Mastering the electronic correlations: a challenge for the

Report
„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
 cirrd,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!

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