the dft+dmft method and its implementation in abinit · the dft+dmft method and its implementation...

The DFT+DMFT method and itsimplementation in Abinit

Bernard Amadon

CEA, DAM, DIF, F-91297 Arpajon, France

New trends in computational approaches for many-body systems, June 2012, Sherbrooke – p.1/70

Outline of the presentation

• DFT+DMFT: a short reminder/reformulation.• Choice of the basis and local orbitals.

• Formalism.• Physical considerations.

• Interaction and double counting and solver• Description of the actual calculation

• Practical issues/details to carry out the calculation (PAW)• Self-consistency issues: explanation, what has to be done.

calculation of non diag occupancies to compute local densities• Outcome of the calculation

• Spectral functions, self-energy.• Energy.

• Some tests/examples.• Conclusion.


Localization of 3d, 4f and 5f electrons

3d and4f are orthogonal to other orbitals only through theangularpart.

-0.50

0.51

spd

φ(r)

-0.50

0.51

0 0.5 1 1.5r (Å)

-0.50

0.51

V (3d)

Nb (4d)

Ta (5d)

-0.50

0.51

spdf

φ(r)

0 1 2r(Å)

-0.50

0.51

Ce (4f)

Th (5f)


Strong local correlations: Introduction (I)

• 3d, 4f and5f elements: localized atomic wavefunctions⇒strong correlations.• Small overlap: bands arenarrow (width: W).• Stronginteractions "U"between electrons.• Competition between band effect and correlation have to be

described.


Strong local correlations: Introduction (II)

• Hamiltonian to solve (i: électrons ):

H =

N∑

i=1

[−1

2∇2

ri+ Vext(ri)] +

1

2

∑

i 6=j

1

|ri − rj |

• Strong correlation in Localized orbitals (f, d)

• Other orbitals: DFT(LDA/GGA) could be tried..

HLDA+Manybody = one electron term (DFT/LDA)+U

2

∑

i 6=j

ninj

︸︷︷︸many body interaction

ELDA+U = ELDA − UN(N − 1)

2+

U

2

∑

i 6=j

ninj

• But in DFT+U: Electrons are frozen, only one Slater determinant,

metastable states.New trends in computational approaches for many-body systems, June 2012, Sherbrooke – p.5/70

DFT+U and the Dynamical Mean Field Theory (DMFT)

• DFT+U⇒ To describe interactions in solids with static mean fieldtheoryMain idea: An electron is described, in the effective field ofall theother electrons.⇒ Self-consistent Hartree Fock problem.

[Anisimov V I, Poteryaev A I, Korotin M A, Anokhin A O and Kotliar G J. Phys.: Condens. Matter 9

735967 (1997)]



• DFT+U⇒ To describe interactions in solids with static mean fieldtheoryMain idea: An electron is described, in the effective field ofall theother electrons.⇒ Self-consistent Hartree Fock problem.

• DMFT ⇒ To describe correlation in solids beyond static mean fieldtheory.Main idea: An atom is isolated (in red),local corre-lations are described exactly, in the effective field ofother atoms.⇒ Self-consistent Impurity problem: Andersonmodel

[see review A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg Rev. Mod. Phys. 68, 13 (1996)]



• DFT+U⇒ Unrestricted Hartree Fock

[Anisimov V I, Poteryaev A I, Korotin M A, Anokhin A O and Kotliar G J. Phys.: Condens. Matter 9

735967 (1997)]



• DFT+U⇒ Unrestricted Hartree Fock• DMFT ⇒ Configuration Interaction for correlated electron in a bath..

Only for correlated electrons on a single atom coupled to a bath !

[see review A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg Rev. Mod. Phys. 68, 13 (1996)]


The anderson model: general case

W

tkf tkftkf

fε +U

ε f

KT

E

Γ

<n −n >

[cf L. Kouwenhoven et L. Glazman, Physics World Jan 2001 p33]


The anderson model: general case

W

tkf tkftkf

fε +U

ε f

KT

E

Γ

<n −n >


The DFT+DMFT schemeA Self-consistent DFT+DMFT scheme in the Projector Augmented Wave : Applications to Cerium, Ce O and Pu

Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity

Self-consistencycondition

DFT+DMFT Loop over density

〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp

Figure 1. Fully Self-consistent DFT+DMFT scheme adapted to the Projected Local Orbital schemeused in our implementation. For a fixed electronic density, the DMFT Loop is represented in blue. Itcontains two steps: Firstly, the impurity model is solved to compute the Green function and the self-energy . Secondly the lattice Green function is computed from the impurity self-energy, andthe self-consistency condition states that the local Green function is also the impurity Green function.When this DMFT loop is converged, one can compute the lattice Green function and the non diagonaloccupations in the Kohn Sham Bloch basis. The occupations are used to compute the total electronicdensity (in practice, its PW and PAW components) thanks to Eq. 2.7. From this density, the Kohn ShamDFT Hamiltonian is build and diagonalized. The new KS bloch wave functions, and the eigenvalues arethen used to compute the Green function (Eq. 2.5). Then a new DMFT loop is performed. This cycle(in pink) is repeated until convergence of the density.

is the density operator in these equations.Then, the DFT Hamiltonian is built and diagonalized: new KS eigenvalues and eigenfunctions

are extracted. A special care is taken to obtain the new KS bands with a good accuracy for eachnew electronic density, especially for unoccupied KS states. Then, projections 2.1 are recomputed tobuild Wannier functions for the next DMFT loop. KS eigenvalues are also used to compute the Greenfunction using 2.5.

Peculiarities of the PAW formalism for the computation of the electronic density are describedin Appendix A.

2.3. Calculation of Internal Energy in DFT+DMFTThe DFT+DMFT formalism can be derived from a functional [45] of both the local density andthe local Green’s function [46]. The internal energy can be derived and one obtain the general


DMFT in an arbitrary basis (1)

• What is necessary to carry out effectively the DMFT calculation ?• The band structure and the one electron Hamiltonian

H =∑

ν,k

|Ψkν〉 εkν 〈Ψkν |

• A basis set for one electron quantities:|Bkα〉.

• The matrix element on the Hamiltonian can thus be computed as:

HKS,αα′(k) = 〈Bkα|H|Bkα′〉

=∑

ν

〈Bkα|Ψkν〉 εkν 〈Ψkν |Bkα′〉 .

• A definition for d/f local orbitals, let’s call them|χR

km〉. (m is anangular momentum index). DFT+DMFT is alocal orbital dependentmethod.



• All one electron quantities can be expressed in the basis set|Bkα〉.

• Lets define projectors:

Projectors:PR

mα(k) ≡ 〈χR

km|Bkα〉 , PR

mα(k)∗ ≡ 〈Bkα|χ

R

km〉 .

• The Self energy is a local quantity in DMFT, thus:

Σ =∑

mm′TR

|χR

Tm〉ΣRimpmm′ 〈χ

R

Tm′ |



Thus

∆Σαα′(k, iωn) = 〈Bkα|Σ|Bkα〉

=∑

mm′

PR

mα(k)∗ΣRimp

m,m′ PR

m′α′(k).

Note that the self-energy in this basis has a k-dependence.The lattice Green’s functionGα,α′(k, iωn) can be expressed in thecomplete Bloch BasisBkα. and projected to compute the local (impurity)Green’s function.

Gimpmm′(iωn) =

∑

k

∑

αα′

PR

mα(k)PR

m′α′(k)∗ ×

×[(iωn + µ)I−HKS(k)−∆Σ(k, iωn)]

−1αα′

,


DMFT in an arbitrary basis (4): main idea

• A definition for d/f local orbitals|χR

km〉. (m is an angular momentumindex)• A physically motivated choice.• Non local correlations have to be minimized.

• A basis set for one electron quantities:|Bkα〉.• Results should be independant of the basis.• Numerical efficiency do limit the size of the basis.


Formalism: DFT+DMFT implementation, LMTO

• LMTO ASA |χl,l′,m,m′(k)〉 (Andersen 1975).• LMTO’s are an optimized atomic orbital Bloch basis.• The local correlated subspace is thus asubset of the LMTO basisBkα.• |χl=2,l′=2,m,m′(k)〉 for d orbitals.

• First implementation in DMFT: Lichtenstein and Katsnelson1998• Fast, efficient, but method dependent.


Why DFT+DMFT in a plane wave code ?

• Plane wave code do not make any approximation on the shapeof the potential: the basis is complete.

• Supplemented by the PAW method, plane wave code areefficient even for localized systems (with d/f electrons) (seelater)


Formalism: DFT+DMFT implementation, MLWF

• Maximally LocalizedWannier Functions.• A Wannier function is by definition unitarily related to Bloch

functions and form an orthonormal basis.

|wR

km〉(r) =∑

ν

Um,ν(k)|Ψk,ν(r)〉 (-6)

• From the Bloch Wannier function, one can construct the Wannierfunction on siteT.

|wR

Tm〉 =

(V

2π

)3 ∫

BZdke−ikT|wR

km〉 (-6)

(depends onr−T only).

