Constrained RPA method for calculating the Hubbard Ufrom first principles.

Takashi Miyake (AIST Tsukuba Japan)Rei Sakuma (Chiba)Christoph Friedrich (Juelich)

Collaborators

Ferdi AryasetiawanChiba University

Juelich 2011-10-0407

Tutorials (Friday) given byRei Sakuma and Christoph Friedrich

SCREENING

),( tr!" ),( tr!"# ! "=# ),'()'('),( trrrvdrtrVH

$%$

),(),(),( trVtrtrVHtot

!!"! +=

The external perturbation is screened by the electrons.

Proton Z Electron gas

Screening charge (neglect Friedel oscillations)r

Ztr !=),("#

Example:

External perturbation

r

Z!

!/rer

Z ""

SCREENED POTENTIAL(neglect Friedel oscillations)

In semiconductors or insulators the screening, due to the band gap, is not complete

r

Z

!"

=! screening length

SCREENING IN TERMS OFLINEAR RESPONSE FUNCTION

)()()( rtVrtrtVHtot

!!"! +=

!"!#! vVtot

+=

Linear response function

!= )''()'',('')( trtrrtRdtdrrt "#"$)''(

)()'',(

tr

rttrrtR

!"

!#=$

!"#!" 1]1[

$=+= vR

]1[1

vRVtot +==

!

"#

"$

vRvvvW +==!1"

Bare Coulomb interaction

W(1,2) is the screened Coulomb potential at 2 of a point charge located at 1

||

1)21(

21 rrv

!=!

Coulomb potential at 2from a point charge at 1

GREEN FUNCTION

00

00

|ˆ|

|)]2(ˆ)1(ˆˆ[|)2,1(

!!

!!=

+

S

ST

iG

""

!"#

$%&'= ( )3(ˆ)3(3expˆ )*diTS

! is an external perturbation

tHitHierert

ˆˆ)(ˆ)(ˆ !

= ""

The field operator in the interaction picture is independent of !

000

00

0 |ˆ|

|)]1(ˆˆ[|

)2()2(

)1,1(

)2(

)1()2,1(

==

+

!!

!!="==

##

$

%#

%

%#

%

%#

%$

S

STGiR

111 tr=

0! is the exact ground state without the perturbation:

[ ] )2(ˆ)2(ˆˆ)2(

ˆ

00

!!"#

"

##

iSiTS

$=$==

=

[ ])2()1(|)]2(ˆ)1(ˆ[|)2,1( 00 !!!! "##"= TiRThe Kuboformula

00 |)]2(ˆ)1(ˆ[| !""!#= $$Ti !!! "=# ˆˆ

000

ˆ !=! EH

1)0(ˆ ==!S

tHitHierert

ˆˆ)(ˆ)(ˆ !"=" ##Using

and inserting a complete set of eigenstates of H in betweenthe density fluctuation operators and Fourier transforming yieldsthe Lehmann or spectral representation of the response function:

!"#$

%&'

((+

)**)(

++(

)**)=

n nniEE

rnnr

iEE

rnnrrrR

+,

--

+,

--,

0

00

0

00 |)'(ˆ||)(ˆ||)(ˆ||)'(ˆ|);',(

TIME-DEPENDENT HARTREE APPROXIMATION(RANDOM-PHASE APPROXIMATION)

)2(

)1,1(

)2(

)1()2,1(

!"

!

!"

!# +

$==G

iR

GG

GGG

GGG

GG!"

!

!"

!

!"

!

!"

! 11

1101

#### #=$=+$=

[ ] !"

#$%

& '++(=)'+++(=)=!"

#$%

&'((((

*

* ((

+,

+

+,

+

+,

+,, H

HH

VGVhGGVh

ti 11

1

0

1

0

GV

iGGG

iGRH

!"

#$%

& '++(==

(

)*

)

)*

)

)*

)1

1

GV

iGRH

!"

#$%

&+'=()

(1

Time-dependent Hartree(Random-Phase Approximation) !=

"0

#$

#

From the equation of motion

)1,3()2(

)3()3,1(3)1,2()2,1()2,1( ++

!""= GV

GdiGiGRH

#$

#

)2,4()43()3,1(43)2,1()2,1( RvPddPR !+= "

)1,2()2,1()2,1( +!= GiGP

!! "="= )2,4()43(4)2(

)4()43(4

)2(

)3(Rvdvd

VH

#$

#%

#$

#

Polarisation function 1 2

= +R P RPv

)';,'()';',();',( !!!!! rrGrrGdirrP +"= #

!!+"

+""

