numerical aspects of many-body theory

Numerical Aspects of Many-Body Theory

• Choice of basis for crystalline solids• Local orbital versus Plane wave

• Plane waves ei(q+G).r

• Complete (in practice for valence space)• No all electron treatment (PAW?)• Large number of functions x.104

• Slow for HF exchange• Straightforward to code (abundance of Dirac delta’s)

• Local orbital (x - Ax)i(y - Ay)j e–(x - A)2

• Incomplete (needs care in choice of basis)• All electron possible and relatively inexpensive• Relatively small number of functions permits large unit cells to be treated• Relatively fast for HF exchange in gapped materials• Difficult to code (lattice sum convergence, exploitation of symmetry, ..)

G q

q+G

IBZ


• Coulomb Energy in real and reciprocal spaces

• Coulomb interaction

• Ewald form of Coulomb interaction

'

)'()('ddECoul rr

rrrr

G q Gq

rrGqq

q

rrqq

rr BZ23

space reciprocal all

23

)').(i(e

2

d)'.(ie

2

d

'

1

dxdk

2

(k)I dx I dx Idx

x'e

2

'

1

R3

R

000

22

rr

rr

r r’

r r’


• Density Matrix Representation of Charge Density

kkk

GkGkGkGG'

GG'GG'

gigikgik

GGkkGkk

kkkk

k

rrgkkrrr

k

kkr

rrr

rr

rrrr

rG GrGkrG k

gk

r GrGk

rk

2*2112

g2

0112

g2

01

0g12

,n,*

',n,,n,

,,n,,,n

,n,n,n,n

nnnn

c c c)(P

)()( .ie )(P)()(P)ρ(

c cc)(P

e )(Pe)e(P)ρ(

)(e )( c)(ψ

ec )(uec)(ψ

)(u )(ψe )(u)(ψ

).'().'().(

.

.

).(

.

tscoefficien expansion orbital local

tscoefficien expansion orbital waveplane

tionrepresenta waveplane

cell gth in site ith on orbital local

tionrepresenta waveplane

function cellperiodic function Bloch

iii

i

ii

i

r


• Coulomb Energy with real space representation of charge density

34g

12403

g2

01

34g

12h4

h3

g2

01Coul

PP)'-()'(- '-

1)()(

PP)- - '() -'('-

1)()( E

rrhrr

g-rr

hrhrrr

g-rr

hhrr

lattice space real on functionperiodic - '-

1

h

r r’

e.convergenc absolute rapid, Overall,

R. limit upperby off cut space real in Part

rapidly. converges FT - smooth space reciprocal in Part integral. Split

dxdk F(k) dx I dx Idx x - '

e2

- '

1

R

R

000

22

hrr

hrr hh

0Gk,kkkk

kGkG

kGkG

kGkGkk

kkkGkk

kkkkGk, q

G q

rrGrrrGr

rrrGqr

rrr

rrGqrrrGqr q

rGq

rrGqr

q

GGGq

-GGGq

G--kGkGq

rGkrGqrGk

rGrGkrk

n,'n''n'nn2Coul

3''n

*'n

3''n

*'n

''n*

'nnn

nnnnn

'n'nnnn, BZ

3

2BZ

3Coul

)'('.ie)'()(.ie)(G

1 E

)(2 cc

)(2 cc

eeed cc)().i(e)(

ec)(uec)e(u)(

)'(').i(-e)'()().i(e)(2

d

)'()').(i(e

)(2

d E

''- ' 0

'''

'''

).''().().'(

. ).(.

iii

iii


• Coulomb Energy with reciprocal space representation of interaction

r r’


• Exchange Energy with real space representation of interaction

• No Ewald transformation possible since h sum is split• 3 lattice sums instead of 2• Absolute convergence neither guaranteed nor rapid

34g

12h4

g2

h3

01Exch PP) - - '() - '(

'-

1)()( E hrgr

rrh -r r r r’

34g

12h4

h3

g2

01Coul PP)- - '() - '(

'-

1)()( E hrhr

rrg-rr r r’

• Exchange Energy with reciprocal space representation of interaction

• q + G lattice sum instead of just G• Absolute convergence not guaranteed nor rapid

Gq,k,

Gq

rrGqrGq

rrGqrrrGqrGq

q

n,

2

qnknk2

nknk'nk'nk2k,n, BZ

3Exch

)(.ie)(1

)'(').i(-e)'()().i(e)(1

2

d E


r r’

Quasiparticle energies in solid Ne and Ar

)()(),,()()(ˆ

),(),(),(

)(),(),(),(),(

)(),()()(ˆ

)(),()(),()()(ˆ

1221211

212121

234331432121

212111

23313212111

QPmm

QPm

QPm

o

oo

o

dH

GG

GGddGG

GVht

i

GdGVht

i

E

11

1

1

,

,

h(1) One-body HamiltonianV(1) Hartree potential(1,2) Self energyGo Non-interacting GFG Interacting GF

H(1) Non-interacting Hamiltonianm

QP Quasiparticle amplitudem Quasiparticle energyQuasiparticle equation

Dyson equation

• Dyson and Quasiparticle equations F125

RPA Polarisability and Dielectric Function

tscoefficien Expansion

functions lorthonorma inof Expansion

)(),)F(( ddF

Fδ δ F

)()( d)()( dF)(),)F(( dd

F )()(F),F(

**kk

kjji,

kiij

**j

ji,i

*kij

**k

*j

ji,iij

221121

222111221121

2121

• Projection of functions onto orthogonal bases

notationDirac in Same FF

)()(FFFφ),F(

jiij

*jiijjijijjii

2121


25 Eq Louie and Hybertsen Compare

BZ. first the to restricted is function. cellperiodic

a is as functions Bloch the

expandingby derived be can by differing points k to nrestrictio The

q

qkk q

221 121q

1221122121

GGk

GGkk

qkk

qkqkk

k

rGrGkrk

rGqrGq

2Gq1Gq

ecu eceuψ

) ,(

...)-(

ψeψ ψeψ

...)-(

)(ψ)e(ψ )(ψ)e(ψdd),(

)-(

)()ψ()ψ()ψ(ψ

)-(

)()ψ()ψ()ψ(ψ),,(

.).(.