Wanniers or NMTO (Pavariniet al 04, Anisimovet al 05)

Maximally localized Wanniers and NMTO (Lechermannet al 06)


Formalism: DFT+DMFT implementation, MLWF (2)

• Flexible: one can choose the extension of the Wannierfunctions.

• The correlated orbitals are a subset of the basis set.• Might be problematic

• Disentanglement for large systems.• f-electron systems.

• Other Wannier functions can however be easily built !


DMFT in the Kohn Sham basis

• Let’s choose the Bloch basis as the basis for one electron quantities.• The band structure and the one electron Hamiltonian

HKS =∑

ν,k


• A basis set for one electron quantities:|Bkα〉.


HKS,αα′(k) = 〈Bkα|HKS|Bkα′〉

=∑

ν

〈Bkα|Ψkν〉 εkν 〈Ψkν |Bkα′〉 .


km〉. (m is anangular momentum index)


DMFT in the Kohn Sham basis

• Let’s choose the Bloch basis as the basis for one electron quantities.• The band structure and the one electron Hamiltonian

HKS =∑

ν,k


• A basis set for one electron quantities:|Ψkν〉.


HKS,νν′(k) = 〈Ψkν |H|Ψkν〉

= εkνδν,ν′


km〉. (m is anangular momentum index)


DMFT in the Kohn Sham basis (2)

• All one electron quantities can be expressed in the basis set|Ψkν〉.

• Lets define projectors:

Projectors:PR

mν(k) ≡ 〈χR

km|Ψkν〉 , PR

mν(k)∗ ≡ 〈Ψkν |χ

R

km〉 .

• P matrix is in general nonsquare.• The Self energy is a local quantity in DMFT, thus:

Σ =∑

mm′T

|χR

Tm〉ΣRimpmm′ 〈χ

R

Tm′ |


DMFT in the Kohn Sham Bloch basis (3)

Thus

∆Σαα′(k, iωn) = 〈Ψkν |Σ(k)|Ψkν〉

=∑

mm′

PR

mν(k)∗ΣRimp

m,m′ PR

mν(k).

The lattice Green’s functionGα,α′(k, iωn) can be expressed in thecomplete Bloch BasisΨkν . and projected to compute the local (impurity)Green’s function.

Gimpmm′(iωn) =

∑

k

∑

νν′

PR

mν(k)PR

m′ν′(k)∗ ×

×[(iωn + µ− εkν)I−∆Σ(k, iωn)]

−1

νν′,


Choice of the basis set and local orbitals

• |wR

km〉 is thus obtained though:

|wR

km〉 =∑

R′,m′

[O(k)]−1/2

R,R′

m,m′

|χR′

km′〉 (-15)

• The localized basis depends on the energy extension of the windowW .

• |wR

km〉 areless localized than initialχR

km since the basis has asmallerenergy range.

• The localized basis depends on the energy extension of the basis.

• If |wR

km〉 is chosen (through the window of energy),the wholeframework is exact: There is no finite basis effect.• 〈wR

km|Ψkν〉 = 0 if ν is outside the window of energy whichdefines|wR

km〉.• No convergence as a function of the size of the basis.


Basis set and local orbitals: Physical considerations

A reminder: hybridization in a diatomic molecule (oversimplified)

H or V

F or O


An oversimplified derivation for the diatomic molecule VO

φV

φOε1

Ψ1 = αφO + βφV β ≪ α

ε2

Ψ2 = βφO − αφV β ≪ α

Two windows of energy are possible to compute|χ〉 =

∑W〈Ψi|φV 〉|Ψi〉

• If only ε2 is included, the correlated wavefunction is|χ〉 = |Ψ2〉 = β|φO〉 − α|φV 〉 and contains an Oxygencontribution

• If only ε1 andε2 is included, the correlated wavefunction is|χ〉 =

∑i〈Ψi|φV 〉|Ψi〉 = |φV 〉 and is much more localized.


Example of SrVO3

Γ

-8 -6 -4 -2 0 2 4 6 8eV

0

2

4

6

8

10(e

lect

rons/

eV)

totalO-p

V-t2g

V-eg

• metallic oxide with oned electron on the correlated atom(Vanadium).

• Hybridization between Oxygen and Vanadium.• In a cubic environnement, d orbitals are splitted in two

subgroups called t2g and eg.


SrVO3: band structure

Γ

R Γ X M Γ

-8

-6

-4

-2

0

2

4

6

8

(eV)

O-p

R Γ X M Γ

V-t2g

R Γ X M Γ

V-eg

Fatbands⇒ Hybridization between O-p orbitals and V-t2g orbitals


Correlated orbitals for two windows of energy

t2g bands only t2g bands and Oxygen p bands

More extended Less extendedFrom F. Lecherman, A. Georges, A. Poteryaev, S. Biermann, M.Posternak, A. Yamasaki and O.K.