=unocc

m m

nn

occ

n n

nn

i

rr

i

rrrrG

#$%

&&

#$%

&&%

)'()()'()();',(

**0

! "#

$%&

'

(+)

()=

* *

**

*

**

+++

)'()()'()();',(

**0 rbrbrbrb

rrP

µ!µ!"!!## $$ <>%%=&=nmnmmn

ib ,,,*

!!"#$

%&'

((+(

++(=occ

n

unocc

m nm

mnmn

nm

mnmn

i

rrrr

i

rrrrrrP

)**+

,,,,

)**+

,,,,+

)'()'()()()'()'()()();',(

****0

First-Principles Methods:Parameter-free but insufficientfor strongly correlated systems

Model Approaches:Good for strongly correlated systems

but need parameters

d subspace

r subspace

d subspaced

U !,

U

DMFT

d!

rdr

DMFT

d!+!+!=!

e.g., GW approximation

Full one-particleHilbert space

•Local Density Approximation (LDA) •GW method

•Dynamical Mean-Field Theory(DMFT)

We insist thatthe parameters aredetermined fromfirst-principles

GW+DMFT: PRL 90, 86402 (2003), PRL 102, 176402 (2009)

Typical electronic structure of correlated materials:Partially filled narrow band (3d or 4f) crossing the Fermi level

Slight change of parameters can induce large change in materials properties.E.g., by slight distortion or pressure the ratio of the effective Coulomb interaction U/bandwidth

changes and the materials can undergo phase transitions (metal-insulator).competition between kinetic energy and U.

The main actiontakes place here

E (eV)

many configurations close in energystrong correlationsone-particle description can be problematic

SrVO3 perovskite

La/YTiO3

LDA DOS

!><

"

+# #+=i

ij iiij nnUcc

WH

4

One-band Hubbard model:

The usual approach isto model the narrow bandby a Hubbard model.

Kotliar and Vollhardt, Physics Today 2004

Iron-based high-temperature superconductors

Fe 3d(Se 4p)

(Fe 3d) Se 4p

Fe 3d(As 4p)

(Fe 3d) As 4p O 2p

2

2

)(

)(

NCSCu

BEDTTF!"

32

2

)(

)(

CNCu

BEDTTF!"

Nakamura et al, J. Phys. Soc. Jpn. 78, 083710 (2009)

Exp: metal insulator

BEDT-TTF organic conductors

LDA band structures

Metallic in LDA!

BEDT-TTF=bis(ethylenedithio)tetrathiafulvaene

Zeolites

Nakamura, Koretsune, and Arita, PRB80, 174420 (2009)

Nakamura, Koretsune, and Arita, PRB80, 174420 (2009)

Alkali-cluster-loaded sodalites

Na-sodalite K-sodalite

The Hubbard model

Many-electron Hamiltonian is too complicated to be solved directly.

J. Hubbard, Proc. Roy. Soc. A276, 238 (1963)

What is U and how do we calculate it?

Focus on the correlated subspace

Basic physical idea:The Hubbard U should be obtained without the screening from

the electrons residing in the subspace that defines the Hubbard model.

!! ++++=

all

mmnnR

RmRmmmnnRnRnnRnRRnRn

all

nRRn

ccvccchcH

,',

'','''''',

'', 2

1

!! ++++=

.

,',

'','''''',

.

'', 2

1 correl

mmnnR

RmRmmmnnRnRnnRnRRnRn

correl

nRRn

HubbardccUccchcH

Related works on the Hubbard U

Seminal work on U (constrained LDA):O Gunnarsson, OK Andersen, O Jepsen, J Zaanen, PRB 39, 1708 (1989)VI Anisimov and O Gunnarsson, PRB 43, 7570 (1991)

Improvement on constrained LDAM Cococcioni and S de Gironcoli, PRB 71, 035105 (2005)Nakamura et al (PRB 2005)

Random-Phase Approximation (RPA):M Springer and FA, PRB 57, 4364 (1998)T Kotani, J. Phys.: Condens. Matter 12, 2413 (2000)

Constrained RPA (cRPA)PRB 70, 195104 (2004)PRB 80, 155134 (2009) for entangled bands

,1)]1/([1

)1/(

11

ddr

r

dr

UP

U

PvPv

vPv

vPvP

v

vP

vW

!=

!!

!=

!!=

!=

Constrained RPA (cRPA): A method for calculating the Hubbard U

PRB 70, 195104 (2004)

rdPPP +=

dP

rP

µ

Polarisation:

includes transitions between the d- and r-subspaces (self-screening)

rP

r

d

Fully screened interaction

rP

r

rvP

vU

!=1

Interpret

as the Hubbard U.

