ˆ H − E N )( t − t ′ ) / � ˆ � � N | ˆ ψ † ( x ′ ) | N � = � s | N + 1 , s �� N + 1 , s | − i � s f s ( x ) f ∗ s ( x ′ ) e − iε s ( t − t ′ ) / � = � Giustino, Lecture Wed.1 14/35

  13. The spectral function After carrying out the same operation for t < t ′ and Fourier transform f s ( x ) f ∗ s ( x ′ ) � G ( x , x ′ , ω ) = ∓ occ/unocc � ω − ε s ∓ i 0 + s Giustino, Lecture Wed.1 15/35

  14. The spectral function After carrying out the same operation for t < t ′ and Fourier transform f s ( x ) f ∗ s ( x ′ ) � G ( x , x ′ , ω ) = ∓ occ/unocc � ω − ε s ∓ i 0 + s The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system Giustino, Lecture Wed.1 15/35

  15. The spectral function After carrying out the same operation for t < t ′ and Fourier transform f s ( x ) f ∗ s ( x ′ ) � G ( x , x ′ , ω ) = ∓ occ/unocc � ω − ε s ∓ i 0 + s The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system From the Green’s function we can obtain the spectral (density) function A ( x , ω ) = 1 � s | f s ( x ) | 2 δ ( � ω − ε s ) π | Im G ( x , x , ω ) | = Giustino, Lecture Wed.1 15/35

  16. The spectral function After carrying out the same operation for t < t ′ and Fourier transform f s ( x ) f ∗ s ( x ′ ) � G ( x , x ′ , ω ) = ∓ occ/unocc � ω − ε s ∓ i 0 + s The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system From the Green’s function we can obtain the spectral (density) function A ( x , ω ) = 1 � s | f s ( x ) | 2 δ ( � ω − ε s ) π | Im G ( x , x , ω ) | = The spectra function is the many-body (local) density of states Giustino, Lecture Wed.1 15/35

  17. The spectral function After carrying out the same operation for t < t ′ and Fourier transform f s ( x ) f ∗ s ( x ′ ) � G ( x , x ′ , ω ) = ∓ occ/unocc � ω − ε s ∓ i 0 + s The poles of the Green’s function represent the electron addition/removal energies of the interacting many-body system From the Green’s function we can obtain the spectral (density) function A ( x , ω ) = 1 � s | f s ( x ) | 2 δ ( � ω − ε s ) π | Im G ( x , x , ω ) | = The spectra function is the many-body (local) density of states • Usually it is presented as momentum-resolved A ( k , ω ) Giustino, Lecture Wed.1 15/35

  18. The spectral function Example: a single complex pole ε s = ε − i Γ Giustino, Lecture Wed.1 16/35

  19. The spectral function Example: a single complex pole ε s = ε − i Γ G ( x , x , t − t ′ ) = − i � | f s ( x ) | 2 e − iε ( t − t ′ ) / � e − Γ( t − t ′ ) / � Giustino, Lecture Wed.1 16/35

  20. The spectral function Example: a single complex pole ε s = ε − i Γ G ( x , x , t − t ′ ) = − i � | f s ( x ) | 2 e − iε ( t − t ′ ) / � e − Γ( t − t ′ ) / � | G ( x , x , t − t ′ ) | = 1 � | f s ( x ) | 2 e − Γ( t − t ′ ) / � Giustino, Lecture Wed.1 16/35

  21. The spectral function Example: a single complex pole ε s = ε − i Γ G ( x , x , t − t ′ ) = − i � | f s ( x ) | 2 e − iε ( t − t ′ ) / � e − Γ( t − t ′ ) / � | G ( x , x , t − t ′ ) | = 1 � | f s ( x ) | 2 e − Γ( t − t ′ ) / � � �� � decay Giustino, Lecture Wed.1 16/35

  22. The spectral function Example: a single complex pole ε s = ε − i Γ G ( x , x , t − t ′ ) = − i � | f s ( x ) | 2 e − iε ( t − t ′ ) / � e − Γ( t − t ′ ) / � | G ( x , x , t − t ′ ) | = 1 � | f s ( x ) | 2 e − Γ( t − t ′ ) / � � �� � decay A ( x , x , ω ) = 1 Γ ( � ω − ε ) 2 + Γ 2 | f s ( x ) | 2 π Giustino, Lecture Wed.1 16/35

