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# Theoretical Spectroscopy Lectures: real-time approach 2

In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.

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### Theoretical Spectroscopy Lectures: real-time approach 2

1. 1.  Real­time Spectroscopy for solids Claudio Attaccalite  CNRS/CINaM, Aix­Marseille Universite (FR)  Theoretical Spectroscopy  Lectures October 8­12, 2018,  Lausanne
2. 2. Outline ● Polarization and Berry-phase ● Advanced Hamiltonians ● Density Functional Polarization Theory
3. 3. A bit of theory Berry's phase and non-linear response
4. 4. The Berry phase IgNobel Prize (2000) together with A.K. Geim for flying frogs A generic quantum Hamiltonian with a parametric  dependence … phase difference between two ground eigenstates  at two different x cannot have any physical meaning Berry, M. V. . Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802), 45-57 (1984).
5. 5. ...connecting the dots... the phase difference of a closed-path is gauge-invariant therefore is a potential physical observable g is an “exotic” observable which cannot be expressed in terms of any Hermitian operator
6. 6. Berry's geometric phase Berry's Phase and Geometric Quantum Distance: Macroscopic Polarization and Electron Localization R. Resta, http://www.freescience.info/go.php?pagename=books&id=1437 −i Δ ϕ≃〈ψ(x)∣∇x ψ(x)〉⋅Δ x g=∑s=1 M Δ ϕs, s+1→∫C i〈 ψ(x)∣∇x ψ(x)〉 d x Berry's connection ● Berry's phase exists because the system is not isolated x is a kind of coupling with the “rest of the Universe” ● In a truly isolated system, there can be no manifestation of a Berry's phase
7. 7. Examples of Berry's phases Molecular AB effectAharonov-Bohm effect Correction to the Wannier-Stark ladder spectra of semiclassical electrons Ph. Dugourd et al. Chem. Phys. Lett. 225, 28 (1994) R.G. Sadygov and D.R. Yarkony J. Chem. Phys. 110, 3639 (1999) J. Zak, Phys. Rev. Lett. 20, 1477 (1968) J. Zak, Phys. Rev. Lett. 62, 2747 (1989)
8. 8. The  problem of bulk polarization ● How to define polarization as a bulk quantity? ● Polarization for isolated systems is well defined P= 〈 R〉 V = 1 V ∫d r n(r)= 1 V 〈 Ψ∣̂R∣Ψ〉
9. 9. Bulk polarization, the wrong way 1  1) P= 〈R〉sample V sample
10. 10. Bulk polarization, the wrong way 2  2) P= 〈 R〉cell Vcell Unfortunately Clausius-Mossotti does not work for solids because WF are delocalized
11. 11. Bulk polarization, the wrong way 3  3) P∝∑n, mk 〈ψn k∣r∣ψm k〉 〈ψnk∣r∣ψm k〉 ● intra-bands terms undefined ● diverges close to the bands crossing ● ill-defined for degenerates states
12. 12. Electrons in a periodic system ϕnk (r +R)=e ik R unk (r) Born-von-Karman boundary conditions [ 1 2m p2 +V (r) ]ϕn k(r)=ϵn(k)ϕn k(r) Bloch orbitals solution of a mean-field Schrödinger eq. ϕn k(r+R)=e ik r unk(r) Bloch functions u obeys to periodic boundary conditions [ 1 2m (p+ℏk)2 +V (r)]un k(r)=ϵn(k)unk (r) We map the problem in k-dependent Hamiltonian and k-independent boundary conditions k plays the role of an external parameter
13. 13. What is the Berry's phase related to k? King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) Pα= 2ie (2π) 3 ∫BZ d k∑n=1 nb 〈un k∣ ∂ ∂ kα ∣unk 〉 Berry's connection again!!