Andersen PRB 2006


Interaction parameters U and J

t2g bands only t2g bands and Oxygen p bands

More extended Less extendedThe interaction parameters between two electrons in orbitals m andm′ areUm,m′ = 〈φmφm′ |W (r, r′)|φmφm′〉.In the windows of energy increases, the value ofU used shouldincrease also, because the localization of correlated orbitalsincreases.


Photoemission spectra of d elements.

Lower Hubbard band (~−1.8eV)

Upper Hubbard band (~2.5eV)

Quasiparticle peak

From Morikawa et al (1995)

Sekiyama 1992

⇒ YTiO3 insulator: metalin LDA.⇒ SrVO3 is a metal: metal en LDA, butwithout the peak at -1.8eV.


Comparison of DFT+DMFT results

-4 -2 0 2 4 6 8(eV)0

0.5

1

1.5

2

(eV-1 )

MLWFWannier from PLO

0 5 10τ (eV-1)

-0.8

-0.6

-0.4

-0.2

0

G(τ

)

0 5 10-0.8

-0.6

-0.4

-0.2

0

Comparison of Spectral function for SrVO3 computed:• Maximally Localized Wannier Functions (MLWF)• Wannier functions obtained by Projection of Localized Orbitals

(PLO).


Interactions in DFT+U/DFT+DMFT

Eee =1

2

σ∑

1,2,3,4

[〈13|Vee|24〉n

σ1,2n

−σ3,4 + (〈13|Vee|24〉 − 〈13|Vee|42〉)n

σ1,2n

σ3,4

]

Eee =1

2

σ∑

1,3

[〈13|Vee13〉n

σ1n

−σ3 + (〈13|Vee|13〉 − 〈13|Vee|31〉)δ13n

σ1n

σ3

]

〈13|Vee|24〉 = 4π∑

k=0,2,4,6

Fk

2k + 1

+k∑

m=−k

〈m1|m|m2〉〈m3|m|m4〉

Only density density terms are used in the interaction term:

H =1

3

∑

1,3,σ

Uσ,σ1,3 n

σ1 n

σ3 +

1

3

∑

1,3,σ

Uσ,−σ1,3 nσ

1 n−σ3

With Uσ,−σ′

1,3 = 〈13|Vee13〉 andUσ,σ1,3 = 〈13|Vee|13〉 − 〈13|Vee|31〉


Double counting corrections

Double counting corrections: Atomic limit (or Full localized limit)

[Lichtenstein(1995), Anisimov (1991)]:

EFLLdc =

∑

t

(U

2N(N − 1)−

∑

σ

J

2Nσ(Nσ − 1))

Around mean field version [Czyzyk(1994)] (delocalized limit):

EAMFdc =

∑

t

(UN↑N↓ +1

2(N2

↑ +N2↓ )

2l

2l + 1(U − J))

(Made to correct the delocalized limit.)


Solvers available in the public version: Hubbard I

• Hubbard I solve the Anderson Impurity neglecting the hybridizationbetween the impurity and the other atoms.

H =∑

m

ǫmc†mcm +1

2

∑

m1m2

Um1m2nm1

nm2

• We can compute the effective levels as

ǫm = 〈χm| ˆHlda|χm〉−ΣDC−µ =∑

kν

Pmν(k)ǫvkP∗νm(k)−ΣDC−µ.

• In the Hubbard I, Green’s function is written as

Gatomique(iωn) =1

Zat

∑

AB

|〈A|d†|B〉|2

iωn +EB − EA(e−βEA + e−βEB )

EA =∑

j

njǫj +∑

12

U12n1n2


Example of input/output of HI for NiO

== Print Diagonalized Energy levels-- polarization spin component 1

-2.21126 0.00000 0.00000 0.00000 0.000000.00000 -2.21126 0.00000 0.00000 0.000000.00000 0.00000 -2.19499 0.00000 0.000000.00000 0.00000 0.00000 -2.19499 0.000000.00000 0.00000 0.00000 0.00000 -2.19499

-- polarization spin component 2-2.16239 0.00000 0.00000 0.00000 0.000000.00000 -2.16239 0.00000 0.00000 0.000000.00000 0.00000 -2.16239 0.00000 0.000000.00000 0.00000 0.00000 -2.15234 0.000000.00000 0.00000 0.00000 0.00000 -2.15234

Hubbard I: Energies as afunction of number of electronsNelec Min. Ene. Max. Ener.