  23. The spectral function Example: a single complex pole ε s = ε − i Γ G ( x , x , t − t ′ ) = − i � | f s ( x ) | 2 e − iε ( t − t ′ ) / � e − Γ( t − t ′ ) / � | G ( x , x , t − t ′ ) | = 1 � | f s ( x ) | 2 e − Γ( t − t ′ ) / � ε � �� � A ( k, ω ) decay A ( x , x , ω ) = 1 Γ ( � ω − ε ) 2 + Γ 2 | f s ( x ) | 2 π Γ k � ω Giustino, Lecture Wed.1 16/35

  24. The spectral function A ( k , ω ) = 1 π | Im G ( k , ω ) | Giustino, Lecture Wed.1 17/35

  25. The spectral function A ( k , ω ) = 1 π | Im G ( k , ω ) | DFT density of states energy Giustino, Lecture Wed.1 17/35

  26. The spectral function A ( k , ω ) = 1 π | Im G ( k , ω ) | DFT density of states many-body DOS energy Giustino, Lecture Wed.1 17/35

  27. The spectral function A ( k , ω ) = 1 π | Im G ( k , ω ) | DFT density of states quasiparticle shift many-body DOS energy Giustino, Lecture Wed.1 17/35

  28. The spectral function A ( k , ω ) = 1 π | Im G ( k , ω ) | DFT density of states quasiparticle shift quasiparticle broadening many-body DOS energy Giustino, Lecture Wed.1 17/35

  29. The spectral function A ( k , ω ) = 1 π | Im G ( k , ω ) | DFT density of states quasiparticle shift boson energy quasiparticle broadening many-body DOS energy Giustino, Lecture Wed.1 17/35

  30. How to calculate the Green’s function Equation of motion for field operators i � ∂ � � ˆ ψ ( x , t ) , ˆ ˆ ψ ( x t ) = H ∂t Giustino, Lecture Wed.1 18/35

  31. How to calculate the Green’s function Equation of motion for field operators � � � − � 2 i � ∂ � � ∇ 2 + ˆ ψ ( x , t ) , ˆ ˆ ˆ d r ′ v ( r , r ′ ) ˆ n ( r ′ t ) ψ ( x t ) = H = ψ ( x t ) ∂t 2 m e total charge, electrons & nuclei Giustino, Lecture Wed.1 18/35

  32. How to calculate the Green’s function Equation of motion for field operators � � � − � 2 i � ∂ � � ∇ 2 + ˆ ψ ( x , t ) , ˆ ˆ ˆ d r ′ v ( r , r ′ ) ˆ n ( r ′ t ) ψ ( x t ) = H = ψ ( x t ) ∂t 2 m e total charge, electrons & nuclei � � � − � 2 i � ∂ ˆ ˆ ∇ 2 ψ (1) = 1 + d 2 v (12) ˆ n (2) ψ (1) ∂t 1 2 m e Giustino, Lecture Wed.1 18/35

  33. How to calculate the Green’s function Equation of motion for field operators � � � − � 2 i � ∂ � � ∇ 2 + ˆ ψ ( x , t ) , ˆ ˆ ˆ d r ′ v ( r , r ′ ) ˆ n ( r ′ t ) ψ ( x t ) = H = ψ ( x t ) ∂t 2 m e total charge, electrons & nuclei � � � − � 2 i � ∂ ˆ ˆ ∇ 2 ψ (1) = 1 + d 2 v (12) ˆ n (2) ψ (1) ∂t 1 2 m e Equation of motion for Green’s function � � � + � 2 i � ∂ G (12) + i d 3 v (13) � ˆ ∇ 2 n (3) ψ (1) ψ † (2) � = δ (12) T ˆ 1 ∂t 1 2 m e � Giustino, Lecture Wed.1 18/35

  34. How to calculate the Green’s function Equation of motion for field operators � � � − � 2 i � ∂ � � ∇ 2 + ˆ ψ ( x , t ) , ˆ ˆ ˆ d r ′ v ( r , r ′ ) ˆ n ( r ′ t ) ψ ( x t ) = H = ψ ( x t ) ∂t 2 m e total charge, electrons & nuclei � � � − � 2 i � ∂ ˆ ˆ ∇ 2 ψ (1) = 1 + d 2 v (12) ˆ n (2) ψ (1) ∂t 1 2 m e Equation of motion for Green’s function � � � + � 2 i � ∂ G (12) + i d 3 v (13) � ˆ ∇ 2 n (3) ψ (1) ψ † (2) � = δ (12) T ˆ 1 ∂t 1 2 m e � 4 field operators → 2-particle Green’s function � ˆ Tψ † (3) ψ (3) ψ (1) ψ † (2) � = [Hartree] + [Fock] + G 2 (31 , 32) Giustino, Lecture Wed.1 18/35