14. 14. King­Smith and Vanderbilt formula Pα= −ef 2πv aα Nkα ⊥ ∑kα ⊥ ℑ∑i Nk α −1 tr ln S(ki , ki+qα ) .. discretized King-Smith and Vanderbilt formula.... Phys. Rev. B 47, 1651 (1993) An exact formulation exists also for correlated wave-functions R. Resta., Phys. Rev. Lett. 80, 1800 (1998)
15. 15. From Polarization to the  Equations of Motion L= i ℏ N ∑n=1 M ∑k 〈 vkn∣˙vkn〉−E 0 −v Ε⋅P i ℏ ∂ ∂t ∣vk n〉=Hk 0 ∣vk n〉+i e Ε⋅∣∂k vk n〉 It is an object difficult to calculate numerically due to the gauge freedom of the Bloch functions ∣vk m〉→∑n occ Uk ,nm∣vkn 〉 I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004) r=i∂k
16. 16. How to perform k-derivatives? Solutions: 1) In mathematics the problem has been solved by using second, third,... etc derivatives SIAM, J. on Matrix. Anal. and Appl. 20, 78 (1998) 2) Global-gauge transformation Phys. Rev. B 76, 035213 (2007) 3) Phase optimization Phys. Rev. B 77, 045102 (2008) 4) Covariant derivative Phys. Rev. B 69, 085106 (2004) M (k )vk =λ(k)vk
17. 17. Hamiltonians
18. 18. GW corrections HGW =H KS+∑i Δϵi∣ψi⟩ ⟨ψi∣ GW correction are included in the Hamiltonian  as a non­local operator Contrary to the linear­response case even a simple scissor correction  does not correspond to a rigid  shift in non­linear response!!! AlAs Δ ϵi=ϵi GW −ϵi KS This operator modifies the  eigenvalues only
19. 19. Bethe-Salpeter like response 1 Excitonic effects in the BSE equation are generated  by the variation of the Screened­Exchange (SEX) TD-Hartree
20. 20. Bethe-Salpeter like response 2 Excitonic effects in the BSE equation are generated  by the variation of the Screened­Exchange (SEX) + -= Σ COHSEX [ ^ρ(t =0)−^ρ(t)]∂ Σ COHSEX ∂V ext Mapping BSE in an effective Hamiltonian
21. 21. Bethe-Salpeter like response 3 If you want to include excitonic effect in an Hamiltonian, you just have to include the exchange (like TD­HF). More precisely to be equivalent to the BSE we include the out­equilibrium SEX self­energy. HBSE =HGW +ΣSEX (Δ ^ρ) ΣSEX (Δ ^ρ)i, j=i ∑G, G' , n,n' WG ,G' (ω=0)Δ ^ρn,n' At equilibrium:  Δ ^ρn, n'=δn,n' f (ϵn) ● Non-linear response in extended systems: a real-time approach Claudio Attaccalite, https://arxiv.org/abs/1609.09639 ● Optical absorption spectra of finite systems from a conserving Bethe-Salpeter equation approach. G. Pal et al. EPJ-B (79), 327, (2011)
22. 22. From real­time to linear optics i ∂ ψl ∂t =[ H +U (t)] ψl(t ) l=1, N v with Build density matrix: ^ρi, j (t)=∑l ⟨ϕi|ψl(t )⟩⟨ ψl(t )|ϕj ⟩ Expansion in terms of U: ^ρ(U )ij= ^ρij (0)+ ∂ ^ρij ∂U m, n Um, n+O(U 2 ) i ∂ ^ρ ∂ t =[ H +U (t), ^ρ]Von Neumann eq.: χ=χ 0 +χ 0 Ξ χ Where  depends from the Hamiltonian  you choose at the beginning χi, j, m,n= ∂ ^ρij ∂Um ,n Use EOM of    to find an  equation for  ^ρ
23. 23. Local-fields and excitonic effects in h-BN monolayer IPA independent particles +time-dependent Hartree (RPA) H=
24. 24. Local-fields and excitonic effects in h-BN monolayer IPA IPA + GW independent particles +quasi-particle corrections +time-dependent Hartree (RPA) H=
25. 25. Local-fields and excitonic effects in h-BN monolayer IPA IPA + GW IPA + GW + TDSHF independent particles +quasi-particle corrections +time-dependent Hartree (RPA) +screend Hartree-Fock (excitonic effects) H=
26. 26. When the density is not enough Density functional polarization theory
27. 27. What about TD-DFT? TD­DFT is much faster than Green's functions ...but is not so good for solids ….. Why?
