1. Alessandro Toschi ERC -workshop „Ab-initio DΓΑ“ Baumschlagerberg, 3 September 2013 „Introduction to the two-particle vertex functions and to the dynamical vertex approximation“ „Introduction to the two-particle vertex functions and to the dynamical vertex approximation“

2. I ) Non-local correlations beyond DMFT  overview of the extensions of DMFT  Focus: diagrammatic extensions (based on the 2P-local vertex) Outlook II) Local vertex functions:  general formalisms  numerical results/physical interpretation III) Dynamical Vertex Approximations (D Γ A):  basics of D Γ A  D Γ A results: (i) spectral function & critical regime of bulk 3d-systems (ii) nanoscopic system (  talk A. Valli)

3. Electronic correlation in solids - J multi-orbital Hubbard model Local part only! U Simplest version: single-band Hubbard hamliltonian:

4. No : spatial correlations Yes : local temporal correlations W. Metzner & D. Vollhardt, PRL (1989) A. Georges & G. Kotliar, PRB (1992) the Dynamical Mean Field Theory Σ(ω)Σ(ω) h eff (t) - J Σ(ω)Σ(ω) U non-perturbative in U, BUT purely local self-consistent SIAM „There are more things in Heaven and Earth, than those described by DMFT“ [W. Shakespeare, readapted by AT ]

5. (exact in d = ∞ ) h eff (t) DMFT DMFT applicability: ✔ high connectivity/dimensions  low dimensions (layered-, surface-, nanosystems)  phase-transitions ( ξ  ∞, criticality) Instead: DMFT not enough [  spatial correlations are crucial] ✔ high temperatures U!! ξ

6. Beyond DMFT: several routes 1.Cellular-DMFT (C-DMFT: cluster in real space) 2.Dynamical Cluster Approx. (DCA: cluster in k-space) ★ cluster extensions [ ⌘ Kotliar et al. PRL 2001; Huscroft, Jarrell et al. PRL 2001] ij()ij() ★ high-dimensional (o(1/d)) expansion [ ⌘ Schiller & Ingersent, PRL 1995] (1/d: mathematically elegant, BUT very small corrections ) ★ a complementary route: diagrammatic extensions (C-DMFT, DCA : systematic approach, BUT only “short” range correlation included )

7. Diagrammatic extensions of DMFT ★ Dual Fermion [ ⌘ Rubtsov, Lichtenstein et al., PRB 2008] (DF: Hubbard-Stratonovic for the non-local degrees of freedom & perturbative/ladder expansion in the Dual Fermion space ) ★ Dynamical Vertex Approximation [ ⌘ AT, Katanin, Held, PRB 2007] (D Γ A: ladder/parquet calculations with a local 2P-vertex [ Γ ir ] input from DMFT ) ★ 1Particle Irreducible approach [ ⌘ Rohringer, AT et al., PRB (2013), in press ] (1PI: ladder calculations of diagram generated by the 1PI-functional ) [  talk by Georg Rohringer] all these methods require Local two-particle vertex functions as input ! ★ DMF 2 RG [ ⌘ Taranto, et al., arXiv 1307.3475 ] (D MF 2 RG : combination of DMFT & fRG ) [  talk by Ciro Taranto]

8. 2P- vertex: Who’s that guy? To a certain extent: 2P-analogon of the one-particle self-energy In the following:  How to extract the 2P-vertex (from the 2P-Greens‘ function)  How to classify the vertex functions (2P-irreducibility)  Frequency dependence of the local vertex of DMFT   1 particle in – 1 particle out U Dyson equations: G (1) (ν)   Σ ( ν ) vertex  2 particle in – 2 particle out U BSE, parquet : G (2)   vertex Year: 1987; Source: Wikipedia

9. How to extract the vertex functions? 2P-Green‘s function: 2P-vertex functions:  numerically demanding, but computable, for AIM (single band: ED still possible; general multi-band case: CTQMC, work in progress) =+ F + Full vertex (scattering amplitude) What about 2P-irreducibility? = Γ (fRG notation) = γ 4 (DF notation)

10. Decomposition of the full vertex F 1) parquet equation: 2) Bethe-Salpeter equation (BS eq.) : Γ ph e.g., in the ph transverse ( ph ) channel: F = Γ ph + Φ ph

11. Types of approximations: 2P- irreducibility *) LOWEST ORDER (STATIC) APPROXIMATION: U

12. F ν + ω ν ‘ + ω ν‘ν‘ν Dynamic structure of the vertex: DMFT results = F( ν, ν ‘, ω ) spin sectors: density/charge magnetic/spin background = 0 intermediate coupling (U ~ W/2) (2n+1)π/β (2n‘+1)π/β for the vertex asymptotics : see also J. Kunes, PRB (2011)

13. full vertex F Frequency dependence: an overview irreducible vertex Γ fully irreducible vertex Λ background and main diagonal (ν=ν‘) ≈ U 2 χ m (0)  ∞ at the MIT No-high frequency problem ( Λ  U) BUT low-energy divergencies

14. full vertex F Frequency dependence: an overview irreducible vertex Γ fully irreducible vertex Λ background and main diagonal (ν=ν‘) ≈ U 2 χ m (0)  ∞ at the MIT No-high frequency problem ( Λ  U) BUT low-energy divergencies MIT [  talk by Thomas Schäfer ] Γ d & Λ   ∞ singularity line ⌘ T. Schäfer, G. Rohringer, O. Gunnarsson, S. Ciuchi, G. Sangiovanni, AT, PRL (2013)