  35. How to calculate the Green’s function � � � + � 2 i � ∂ ∇ 2 1 − V tot (1) G (12) − d 3 Σ(13) G (32) = δ (12) ∂t 1 2 m e � V tot (1) = d 2 v (12) � ˆ n (2) � rewrite 2-particle Green’s function using self-energy Σ Giustino, Lecture Wed.1 19/35

  36. How to calculate the Green’s function � � � + � 2 i � ∂ ∇ 2 1 − V tot (1) G (12) − d 3 Σ(13) G (32) = δ (12) ∂t 1 2 m e � V tot (1) = d 2 v (12) � ˆ n (2) � rewrite 2-particle Green’s function using self-energy Σ Express the Green’s function in terms of Dyson’s orbitals � � � − � 2 ∇ 2 + V tot ( r ) d x ′ Σ( x , x ′ , ε s / � ) f s ( x ′ ) = ε s f s ( x ) f s ( x ) + 2 m e Giustino, Lecture Wed.1 19/35

  37. How to calculate the Green’s function � � � + � 2 i � ∂ ∇ 2 1 − V tot (1) G (12) − d 3 Σ(13) G (32) = δ (12) ∂t 1 2 m e � V tot (1) = d 2 v (12) � ˆ n (2) � rewrite 2-particle Green’s function using self-energy Σ Express the Green’s function in terms of Dyson’s orbitals � � � − � 2 ∇ 2 + V tot ( r ) d x ′ Σ( x , x ′ , ε s / � ) f s ( x ′ ) = ε s f s ( x ) f s ( x ) + 2 m e Sources of electron-phonon interaction Giustino, Lecture Wed.1 19/35

  38. How to calculate the Green’s function � � � + � 2 i � ∂ ∇ 2 1 − V tot (1) G (12) − d 3 Σ(13) G (32) = δ (12) ∂t 1 2 m e � V tot (1) = d 2 v (12) � ˆ n (2) � rewrite 2-particle Green’s function using self-energy Σ Express the Green’s function in terms of Dyson’s orbitals � � � − � 2 ∇ 2 + V tot ( r ) d x ′ Σ( x , x ′ , ε s / � ) f s ( x ′ ) = ε s f s ( x ) f s ( x ) + 2 m e Sources of electron-phonon interaction Giustino, Lecture Wed.1 19/35

  39. How to calculate the self-energy Electron self-energy from Hedin-Baym’s equations � d (34) G (13) Γ(324) W (41 + ) Σ(12) = i � Giustino, Lecture Wed.1 20/35

  40. How to calculate the self-energy Electron self-energy from Hedin-Baym’s equations � d (34) G (13) Γ(324) W (41 + ) Σ(12) = i � Green’s function Giustino, Lecture Wed.1 20/35

  41. How to calculate the self-energy Electron self-energy from Hedin-Baym’s equations � d (34) G (13) Γ(324) W (41 + ) Σ(12) = i � Green’s function Vertex Giustino, Lecture Wed.1 20/35

  42. How to calculate the self-energy Electron self-energy from Hedin-Baym’s equations � d (34) G (13) Γ(324) W (41 + ) Σ(12) = i � Green’s function Vertex Screened Coulomb interaction Giustino, Lecture Wed.1 20/35

  43. How to calculate the self-energy Electron self-energy from Hedin-Baym’s equations � d (34) G (13) Γ(324) W (41 + ) Σ(12) = i � Green’s function Vertex Screened Coulomb interaction W = W e + W ph � d 3 ǫ − 1 W e (12) = e (13) v (32) Giustino, Lecture Wed.1 20/35

  44. How to calculate the self-energy Electron self-energy from Hedin-Baym’s equations � d (34) G (13) Γ(324) W (41 + ) Σ(12) = i � Green’s function Vertex Screened Coulomb interaction W = W e + W ph � d 3 ǫ − 1 W e (12) = e (13) v (32) Basically the standard GW method + screening from nuclei Giustino, Lecture Wed.1 20/35