!!±+"

=unocc

j ij

jijiocc

i i

rrrrrrP

#$$%

&&&&%

)'()'()()();',(

**

Polarisation function in RPA: vertex Γ = 1 P=-iGG

!!"" ±+#

=unocc

dj ij

jijiocc

di

di

rrrrrrP

$%%&

''''&

)'()'()()();',(

**

Full system

Correlated bands

! "+"= );',();,()()'();',( 221121 ### rrUrrPrrvdrdrrrvrrUr

drPPP !=

Basis independent, can use any bandstructure method.

)'()'();',()()(')(, rrrrUrrdrdrU lkjiklij !!"!!" #=

The matrix elements of U evidently depend on the choice of the orbitals

Maximally localised Wannier orbitals

Minimise the extent of the orbitals

gt2 pgg Oet ++ )( 2model model

xz orbital

Wannier orbitals of SrVO3

• Full matrix U•Energy-dependent U•Onsite and offsite U•U(r,r’;ω) is basis-independent for a givensubspace: Can use any band-structure method

rU

!

1~ is long range,

because metallic screening is absent whencalculating U.

Justification of RPA:U is determined mainly by long-range screening.

Short-range screening is taken care of by the Hubbard model.cRPA is general, can go beyond RPA .

Advantages of cRPA:

1>!

Constrained LDA

Hopping from and tothe 3d orbitals is cut off

Super Cell

Transition metal orrare earth atom

“impurity”

Change the 3d charge on the impurity, keeping the system neutral,do a self-consistent calculation

and calculate the change in the 3d energy level U(3d).

ggtt22

!

allt g !2

allt g !2

ggp etO ,22

!

allt g !2

allO p !2

bareinteraction

Controlling the screening channels:U as a function of (eliminated transitions)

SrVO3

U=3.5 eV

gt2

ge

The O 2p plays a crucial rolein determining U

PRB 74, 125106 (2006) c.f. Solovyev PRB 2007

dP

SrVO3

Miyake et al, unpublished

ReU ImU

Plasmonexcitation

!"

##=

0 22 '

)'(Im''

2)(Re

$$

$$$

%$

UdvU

2

2

)(

)(

NCSCu

BEDTTF!"

32

2

)(

)(

CNCu

BEDTTF!"

Nakamura et al, J. Phys. Soc. Jpn. 78, 083710 (2009)

bare v

2

2

)(

)(

NCSCu

BEDTTF!"

32

2

)(

)(

CNCu

BEDTTF!"

The dielectric constant is anisotropic. U is almost isotropic and long ranged.Nearest-neighbour U/onsite U ~0.45Maximally localised Wannier orbitals of

22 )()( NCSCuBEDTTF!"

Exp: metal insulator

BEDT-TTF organic conductors

)5/(1~ rU

cRPA for entangled bands

Disentangled 3d band structure frommaximally localised Wannier orbitals

(using the procedure ofSouza, Marzari and Vanderbilt)

In many materials the correlated bandsof interest are entangled with other more

extended bands.

Paramagnetic nickel

PRB 80, 155134 (2009), also Sasioglu, Friedrich, and Bluegel PRB 2011

Approximation: The off-diagonalelements are set to zero

U and J of Ni as a function of frequency Static W for the 3d series

Nearest-neighbour U and J

J

Difficult to screen a charge distributionwithout l=0 component

Application to LaFeAsO and FeSe

1111: LaFePO, LaFeAsO, …122: BaFe2As2, …111: LiFeAs, …11: FeSe, FeTe…

Fe 3d(Se 4p)

(Fe 3d) Se 4p

Fe 3d(As 4p)

(Fe 3d) As 4p O 2p

Effective interaction in the cRPAT.Miyake, K.Nakamura, R.Arita and M.Imada, JPSJ (2010)

- strong family dependence in U (the 11 family is substantially more correlated)- t = 0.3-0.4 eV in the d model- strongly orbital dependent in the d model, due to the different extents of the Wannier orbitals

d model

FeSe is probablythe most correlated

U matrix in the d model (in eV)

xy yz 3z2 - r2 xz x2-y2

c.f. K.Nakamura, R.Arita and M.Imada, J.Pys.Soc.Jpn.77, 093711(2008). T.Miyake et al., J.Phys.Soc.Jpn.77 Suppl.C99(2008).

Re and Im U of BaFe2As2 Spectral functions of BaFe2As2

Talk by Philipp Werner on Thursday

Werner et al, arXiv:1107.3128 (2011)

4f band

Summary:We do have a reliable scheme to calculate U from first principles

Tutorials (Friday) given byRei Sakuma and Christoph Friedrich