Alessandro Toschi
ERC -workshop „Ab-initioDΓΑ“ Baumschlagerberg, 3 September 2013
„Introduction to the two-particle vertex functions and tothe dynamical vertex
approximation“
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 (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
- Jmulti-orbital
Hubbard model
V(r) e2
r
Local part only!
U
Simplest version: single-band Hubbard hamliltonian:
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)
heff(t)
- J
Σ(ω)
U
non-perturbative in U, BUT purely local
self-consistentSIAM
„There are more things in Heaven and Earth, than those described by DMFT“ [W. Shakespeare , readapted by AT ]
(exact in d = ∞)heff(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
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()
★ 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)
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]
(DΓA: ladder/parquet calculations with a local 2P-vertex [ Γir ] 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 !
★ DMF2RG [⌘ Taranto, et al. , arXiv 1307.3475](DMF2RG: combination of DMFT & fRG)
[ talk by Ciro Taranto]
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
S 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
How to extract the vertex functions?
2P-Green‘s function:
2P-vertex functions:
Gloc(2)(,, ') 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)
Gloc(2)(,, ') Gloc
(1 )Gloc(1 ) ... Gloc
(1 )Gloc(1 )F(,, ')Gloc
(1 )Gloc(1 )
BSE
cd,m,s,tirr (,, ')
parquet
irr(,, ')
= + F+
Full vertex(scattering amplitude)
What about 2P-irreducibility?= Γ
(fRG notation)= γ4
(DF notation)
Decomposition of the full vertex F1) parquet equation:
2) Bethe-Salpeter equation (BS eq.):
Γph
e.g., in the ph transverse ( ph ) channel: F = Γph + Φph
Types of approximations:
2P-
irre
du
cib
ilit
y*) LOWEST ORDER (STATIC) APPROXIMATION: U
F
ν + ω ν‘ + ω
ν‘ν
Dynamic structure of the vertex: DMFT results
= F(ν,ν‘,ω)
spin sectors:
density/charge
magnetic/spin
Fd F F
Fm F F
background
= 0
intermediatecoupling
(U ~ W/2)
(2n+1)π/β(2n‘+1)π/β
for the vertex asymptotics: see also J. Kunes, PRB (2011)
full vertex F
Frequency dependence: an overview
irreducible vertex Γ
fully irreducible vertex Λ
backgroundand maindiagonal (ν=ν‘) ≈ U2 χm(0)
∞ at the MIT
No-highfrequency problem (Λ U)BUT
low-energydivergencies
full vertex F
Frequency dependence: an overview
irreducible vertex Γ
fully irreducible vertex Λ
backgroundand maindiagonal (ν=ν‘) ≈ U2 χm(0)
∞ at the MIT
No-highfrequency problem (Λ U)BUT
low-energydivergencies
MIT
[ talk by Thomas Schäfer]
Γd & Λ ∞
singularity line
⌘ T. Schäfer, G. Rohringer, O. Gunnarsson, S. Ciuchi,
G. Sangiovanni, AT, PRL (2013)
Types of approximations:
2P-
irre
du
cib
ilit
y*) DIAGRAMMATIC EXTENSIONS OF DMFT: dynamical local vertices
F(ν,ν‘,ω)
Γ(ν,ν‘,ω)
Λ(ν,ν‘,ω)
Dual Fermion, 1PI approach,
DMF2RG
Dynamical Vertex Approx. (DΓA)
methods based on F
methods based on Γc , Λ
more directcalculation
Locality of F?
Locality of ΓC, Λ
inversion of BS eq.or parquet needed
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
Λir
Algorithm (flow diagrams):
SIAM, G0
-1()
Dyson equation
Gloc=Gii
GA
IM = G
loc
ii()
Gij
★ DMFT
SIAM, G0
-1()
ParquetSolver
Gloc=Gii
GA
IM = G
loc
Λir(ω,ν,ν’)
Gij, ij
★ DΓA
(⌘ Parquet Solver : Yang, Fotso, Jarrell, et al. PRB 2009)
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
Λirr is constant
in k-space
fully LOCAL in real space
[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:
low dimensions (layered-, surface-, nanosystems)
phase-transitions (ξ ∞,criticality)
DMFT not enough [ spatial correlations are crucial]
U!! ξ
critical exponents of theHubbard model in d=3
DΓA(with ladder approx.)
Ladder approximation:
SIAM, G0
-1()
ParquetSolver
Gloc=Gii
Λir(ω,ν,ν’)
Gij, ij
★ DΓA algorithm :
Γir(ω,ν,ν’)
) local assumptionalready at the level of Γir (e.g., spin-channel)
) working at the level of the Bethe-Salpeter eq.
(ladder approx.)
Ladderapprox.
) full self-consistency not possible!
Moriya 2P-constraint
GA
IM = G
loc
Moriya constraint:
χloc = χAIM
Changes:
(⌘ Ladder-Moriya approx.: A.Katanin, et al. PRB 2009)
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)
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
TN
Quantitatively:
good agreement with extrapolated DCA and lattice-QMC at intermediate coupling (U > 1)
✗ underestimation of TN at weak-coupling
G. Rohringer, AT, et al., PRL (2011)
✔ phase diagram: one-band Hubbard model in d=3 (half-filling)
DΓA results: 3 dimensions
spectral functionA(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)
G. Rohringer, AT, A. Katanin, K. Held, PRL (2011)
✔ phase diagram: one-band Hubbard model in d=3 (half-filling)
DΓA results: 3 dimensions
DΓA results: the critical region
DMFT
MFT result!wrong in d=3
DΓAγDMFT= 1
γDΓA= 1.4
AF 1 S
1(q (, , ))
TN
correct exponent !!
AF 1 (T TN )
✔ 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!
TN = 0 Mermin-Wagner Theorem in d = 2!
Summary:Going beyond
DMFT(non-perturbative but only LOCAL)
DΓA 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, DMF2RG, & DΓA)
(based on 2P-vertices )
2. critical exponents unbiased treatment of QCPs(on-going work)
talk A. Valli
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/DΓA nanoDΓA DMF2RG
✔ all of you for the attention!