HI 0 0.0000000 0.0000000HI 1 -2.2112560 -2.1523429HI 2 -4.1225121 -4.0046858HI 3 -5.7175036 -5.5670744HI 4 -7.0124951 -6.8294630HI 5 -8.0074866 -7.7918517HI 6 -8.6698752 -8.4868432HI 7 -9.0322638 -8.8818347HI 8 -9.0946524 -8.9768262HI 9 -8.8469954 -8.7880822HI 10 -8.2993383 -8.2993383


Example of input/output of HI for NiO

=== Integrate green function-------> For Correlated Atom 1

Nb of Corr. elec. is: 7.998950

== The occupations (integral of the Green function) are ==

= In the atomic basis

-------> For Correlated Atom 1

-- polarization spin component 10.99979 0.00000 0.00000 0.00000 0.000000.00000 0.99979 -0.00000 0.00000 0.000000.00000 -0.00000 0.99980 0.00000 0.000000.00000 0.00000 0.00000 0.99979 0.000000.00000 0.00000 0.00000 0.00000 0.99980

-- polarization spin component 20.90117 0.00000 0.00000 0.00000 -0.000000.00000 0.90117 -0.00000 0.00000 0.000000.00000 -0.00000 0.14822 0.00000 0.000000.00000 0.00000 0.00000 0.90117 0.00000

-0.00000 0.00000 0.00000 0.00000 0.14822


Solvers available in the public version: Hubbard I

• Drawback: number of electrons computed from the atom is aninteger, no hybridization, .....

• In progress (J. Bieder and BA): Implementation of CTQMCStrong coupling solver (see presentation of P. Werner) .Doneand under test.


The DFT+DMFT scheme (1): a practical calculationA Self-consistent DFT+DMFT scheme in the Projector Augmented Wave : Applications to Cerium, Ce O and Pu

Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity



〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp






• First step: Make a converged DFT calculation.• WavefunctionsΨkν and Eigenvaluesεkν are available.• Unoccupied wavefunctions must be accurately computed in

order to be used in the DMFT calculation.


Diagonalise the Hamiltonian

• Solve DFT self-consistently• Diagonalize the Hamiltonian with empty states• Get Eigenvalues and eigenvectors.


The Projector Augmented Wave Method

12PAW method and DFPT

-48

nn ~ =

n nnnnnnn AfAfA ~~ *

!"

6- ,

$ ! $!

RRR nnnn rrrr 11

~~R

RR EEEE 11~~

* 48

%&

n~τ=(

iiφ+ nip ~~

)i~φ−

PAW• Efficiency with respect to usual plane wave calculations.• Accuracy: nodal structure of valence wfc are reproduced.• Flexibility: energy of several structures can be compared with the

same basis.• Frozen core approximation can be controlled.

From M. Torrent and F. Jollet Torrent, Jollet, Bottin, Zerah, Gonze CMS 2008


The Projector Augmented Wave Method

FIG. 3: Bonding p-σ orbital of the Cl2 molecule and its decomposition of the wavefunction into

auxiliary wavefunction and the two one-center expansions. Top-left: True and auxiliary wave

function; top-right: auxiliary wavefunction and its partial wave expansion; bottom-left: the two

partial wave expansions; bottom-right: true wavefunction and its partial wave expansion.

Ψ =|Ψ〉

+∑

i〈pi|Ψ〉|φi〉

−∑

i〈pi|Ψ〉|ϕi〉

Handbook of Materials Modeling; Bloechl, Kaestner, Foerst

Implemented in Abinit. Torrent, Jollet, Bottin, Zerah, Gonze,CMS, 2008


Calculation of projections within PAW

Projectors:PR

mν(k) ≡ 〈χR

km|Ψkν〉 , PR

mν(k)∗ ≡ 〈Ψkν |χ

R

km〉 .

Pmν(k) = 〈χm|Ψkν〉+∑

i

〈pi|Ψkν〉(〈χm|ϕi〉 − 〈χm|ϕi〉)

However, atomicd or f wavefunctions are mainly localized insidespheres.

Pmν(k) =∑

ni

〈pni|Ψkν〉〈χm|ϕni

〉

• Check completness during contruction of atomic data.• Check that PAW radius is not too small.• Logarithmic derivatives• Base completness relative to energy

Data constructed with Atompaw (Holzwarth, Torrent and Jollet 2007.)


