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

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Alessandro Toschi ERC -workshop „Ab-initioDΓΑ“ Baumschlagerberg, 3 September 2013 „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“. ERC -workshop „Ab-initio D ΓΑ “ Baumschlagerberg, 3 September 2013. Alessandro Toschi. Outlook. I ) Non-local correlations beyond DMFT overview of the extensions of DMFT - PowerPoint PPT Presentation

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

Page 1: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

Alessandro Toschi

ERC -workshop „Ab-initioDΓΑ“ Baumschlagerberg, 3 September 2013

„Introduction to the two-particle vertex functions and tothe dynamical vertex

approximation“

Page 2: „Introduction to the two-particle vertex functions and to the  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)

Page 3: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

Electronic correlation in solids

- Jmulti-orbital

Hubbard model

V(r) e2

r

Local part only!

U

Simplest version: single-band Hubbard hamliltonian:

Page 4: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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 ]

Page 5: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

(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

Page 6: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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)

Page 7: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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]

Page 8: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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

Page 9: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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)

Page 10: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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

Page 11: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

Types of approximations:

2P-

irre

du

cib

ilit

y*) LOWEST ORDER (STATIC) APPROXIMATION: U

Page 12: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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)

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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

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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)

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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

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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

Page 17: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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)

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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 ]

Page 19: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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]

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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.)

Page 21: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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)

Page 22: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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)

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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

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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)

Page 25: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

G. Rohringer, AT, A. Katanin, K. Held, PRL (2011)

✔ phase diagram: one-band Hubbard model in d=3 (half-filling)

DΓA results: 3 dimensions

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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 )

Page 27: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

✔ 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!

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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

Page 29: „Introduction to the two-particle vertex functions and to the  dynamical vertex approximation“

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!