28. 28. Runge-Gross theorem does not hold in periodic system (PRB 68, 045109) Current-density functional theory of the response of solids Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68, 045109(2003) charge densityexternal potential current This part of the RG does not hold jq (ω)=ω nq(ω) q
29. 29. A simple example L Independent Electrons in 1D Current-density functional theory of the response of solids Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68, 045109(2003) ϕm(x ,0)= e i2π m x/L √(L) At equilibrium the solution is H (t )= ^p2 2
30. 30. L Independent Electrons in 1D ϕm(x ,0)= e i2π m x/L √L At equilibrium the solution is Turn on a constant electric field: A=−cεt E= −1 c ∂ A ∂t H (t )= ( ^p−εt) 2 2 ϕm(x ,t)= e −ikm t/2−km εt 2 /2+ε 2 t 3 /6 e i2π m x/L √L Current-density functional theory of the response of solids Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68, 045109(2003) A simple example km=2 πm/ L
31. 31. L Independent Electrons in 1D ϕm(x ,0)= e 2π m x/L √L At equilibrium the solution is Turn on a constant electric field: A=−cεt E= −1 c ∂ A ∂t H (t )= ( ^p−εt) 2 2 ϕm(x ,t)= e−ikmt/2−km εt 2 /2+ε 2 t 3 /6 e2πm x/ L √L n(r ,t)=|ϕm( x,t) 2 |=n(r ,0) The density does not change!!!!! Current-density functional theory of the response of solids Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68, 045109(2003) A simple example … but the current does!
32. 32. When the density is not enough Current­density­Funtional Theory works :­) If there are not magnetic fields and transverse  reponse is not important it reduces to Density­Polarization­FT Exc (P ,n)
33. 33. Different functional are  implemented in Lumen  in the form Exc (t )=f xc (n)+α P (t )+β P 2 (t )+ g P 3 (t)+.... For χ2, χ3... For χ3,χ4,....For linear optics   has been derived from LRC, Jellium  with gap, current­density etc.. Dielectrics in a time-dependent electric field: A real-time approach based on density-polarization functional theory M. Grüning, D. Sangalli, and C. Attaccalite, Phys. Rev. B 94, 035149
34. 34. Some result for the functional IPA IPA + GW JGM opt-PF Exc (t )=f xc (n)+α P (t )
35. 35. MoS2 single-layer 10) Second harmonic microscopy of monolayer MoS2 N. Kumar et al. Phys. Rev. B 87, 161403(R) (2013) 100) Observation of intense second harmonic generation from MoS2 atomic crystals L. Malard et al. Phys. Rev. B 87, 201401(R) (2013) 1000) Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation Y. Li et al. NanoLetters, 13, 3329 (2013)
36. 36. Two Photon Absorption  coefficients  Richardson extrapolation P(ω)=χ (1) (ω) E(ω)+χ (3) (ω;ω ,−ω ,ω) E(ω)E(−ω) E(ω)
37. 37. Two-photons absorption in hexagonal boron nitride C. Attaccalite et al., arXiv preprint arXiv:1803.10959 Bulk h­BN  Two­photon absorption in hBN Hexagonal boron nitride is an indirect band­gap  semiconductor G. Cassabois et al.,  Nature Photonics, 10, 262 (2016)
38. 38. ● Non-linear response in extended systems: a real-time approach Claudio Attaccalite, https://arxiv.org/abs/1609.09639 ● Geometry and Topology in Electronic Structure Theory Reffaele Resta, http://www-dft.ts.infn.it/~resta/gtse/draft.pdf ● TD-DFT on Youtube by N. Maitra https://www.youtube.com/watch?v=ItWonHYm-ls
39. 39. Something More
40. 40. Something More flying frogs
41. 41. The King-Smith and Vanderbilt formula We introduce the Wannier functions Blount, 1962 We express the density in terms of Wannier functions Polarization in terms of Wannier functions [Blount 62]
42. 42. King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993) Pα= 2ie (2π) 3 ∫BZ d k∑n=1 nb 〈un k∣ ∂ ∂ kα ∣unk 〉 The idea of Chen, Lee, Resta..... Berry's phase and Green's functions Z. Wang et al. PRL 105, 256803 (2010) Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011) R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf
43. 43. Electron gas with gap (JGM) P. Trevisanutto et al.  PRB 87, 205143(2013) Dielectric constant Corresponding exchange­corr functional (with a Micro and Macroscopic contribution) We map the Macroscopic part in a  functional of the polarization and the  microscopic part in a functional of the  density N. T. Maitra, I. Souza, and K. Burke PRB 68, 045109
44. 44. What we miss?  Exc (t )=f xc (n)+α P (t )+β P 2 (t )+ g P 3 (t)+.... For linear optics For χ2, χ3... For χ3,χ4,.... It can be shown that second term is as large as  local field effects(PRB 54, 8540)   First problem Second problem α(t) In principle all the coefficients depend on time. This is important to describe complex excitons, etc...    β(t) g(t)

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