  45. How to calculate the self-energy Screened Coulomb interaction from the nuclei � e (13) ∂V κ ( r 3 ) e (24) ∂V κ ′ ( r 4 ) � d (34) ǫ − 1 · D κκ ′ ( t 3 t 4 ) · ǫ − 1 W ph (12) = ∂ τ κ ∂ τ κ ′ κκ ′ Giustino, Lecture Wed.1 21/35

  46. How to calculate the self-energy Screened Coulomb interaction from the nuclei “electron-phonon matrix elements” � e (13) ∂V κ ( r 3 ) e (24) ∂V κ ′ ( r 4 ) � d (34) ǫ − 1 · D κκ ′ ( t 3 t 4 ) · ǫ − 1 W ph (12) = ∂ τ κ ∂ τ κ ′ κκ ′ Giustino, Lecture Wed.1 21/35

  47. How to calculate the self-energy Screened Coulomb interaction from the nuclei “electron-phonon matrix elements” � e (13) ∂V κ ( r 3 ) e (24) ∂V κ ′ ( r 4 ) � d (34) ǫ − 1 · D κκ ′ ( t 3 t 4 ) · ǫ − 1 W ph (12) = ∂ τ κ ∂ τ κ ′ κκ ′ Displacement-displacement correlation function of the nuclei, a.k.a. the phonon Green’s function D κκ ′ ( tt ′ ) = − i � � ˆ τ T κ ′ ( t ′ ) � T ∆ˆ τ κ ( t ) ∆ˆ Giustino, Lecture Wed.1 21/35

  48. Diagrammatic representation of the self-energy Standard GW self-energy (we will ignore this from now on) Figure from Giustino, Rev. Mod. Phys. 89, 015003 (2017) Giustino, Lecture Wed.1 22/35

  49. Diagrammatic representation of the self-energy Standard GW self-energy (we will ignore this from now on) Fan-Migdal self-energy Figure from Giustino, Rev. Mod. Phys. 89, 015003 (2017) Giustino, Lecture Wed.1 22/35

  50. Diagrammatic representation of the self-energy Standard GW self-energy (we will ignore this from now on) Fan-Migdal self-energy Debye-Waller self-energy Figure from Giustino, ( Lecture Thu.2 ) Rev. Mod. Phys. 89, Improper self-energy: comes form 015003 (2017) � V tot (1) = d 2 v (12) � ˆ n (2) � term Giustino, Lecture Wed.1 22/35

  51. Diagrammatic representation of the self-energy Standard GW self-energy (we will ignore this from now on) Fan-Migdal self-energy Debye-Waller self-energy Figure from Giustino, ( Lecture Thu.2 ) Rev. Mod. Phys. 89, Improper self-energy: comes form 015003 (2017) � V tot (1) = d 2 v (12) � ˆ n (2) � term Giustino, Lecture Wed.1 22/35

  52. Fan-Migdal self-energy Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons � d q n k ( ω ) = 1 � | g mnν ( k , q ) | 2 Σ FM Ω BZ � mν � � 1 − f m k + q f m k + q × ω − ε m k + q / � − ω q ν + iη + ω − ε m k + q / � + ω q ν + iη Giustino, Lecture Wed.1 23/35

  53. Fan-Migdal self-energy Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons � d q n k ( ω ) = 1 � | g mnν ( k , q ) | 2 Σ FM Ω BZ � mν � � 1 − f m k + q f m k + q × ω − ε m k + q / � − ω q ν + iη + ω − ε m k + q / � + ω q ν + iη Dynamical structure on the scale of the phonon energy Giustino, Lecture Wed.1 23/35

  54. Fan-Migdal self-energy Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons Summation over all phonon branches and wavevectors � d q n k ( ω ) = 1 � | g mnν ( k , q ) | 2 Σ FM Ω BZ � mν � � 1 − f m k + q f m k + q × ω − ε m k + q / � − ω q ν + iη + ω − ε m k + q / � + ω q ν + iη Dynamical structure on the scale of the phonon energy Giustino, Lecture Wed.1 23/35

  55. Fan-Migdal self-energy Fan-Migdal self-energy using Kohn-Sham states and DFPT phonons Summation over all phonon branches and wavevectors � d q n k ( ω ) = 1 � | g mnν ( k , q ) | 2 Extension to Σ FM Ω BZ � finite temperature mν � � 1 − f m k + q + n q ν f m k + q + n q ν × ω − ε m k + q / � − ω q ν + iη + ω − ε m k + q / � + ω q ν + iη Dynamical structure on the scale of the phonon energy Giustino, Lecture Wed.1 23/35