15. Types of approximations: 2P- irreducibility *) DIAGRAMMATIC EXTENSIONS OF DMFT: dynamical local vertices F( ν, ν ‘, ω ) Γ ( ν, ν ‘, ω ) Λ ( ν, ν ‘, ω )  Dual Fermion, 1PI approach, DMF 2 RG Dynamical Vertex Approx. (DΓA) methods based on F methods based on Γ c, Λ more direct calculation Locality of F? Locality of Γ C, Λ inversion of BS eq. or parquet needed

16.  DMFT: all 1-particle irreducible diagrams (=self-energy) are LOCAL !!  DΓA : all 2 -particle irreducible diagrams (=vertices) are LOCAL !! the self-energy becomes NON-LOCAL the dynamical vertex approximation ( DΓA ) AT, A. Katanin, K. Held, PRB (2007) See also: PRB (2009), PRL (2010), PRL (2011), PRB (2012) i j Λ ir

17. Algorithm (flow diagrams): SIAM, G 0 -1 (  ) Dyson equation G loc =G ii G AIM = G loc  ii (  ) G ij ★ DMFT SIAM, G 0 -1 (  ) Parquet Solver G loc =G ii G AIM = G loc Λ ir (ω,ν,ν’) G ij,  ij ★ DΓA ( ⌘ Parquet Solver : Yang, Fotso, Jarrell, et al. PRB 2009 )

18. DCA, 2d-Hubbard model, U=4t, n=0.85, ν = ν ‘= π / β, ω =0 Th. Maier et al., PRL (2006) k-dependence of the irreducible vertex Differently from the other vertices Λ irr is constant in k-space  fully LOCAL in real space [ BUT… is it always true?  on-going project with J. Le Blanc & E. Gull ]

19. Applications: Applications: DMFT not enough [  spatial correlations are crucial]  low dimensions (layered-, surface-, nanosystems) U!!  phase-transitions ( ξ  ∞, criticality) ξ non-local correlations in a molecular rings nanoscopic DΓA [  talk by Angelo Valli]

20. Applications: Applications:  low dimensions (layered-, surface-, nanosystems)  phase-transitions ( ξ  ∞, criticality) DMFT not enough [  spatial correlations are crucial] U!! ξ critical exponents of the Hubbard model in d=3 DΓA (with ladder approx.)

21. Ladder approximation: SIAM, G 0 -1 (  ) Parquet Solver G loc =G ii Λ ir (ω,ν,ν’) G ij,  ij ★ DΓA algorithm : Γ ir (ω,ν,ν’)  ) local assumption already at the level of Γ ir (e.g., spin-channel )  ) working at the level of the Bethe-Salpeter eq. (ladder approx.) Ladder approx.  ) full self-consistency not possible!  Moriya 2P-constraint G AIM = G loc Moriya constraint: χ loc = χ AIM Changes : ( ⌘ Ladder-Moriya approx.: A.Katanin, et al. PRB 2009 )

22. DΓA results in 3 dimensions G. Rohringer, AT, A. Katanin, K. Held, PRL (2011) ✔ phase diagram : one-band Hubbard model in d=3 (half-filling )

23. G. Rohringer, AT, A. Katanin, K. Held, PRL (2011) ✔ phase diagram : one-band Hubbard model in d=3 (half-filling ) DΓA results in 3 dimensions TNTN Quantitatively: good agreement with extrapolated DCA and lattice-QMC at intermediate coupling (U > 1) ✗ underestimation of T N at weak-coupling

24. G. Rohringer, AT, et al., PRL (2011) ✔ phase diagram : one-band Hubbard model in d=3 (half-filling ) DΓA results: 3 dimensions spectral function A(k, ω ) in the self-energy (@ the lowerst ν n ) not a unique criterion!! (larger deviation found in entropy behavior) See: S. Fuchs et al., PRL (2011)

25. G. Rohringer, AT, A. Katanin, K. Held, PRL (2011) ✔ phase diagram : one-band Hubbard model in d=3 (half-filling ) DΓA results: 3 dimensions

26. DΓA results: the critical region DMFT MFT result! wrong in d= 3 DΓA γ DMFT = 1 γ DΓA = 1.4 TNTN correct exponent !!

27. ✔ phase diagram : one-band Hubbard model in d=2 (half-filling ) A. Katanin, AT, K. Held, PRB (2009) DΓA results in 2 dimensions DΓA exponential behavior! T N = 0  Mermin-Wagner Theorem in d = 2!

28. Summary: Going beyond DMFT (non-perturbative but only LOCAL) D Γ A results 1. spectral functions in d=3 and d=2 γ =1.4 & more... spatial correlation in nanoscopic systems cluster extensions (DCA, C-DMFT) diagrammatic extensions (DF, 1PI, DMF 2 RG, & D Γ A) (based on 2P-vertices ) 2. critical exponents  unbiased treatment of QCPs (on-going work)  talk A. Valli

29. Thanks to: ✔ all collaborations A. Katanin (Ekaterinburg), K. Held (TU Wien), S. Andergassen (UniWien) N. Parragh & G. Sangiovanni (Würzburg), O. Gunnarsson (Stuttgart), S. Ciuchi (L‘Aquila), E. Gull (Ann Arbor, US), J. Le Blanc (MPI, Dresden), P. Hansmann, H. Hafermann (Paris). ✔ PhD/master work of G. Rohringer, T. Schäfer, A. Valli, C. Taranto (TU Wien) local vertex/D Γ A nanoD Γ ADMF 2 RG ✔ all of you for the attention!