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

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

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„Introduction to the two-particle vertex functions and tothe 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 VertexApproximations (DΓA):
- basics of DΓA
- DΓA results: (i) spectral function & critical regime of bulk 3d-systems
- (ii) nanoscopic system ( talk A. Valli)

Local part only!

U

multi-orbital

Hubbard model

- J

Simplest version: single-band Hubbard hamliltonian:

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

- 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!!

- 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()

- 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 !

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

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?

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

Types of approximations:

U

*) LOWEST ORDER (STATIC) APPROXIMATION:

2P- irreducibility

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

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

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]

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?

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

- 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)

- 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]

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]

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.)

- 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)

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 TNat weak-coupling

TN

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)

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

γDMFT= 1

γDΓA= 1.4

DMFT

DΓA

correct

exponent !!

TN

MFT result!

wrong in d=3

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!

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

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!