The DFT+DMFT scheme (2)A Self-consistent DFT+DMFT scheme in the Projector Augmented Wave : Applications to Cerium, Ce O and Pu

Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity



〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp






• Fromεkν, compute DFT (LDA) Green’s function:

GBlochν,ν′ (iωn,k) = [(iωn + µ− εkν)I]

−1νν′

Gimpmm′(iωn) =

∑

k

∑

νν′

PR

mν(k)PR

m′ν′(k)∗GBloch

ν,ν′ (iωn,k)



Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity



〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp






• 1st iteration: DFT LDA Green’s function used as Weiss field• From Weiss FieldG0m,m′ , Anderson Impurity Model is solved

to get impurity Green’s functionGm,m′ .



Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity



〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp






• Self-energy is computed from Dyson Equation :Σ = G−10 −G−1

• The Lattice Green’s function is computed with the self-energy.• Fermi level ajusted to the number of electrons• The Impurity Green’s function is obtained by projection.• Dyson equation is again used to recover new Weiss Field.


Self energy and Lattice Green function

Σimpm,m′ is obtained from the Dyson equation, then:

Σαα′(k, iωn) = 〈Ψkν |Σ(k)|Ψkν〉

=∑

mm′

PR

mν(k)∗Σimp

m,m′PR

mν(k).

The lattice Green’s functionGν,ν′(k, iωn) can be expressed in thecomplete Bloch BasisΨkν and projected to compute the local (impurity)Green’s function.

Gν,ν′(k, iωn) =[(iωn + µ− εkν)I−∆Σ(k, iωn)]

−1

νν′,

Gimpmm′(iωn) =

∑

k

∑

νν′

PR

mν(k)PR

m′ν′(k)∗ ×

×[(iωn + µ− εkν)I−∆Σ(k, iωn)]

−1νν′

,



Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity



〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp






• The DMFT Loop is done until convergency (ofG, Σ).• Then Occupations of electrons inside orbitals, have changed and thus,

the local density must be updated.


The self-consistency over electronic density

The number of electron are recovered by a Fourier transform:

fν,ν′(k) =∑

iωn

Gν,ν′(k, iωn)eiωn0+

The Green function can be written in real space basis as:

G(r, r′) =∑

iωn

〈r|[G(iωn)

]|r〉eiωn0+

G(r, r′) =∑

iωn

〈r|

∑

k,ν,ν′

|Ψkν〉Gν,ν′(k, iωn)〈Ψkν′

|r〉eiωn0+

The local density can be written as

n(r) = G(r, r)

n(r) =∑

k,ν,ν′

Ψ∗kν(r)fν,ν′(k)Ψkν′(r

′)



Σ = G−10 − G−1

impG−10 = Σ + G−1

imp

Gimpn(r) Σ

Glatt

GHKS−DFT problem

Impurity



〈χRkm|Ψkν〉

fDFT+DMFTν,ν′,k

DMFT for fixed n(r)

ǫKS−DFTν,k G0

Gimp






• From total density, the HamiltonianH[n(r)] is diagonalized and newKS wavefunctions, projectors, and eigenvalues are computed for thenext DMFT loop.

• At convergence of the electronic densityn(r), the calculation isstopped.


Analysis of DMFT calculations

• Spectral Function• Impurity spectral function: From Green’s function projected over

local correlated orbitals• Bloch states spectral function (k-resolved or sum): Green’s

function of Kohn Sham states.• Self-energy• Total energy


SrVO3: band structure

Γ

R Γ X M Γ

-8

-6

-4

-2

0

2

4

6

8

(eV)

O-p

R Γ X M Γ

V-t2g

R Γ X M Γ

V-eg

Fatbands⇒ Hybridization between O-p orbitals and V-t2g orbitals


Spectral functions

-4 -2 0 2 4 6 8(eV)0

0.5

1

1.5

2

(eV-1 )

MLWFWannier from PLO

0 5 10τ (eV-1)

-0.8

-0.6

-0.4

-0.2

0

G(τ

)

0 5 10-0.8

-0.6

-0.4

-0.2

0

Comparison of Spectral function for SrVO3 computed:• Maximally Localized Wannier Functions (MLWF)• Wannier functions obtained by Projection of Localized Orbitals

(PLO).


Formalism: Spectral function of SrVO3

00.5

11.5

22.5

(eV-1 )

-8 -6 -4 -2 0 2 4 6w (eV)

00.5

11.5

22.5

(eV-1 )

Dos partielle V-t2g Bloch O2pBloch Vt2g

LDA+DMFT

LDA

O-p Bloch

V-t2g projection

V-t2g Bloch

(hybridization)Full orbitals.

