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„Introduction to the two-particle vertex functions and to the dynamical vertex approximation“

This workshop covers non-local correlations beyond DMFT, diagrammatic extensions, and vertex functions in electronic systems. It delves into Dynamical Vertex Approximations, discussing formalisms, numerical results, and physical interpretations.

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„Introduction to the two-particle vertex functions and to the dynamical vertex approximation“

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

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

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

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

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

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

  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) • 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] • DMF2RG [⌘Taranto, et al. , arXiv 1307.3475] • (DMF2RG: combination of DMFT & fRG) • [talk by Ciro Taranto] • Dynamical Vertex Approximation [⌘AT, Katanin, Held, PRB 2007] • (DΓA: ladder/parquet calculations with a local 2P-vertex [ Γir ] input from DMFT) all these methods require Local two-particle vertex functionsas input !

  8. 2P- vertex: Who’s that guy? To a certainextent: 2P-analogon of theone-particleself-energy vertex 2 particle in–2 particle out Year: 1987; Source: Wikipedia •  1 particle in–1 particle out U Dyson equations: G(1) (ν) Σ(ν) BSE, parquet : G(2) vertex U 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

  9. How to extractthevertexfunctions? 2P-Green‘s function: • numerically demanding, but computable, for AIM • (single band: ED still possible; general multi-band case: CTQMC, work in progress) 2P-vertex functions: Full vertex (scatteringamplitude) = + + F = Γ (fRG notation) = γ4 (DF notation) Whatabout2P-irreducibility?

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

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

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

  13. Frequency dependence: an overview full vertex F background and main diagonal (ν=ν‘) ≈ U2 χm(0) ∞at the MIT irreducible vertexΓ No-high frequency problem (Λ U) BUT low-energy divergencies fully irreducible vertexΛ

  14. Frequency dependence: an overview full vertex F background and main diagonal (ν=ν‘) ≈ U2 χm(0) ∞at the MIT singularity line Γd&Λ∞ irreducible vertexΓ MIT No-high frequency problem (Λ U) BUT low-energy divergencies ⌘ T. Schäfer, G. Rohringer, O. Gunnarsson, S. Ciuchi, G. Sangiovanni, AT,PRL (2013) fully irreducible vertexΛ [talk by Thomas Schäfer]

  15. Types of approximations: *) DIAGRAMMATIC EXTENSIONS OF DMFT: dynamical local vertices 2P- irreducibility F(ν,ν‘,ω) • Dual Fermion, 1PI approach, • DMF2RG Γ(ν,ν‘,ω) Dynamical VertexApprox. (DΓA) Λ(ν,ν‘,ω) moredirect calculation Locality of ΓC, Λ methods based onΓc ,Λ methods based onF inversion of BS eq. or parquet needed Locality of F?

  16. the dynamical vertex approximation (DΓA) AT, A. Katanin, K. Held, PRB (2007) See also: PRB (2009), PRL (2010), PRL (2011), PRB (2012)  DMFT:all1-particle irreducible diagrams (=self-energy) are LOCAL!!  DΓA:all2-particle irreducible diagrams (=vertices) are LOCAL!! Λir j i the self-energy becomes NON-LOCAL

  17. Algorithm(flowdiagrams): • DΓA • DMFT SIAM, G0-1() SIAM, G0-1() Λir(ω,ν,ν’) ii() GAIM = Gloc GAIM = Gloc Parquet Solver Dyson equation Gij Gij, ij Gloc=Gii Gloc=Gii (⌘ Parquet Solver :Yang,Fotso, Jarrell, et al. PRB 2009)

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

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

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

  21. Ladderapproximation: • DΓA algorithm : Changes: SIAM, G0-1() • ) local assumption • already at the level • of Γir (e.g., spin-channel) Λir(ω,ν,ν’) Γir(ω,ν,ν’) GAIM = Gloc • ) working at thelevel • of theBethe-Salpetereq. • (ladderapprox.) Parquet Solver Ladder approx. Gij, ij • ) fullself-consistency • notpossible! • Moriya 2P-constraint Gloc=Gii Moriyaconstraint: χloc =χAIM (⌘ Ladder-Moriya approx.:A.Katanin, et al. PRB 2009)

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

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

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

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

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

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

  28. Summary: cluster extensions (DCA, C-DMFT) Goingbeyond DMFT (non-perturbativebutonly LOCAL) diagrammatic extensions (DF, 1PI, DMF2RG, &DΓA) (based on 2P-vertices ) 1. spectral functions in d=3 and d=2 DΓA results • unbiased • treatment of • QCPs • (on-goingwork) 2. critical exponents γ=1.4 & more ... spatial correlation in nanoscopic systems • talk A. Valli

  29. Thanks to: ✔PhD/master work of G. Rohringer, T. Schäfer, A. Valli, C. Taranto (TU Wien) local vertex/DΓA nanoDΓA DMF2RG ✔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). ✔all of you for the attention!

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