  56. Fan-Migdal self-energy ε F Example: A single dispersionless phonon (Holstein model) Giustino, Lecture Wed.1 24/35

  57. Fan-Migdal self-energy ε F Example: A single dispersionless phonon (Holstein model) Energy y g r e n E Wavevector Wavevector Giustino, Lecture Wed.1 24/35

  58. Fan-Migdal self-energy ε F Example: A single dispersionless phonon (Holstein model) Energy y g r e n E Wavevector Wavevector Giustino, Lecture Wed.1 24/35

  59. Fan-Migdal self-energy ε F Example: A single dispersionless phonon (Holstein model) Energy y g r e n E Wavevector Wavevector Giustino, Lecture Wed.1 24/35

  60. Fan-Migdal self-energy ε F Example: A single dispersionless phonon (Holstein model) phonon energy Energy y g r e n E Wavevector Wavevector Giustino, Lecture Wed.1 24/35

  61. Fan-Migdal self-energy ε F Example: A single dispersionless phonon (Holstein model) change of velocity/mass phonon energy Energy y broadening g r e n E Wavevector Wavevector Giustino, Lecture Wed.1 24/35

  62. Examples from experiments • Velocity renormalization in MgB 2 v = v 0 / 2 . 4 Right figure from Mou et al, Phys. Rev. B 91, 140502(R) (2015) Giustino, Lecture Wed.1 25/35

  63. Examples from experiments • Velocity renormalization in Ca-decorated graphene on Au v = v 0 / 1 . 25 Right figure adapted from Fedorov et al, Nat. Commun. 5, 3257 (2014) Giustino, Lecture Wed.1 26/35

  64. Examples from calculations • Velocity renormalization in C 6 CaC 6 (EPW) Figure adapted from Margine et al, Sci Rep. 6, 21414 (2016) Giustino, Lecture Wed.1 27/35

  65. Examples from calculations • Velocity renormalization and broadening in MgB 2 Figure from Eiguren et al, Phys. Rev. B 79. 245103 (2009) Giustino, Lecture Wed.1 28/35

  66. Quasiparticle shift and broadening Spectral function from the self-energy A ( k , ω ) = − 1 1 � π Im � ω − ε n k − Σ n k ( ω ) n Giustino, Lecture Wed.1 29/35

  67. Quasiparticle shift and broadening Spectral function from the self-energy A ( k , ω ) = − 1 1 � π Im � ω − ε n k − Σ n k ( ω ) n Quasiparticle approximation: assume Lorentzian peaks centered near � ω = E n k � Σ n k ( ω ) = Σ n k ( E n k ) + 1 ∂ ReΣ n k � ( � ω − E n k ) + · · · � ∂ω � � ω = E n k / � Giustino, Lecture Wed.1 29/35

  68. Quasiparticle shift and broadening Spectral function from the self-energy A ( k , ω ) = − 1 1 � π Im � ω − ε n k − Σ n k ( ω ) n Quasiparticle approximation: assume Lorentzian peaks centered near � ω = E n k � Σ n k ( ω ) = Σ n k ( E n k ) + 1 ∂ ReΣ n k � ( � ω − E n k ) + · · · � ∂ω � � ω = E n k / � Define the quasiparticle strength � � − 1 � 1 − 1 ∂ ReΣ n k ( ω ) � Z n k = � � ∂ω � ω = E n k / � Giustino, Lecture Wed.1 29/35

  69. Quasiparticle shift and broadening Replace the Taylor expansion inside the spectral function A ( k , ω ) = − 1 1 � π � ω − ε n k − Σ n k ( E n k ) − (1 − 1 /Z n k )( � ω − E n k ) n Giustino, Lecture Wed.1 30/35

  70. Quasiparticle shift and broadening Replace the Taylor expansion inside the spectral function A ( k , ω ) = − 1 1 � π � ω − ε n k − Σ n k ( E n k ) − (1 − 1 /Z n k )( � ω − E n k ) n After rearranging ( ∗ ) : A ( k , ω ) = − 1 Z n k � π � ω − ( E n k + i Γ n k ) n ( ∗ ) Requires the additional approximation | ∂ ImΣ n k /∂ω | ≪ | ∂ ReΣ n k /∂ω | Giustino, Lecture Wed.1 30/35

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