⇒ The position of Op bands % Vd

Redefinition of local orbitals

⇒ More localized

Amadon, Lechermann, Georges, Jollet, Wehling and Lichtenstein Phys. Rev. B 77, 205112 (2008)

Anisimov, Kondakov, Kozhevnikov, Nekrasovet al PRB (2005)


Functionals

A functional is built from the partition function (Kotliar and Savrasov (2004)):

Ω[ρ(r), Gab; vKS(r),∆Σ ab]LDA+DMFT = −tr ln[iωn + µ+1

2∇2 − vKS(r)− χ∗.∆Σ .χ]

−

∫

dr (vKS − vc)ρ(r)− tr [G.∆Σ ] +1

2

∫

dr dr′ρ(r)U(r− r′)ρ(r′) + Exc[ρ(r)]

+∑

R

(

Φimp[GRR

ab ]− ΦDC [GRR

ab ])

minimisation⇒ equations of LDA+DMFT.

⇒ Total energy.

ELDA+DMFT = EDFT −∑

λ

εLDAλ + 〈HKS〉+ 〈HU 〉 − EDC

The one electron part of DFT is replaced by one electron part of DMFT,and DMFT interaction terms are added.


Total energy: first, reminder of total energy in DFT

EDFT = TDFT0 + Exc+Ha[n(r)] +

∫drvext(r)n(r)

Alternatively

EDFT =∑

ν,k

fDFTν,k ǫDFT

ν,k + EDFT DC[n(r)]

with EDFT DC[n(r)] = −EHa[n(r)] + Exc[n(r)]−∫vxcn(r)dr


Total energy in the DFT+DMFT framework

Thus, we have the following expression for the total energy inDFT+DMFT:

E2DFT+DMFT = TDFT+DMFT

0 +Exc+Ha[n(r)]+

∫drvext(r)n(r)+〈HU 〉−EDC

with TDFT+DMFT0 = −

∑ν,ν′,k f

DFT+DMFTν,ν′,k

∫Ψ∗

kν∇2

2 Ψkν′

Alternatively, one can write:

E1DFT+DMFT =

∑

ν,k

fDFT+DMFTν,k ǫKS−DFT

ν,k +EDFT DC[n(r)]+〈HU 〉−EDC

with EDFT DC[n(r)] = −EHa[n(r)] + Exc[n(r)]−∫vxcn(r)dr

Should give the same results at convergence.


Input variables

# == LDA+Uusepawu 1 # Use U Hamiltoniandmatpuopt 3 # Choose density matrix (only for print out)lpawu 2 -1 # Use Hubbard U for l=lpawuupawu 4.0 0.0 eV # Value of U for the two speciesf4of2_sla 0.0 0.0 # Value of F4/F2 to compute <m1m2|1/r|m3m4>

# == LDA+DMFTusedmft 1 # Enable DMFTdmftbandi 6 # First KS Band to usedmftbandf 12 # Last KS Band to usedmft_nwlo 100 # Nb of Freq in the Log Griddmft_nwli 100000 # Nb of Freq in the Linear Matsubaradmft_iter 10 # Nb of Iter for the DMFT Loopdmftcheck 1 # Enable checks (Symmetry, Projection, Fourier)dmft_solv 2 # Choice of the solverdmft_rslf 1 # Read Self Energy from File (for Restart)dmft_mxsf 0.7 # Mixing Coefficient for Self Energydmft_dc 1 # Double counting for DMFT#dmft_tollc 0.00001 # Tolerance over local charge# # for convergence of the DMFT Loop# (in the next version of ABINIT)occopt 3 # Fermi Dirac smearingtsmear 500 K # Temperature


Automatic tests

v6/t07 LDA calculation through the DMFT Loop (check)v6/t45 DMFT calculation with U=0, and U/=0 for several simple solversv6/t46 DMFT calculation with two Ni atoms.v6/t47 DMFT calculation for f-orbitals (Gd)

Several Internal checks:ProjectionsComparison of LDA occupations with wavefunctions or Green’s functionNevertheless, always read the whole log file to check.


Parallelization

In the Hubbard one implementation, the most expensive task is thecalculation and integration of Green’s function.

Gν,ν′(k, iωn) =[(iωn + µ− εkν)I−∆Σ(k, iωn)]

−1νν′

,

This part is thus parallelized over logarithmic frequencies.

Computed on a log frequency


Comparison DFT+U/DFT+DMFT(HF)

If three conditions are met, DFT+U and DFT+DMFT calculationscan be compared for testing purposes.

• a large number of bands have to be used in order that Wannierfunction used in DMFT looks like atomic orbitals.

• a specific density matrix have to be used in DFT+U, namely:

nR,σm,m′ =

∑

kν

fσk,ν

〈χR

m|Ψσk,ν〉〈Ψ

σk,ν |χ

R

m′〉

〈χR|χR〉

The allows for the renormalization of the projections as it isdone in DFT+DMFT.

