„Introduction to the two-particle vertex functions and to the dynamical vertex approximation“ ERC -workshop „Ab-initioDΓΑ“ Baumschlagerberg, 3 September 2013 Alessandro Toschi 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 Vertex Approximations (DΓA): basics of DΓA DΓA results: (i) spectral function & critical regime of bulk 3d-systems (ii) nanoscopic system ( talk A. Valli) Electronic correlation in solids Local part only! e2 V (r) r -J U multi-orbital Hubbard model Simplest version: single-band Hubbard hamliltonian: the Dynamical Mean Field Theory W. Metzner & D. Vollhardt, PRL (1989) A. Georges & G. Kotliar, PRB (1992) -J U h (t) eff described by DMFT“ „There are more things in Heaven and Earth, than those self-consistent SIAM Σ(ω) Σ(ω) [W. Shakespeare , readapted by AT ] No: spatial correlations Yes: local temporal correlations non-perturbative in U, BUT purely local DMFT applicability: high connectivity/dimensions (exact in d = ∞) heff(t) DMFT high temperatures Instead: DMFT not enough [ spatial correlations are crucial] low dimensions (layered-, surface-, nanosystems) U!! phase-transitions (ξ ∞,criticality) ξ 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. ij() PRL 2001; Huscroft, Jarrell et al. PRL 2001] 1. Cellular-DMFT (C-DMFT: cluster in real space) 2. Dynamical Cluster Approx. (DCA: cluster in k-space) (C-DMFT, DCA : systematic approach, BUT only “short” range correlation included) a complementary route: diagrammatic extensions 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 functions as input ! 2P- vertex: Who’s that guy? To a certain extent: 2P-analogon of the one-particle self-energy 1 particle in – 1 particle out vertex 2 particle in – 2 particle out U U Dyson equations: G(1) (ν) Σ(ν) Year: 1987; Source: Wikipedia BSE, parquet : G(2) vertex 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 How to extract the vertex functions? (2) 2P-Green‘s function: Gloc (,, ') FT T c (1)c( 2 )c ( 3 )c(0) numerically demanding, but computable, for AIM (single band: ED still possible; general multi-band case: CTQMC, work in progress) 2P-vertex functions: Full vertex (scattering amplitude) (2) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) Gloc ( , , ') Gloc Gloc ... Gloc Gloc F( , , ')Gloc Gloc = What about 2P-irreducibility? BSE + + F =Γ parquet cirrd,m,s, t ( , , ') (fRG notation) irr ( ,=,γ4') (DF notation) Decomposition of the full vertex F 1) parquet equation: Γph 2) Bethe-Salpeter equation (BS eq.): e.g., in the ph transverse ( ph ) channel: F = Γph + Φph Types of approximations: 2P- irreducibility *) LOWEST ORDER (STATIC) APPROXIMATION: U Dynamic structure of the vertex: DMFT results ν+ω ν‘ + ω density/charge = F(ν,ν‘,ω) F ν spin sectors: ν‘ =0 (2n+1)π/β (2n‘+1)π/β Fd F F magnetic/spin Fm F F intermediate coupling (U ~ W/2) for the vertex asymptotics: see also J. Kunes, PRB (2011) background Frequency dependence: an overview full vertex F irreducible vertex Γ fully irreducible vertex Λ background and main diagonal (ν=ν‘) ≈ U2 χm(0) at the MIT No-high frequency problem (Λ U) BUT low-energy divergencies Frequency dependence: an overview full vertex F irreducible vertex Γ background and main diagonal (ν=ν‘) singularity line 2 ≈ U χm(0) Γd & Λ at the MIT MIT fully irreducible vertex Λ ⌘ T. Schäfer, G. Rohringer, O. Gunnarsson, S. Ciuchi, No-high G. Sangiovanni, AT, PRL (2013) frequency [ talk by Thomas problem Schäfer ] (Λ U) BUT low-energy divergencies Types of approximations: 2P- irreducibility *) DIAGRAMMATIC EXTENSIONS OF DMFT: dynamical local vertices F(ν,ν‘,ω) Dual Fermion, 1PI approach, DMF2RG Γ(ν,ν‘,ω) Λ(ν,ν‘,ω) methods based on F more direct calculation Locality of F? Dynamical Vertex Approx. (DΓA) methods based on Γc , Λ Locality of ΓC, Λ inversion of BS eq. or parquet needed the dynamical vertex approximation (DΓA) AT, A. Katanin, K. Held, PRB (2007) See also: PRB (2009), PRL (2010), PRL (2011), PRB (2012) DMFT: all 1-particle irreducible diagrams (=self-energy) are LOCAL !! DΓA: all 2-particle irreducible diagrams (=vertices) are LOCAL !! Λir the self-energy becomes NON-LOCAL i j Algorithm (flow diagrams): DΓA DMFT SIAM, G0-1() Λir(ω,ν,ν’) Parquet Solver Gij Gij, Gloc=Gii Gloc=Gii ij (⌘ Parquet Solver : Yang, Fotso, Jarrell, et al. PRB 2009) GAIM = Gloc Dyson equation GAIM = Gloc ii() SIAM, G0-1() k-dependence of the irreducible vertex Differently from the other vertices Λirr is 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 ] 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] Applications: DMFT not enough [ spatial correlations are crucial] low dimensions (layered-, surface-, nanosystems) U!! phase-transitions (ξ ∞,criticality) ξ critical exponents of the Hubbard model in d=3 DΓA (with ladder approx.) Ladder approximation: Changes: ) local assumption ) working at the level of the Bethe-Salpeter eq. (ladder approx.) ) full self-consistency not possible! Moriya 2P-constraint algorithm : SIAM, G0-1() Λ Γir(ω,ν,ν’) Ladder Parquet approx. Solver Gij, Gloc=Gii (⌘ Ladder-Moriya approx.: A.Katanin, et al. PRB 2009) GAIM = Gloc already at the level of Γir (e.g., spin-channel) DΓA ij Moriya constraint: χloc = χAIM 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) 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 TN at weak-coupling TN DΓA results: 3 dimensions ✔ phase diagram: one-band Hubbard model in d=3 (half-filling) spectral function G. Rohringer, AT, et al., PRL (2011) A(k, ω) in the self-energy (@ the lowerst νn) not a unique criterion!! (larger deviation found in entropy behavior) See: S. Fuchs et al., PRL (2011) DΓA results: 3 dimensions ✔ phase diagram: one-band Hubbard model in d=3 (half-filling) G. Rohringer, AT, A. Katanin, K. Held, PRL (2011) DΓA results: the critical region (q ( , , )) 1 AF 1 S DMFT DΓA TN γDMFT= 1 γDΓA= 1.4 MFT result! correct wrong in!!d=3 exponent 1 AF (T TN ) 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 in d = 2! Summary: Going beyond DMFT (non-perturbative but only LOCAL) cluster extensions (DCA, C-DMFT) diagrammatic extensions (DF, 1PI, DMF2RG, & DΓA) (based on 2P-vertices ) 1. spectral functions in d=3 and d=2 DΓA results 2. critical exponents γ=1.4 & more ... spatial correlation in nanoscopic systems unbiased treatment of QCPs (on-going work) talk A.Valli Thanks to: ✔ PhD/master work of G. Rohringer, T. Schäfer, A.Valli, local vertex/DΓA nanoDΓA C. Taranto (TU Wien) 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!