• a Hartree Fock self-energy need to be used to carry out thecalculation.


Isostructural transition in Cerium

Isostructural transitionVγ−Vα

Vγ= 15%

α phase: Non-magnetic.

γ phase: Paramagnetic

β phase: Antiferromagnetic

α phase:≃ f 1 electron more delocalized. γ phase:e− is localized

Johansson 1974


Cerium: experimental spectra and LDA

-5 0 5 10w (eV)

Experiment (Wuilloud et al 1983, Wieliczka et al 1984)

-5 0 5 10w (eV)

Ce aCe g

Experimental photoemission spectra. LDA density of states.

• Peak at the Fermi levelonly in theα phase.γ andα phase:high energy bands(-2 eV and 5 eV).

• bands at high energynot described in LDA.• peak at the Fermi levelnot correct in LDA• Edft−lda(V ): γ phasenot stable



-5 0 5 10w (eV)

0

5

10LDA+DMFT(HF)LDA+U

Spectral function ofγ-cerium, LDA+U and LDA+DMFT (Static)

With the LDA+U Self-energy, the DMFT recover the LDA+U results at

convergence of the KS basis.



-5 0 5 10w (eV)

0

5

10LDA+DMFT(HF)LDA+U

a (a.u.) B0 (GPa)

PAW/LDA+U 9.58 32

PAW/LDA+DMFT(HF) 9.59 31


Results: Spectral function of Ce2O3

-6 -4 -2 0 2 4 6w (eV)

0

10

20

30

40

NSCFSCF

LMTO-ASA +DMFT PAW + DMFT

L. Pourovskii, B. A., S. Biermann and A. Georges PRB (2007) B.Amadon JPCM 2012


Results: Spectral function of Ce2O3

-6 -4 -2 0 2 4 6 8w (eV)

0

10

20

30

40

XPS+BISSCF

Spectral function of Ce2O3, in LDA+DMFT (Hubbard I) with and

without self-consistency


Results: Structural parameters of Ce2O3

a (A) B0 (Mbar)

Exp(Barnighausen 1985) 3.89 1.11

PAW/LDA+U(AFM)(Da Silva 07) 3.87 1.3

PAW/LDA+U(AFM) 3.85 1.5

PAW/LDA+DMFT (H-I) NSCF 3.76 1.7

PAW/LDA+DMFT (H-I) SCF 3.83 1.6

ASA/LDA+DMFT(H-I) NSCF (Pourovskii 2007) 3.79 1.6

ASA/LDA+DMFT(H-I) SCF (Pourovskii 2007) 3.81 1.6

Lattice parametera and Bulk modulusB0 of Ce2O3.


Results: Converged Electronic Density.

Difference between electronic densities computed in the LDA+U (left)/LDA+DMFT (right) and inLDA for Ce2O3.

Blue (resp. green-red) area corresponds to positive (resp negative) value of the difference.


Implementation of the LDA+DMFT (self-consistency)

a (a.u.) B0 (GPa)

Exp[Jeong 2004] 9.76 19/21

PAW/LDA+U 9.58 32

PAW/LDA+DMFT NSCF (H-I) 9.41 38

PAW/LDA+DMFT SCF (H-I) 9.58 36

ASA/LDA+DMFT NSCF (H-I) 9.28 50

ASA/LDA+DMFT SCF (H-I) 9.31 48

Lattice parametera and Bulk modulusB0 of γ Cerium according to

experimental data and calculations


Conclusion

• DFT+DMFT physical choices• Interactions U, J• Definition of local correlated orbitals: Wannier functions

• Technical choices• Basis for the one electron Green function: KS in Abinit• Solver: Hubbard one

• In progress and/or not yet distributed:• CT-Quantum Monte Carlo (see talk of P. Werner)• Spin-orbit• Improve parallelism

• Details of the Abinit implementation can be found inAmadon B, Lechermann F, Georges A, Jollet F, Wehling T O and Lichtenstein A I 2008 Phys.Rev. B 77 205112Amadon B, 2012 J. Phys.: Condens. Matter 24 075604


Some Review, lectures notes, and introductary articles

• A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg Rev.

Mod. Phys.68, 13 (1996)

• A. Georges cond-mat/0403123 Lectures on the Physics of

Highly Correlated Electron Systems VIII (2004) 3, AIP Conf.

Proc.715.

• G. Kotliar and D. Vollhardt Physics Today Mars 2004

• G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O.

Parcollet, C. A. Marianetti, Rev. Mod. Phys.78, 865 (2006)

• K. Held, Advances in Physics56 829-926 (2007)


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