').'().(

).'().(

nnn

mn

nmmnBZunocc,occ,

n,m,

mn

n*m

*nm

unoccocc,

nm,

RoGG'

mn

*nn

*mm

mn

*nn

*mm

unoccocc,

nm,

Ro

iii

ii

ii

i

i

ii

• Projection of o onto plane wave basis


function dielectric )(Hermitian dSymmetrise

space! in tionmultiplica becomes space in nConvolutio

function dielectric Hermitian-non

'

4)(v~

),()(v~δ2),(ε~

'

δ 44 )(v

e)'(e'dd2

e),',(εe'dd),(ε

),()(vδ2),(ε

),',''( )''(v''d - )'( ),',(ε

'

R'''o'''

3RPA'

'2'

'3

RPARPA'

R'o''

3RPA'

Ro

RPA

'.'.

'.'.

GqGqq

qqq

GqGqGqq

rrrr

rrrrq

qr

qqq

rrrrrrrrr

GG

GGGGGGGG

GGGG

GG

GG

GGGGGGGG

rGqrGq

rGqrGq

ii

ii

• Projection of o onto plane wave basis

• Dielectric bandstructure () expanded in eigenfunctions of static inverse dielectric function

Plasmon pole approximation for -1(q,)

ii

z

ii

iii

i

ii

i

ii

i

qq

qq

q

q

q

qqq

qqq

GGGG

GGGG'

11

21)(

)()(

1)(),(~

)()()0,(~

1

*1'

Pole strength zq and plasmon frequency q fitted at = 0 and several imaginary frequencies

Baldereschi and Tossatti, Sol. St. Commun. (1979)

Ar 15v Ar 1c

Energy dependence of self-energies in Ar

Nicastro, Galamic-Mulaomerovic and Patterson, J. Phys. Cond. Matt. (2001)

• Dielectric bandstructure and self energy

Self-energy operator matrix elements

Rohlfing, Kruger and Pollmann, Phys. Rev. B (1993)

HF exchange - looks like dynamically screened HFT

• Self-energy calculated from dielectric bandstructure

i

n

iiii

n

iiii

nnnnn

nn

CBnEE

z

VBnEE

z

eee

E

LDA

LDA

ii

' 1

2

' 1

21

4

)(

'

*

'

*

'''

2

qqk

q-q-

qqk

q-q-

kqkqkk

kk

qG'-qG-

qG'-qG-

qGG'

).rG'(qG).r(q

G'q

1

Gq

1

fcc Ne DFT & GW bandstructures

Ne DFT

PP

GW

PP

DFT

AE

GW

AE

Expt.

15 -13.14 -19.37 -13.18 -19.10 -20.21

1c -1.35 0.86 -1.42 1.03 1.3

Wv 0.71 0.93 0.79 0.93 1.3

Eg 11.79 20.23 11.76 20.13 21.51

Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995)DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005)

Ne

DFTmxc

QPm

DFTm

DFTm

QPm VEEE

fcc Ar DFT & GW bandstructures

Ar DFT

PP

GW

PP

DFT

AE

GW

AE

Expt.

15 -9.74 -13.15 -10.27 -13.00 -13.75

1c -0.60 0.72 -0.76 0.81 0.4

Wv 1.35 1.73 1.32 1.85 1.7

Eg 9.14 13.87 9.51 13.81 14.15

Expt. Runne and Zimmerer, Nucl. Instrum. Methods Phys. Res. B (1995)DFT/GW Galamic-Mulaomerovic and Patterson, Phys. Rev. B (2005)

Ar

• Bethe-Salpeter Equation (F 558)

ioG(1,2)G(2,1) i.e. dressed Green’s function product

• K* proper part of electron/hole scattering kernel• o is a special case of the particle-hole Green’s function

• 4-index function

• (1,1,2,2) = o(1,1,2,2) + o(1,1,3,4) K*(3,4,5,6)(5,6,2,2)

Bethe-Salpeter Equation

K*

= +

oo

oo

2

)(ψ̂)(ψ̂)(ψ̂)(ψ̂),,,(G

12341234

Ti

K*

1

4 6

53

2

• Electron-hole scattering kernel K*


babaV kjiijkl

ℓ

k

j

i

bbaaV kjiijkl

ℓ

ik

j

bbabV kjiijkl

ℓ

i

k

j

bbabV kjiijkl

j

ik

ℓ

Time flows from left to right here

• Electron-hole scattering Lego

• Electron-hole pair scattering (summed in BSE)

• Electron-hole scattering (summed in screened electron-hole interaction)


Can’t have dangling ends

• Electron-hole scattering kernel K*

• K*(3,4,5,6) =

• Iteration of the Bethe-Salpeter equation leads to a series of the form

• = o + oK*o + oK*oK*o + oK*oK*oK*o + …

• Generates sums of ring and screened ladder diagrams


3 5

+ + + …

4 6

• Bethe-Salpeter Equation: Solution as an eigenvalue problem

• = o + o K* • (1 - o K* ) = o • = (1 - o K* ) -1 o • = (1 - o K* )-1 ( o

-1)-1

• = (o-1 - K* )-1

• -1= o-1 - K*


.(excitons) pairs hole-electron bound produces it ninteractio

hole-electron strong For strengths. oscillator and energies transition

particle single modifies energies. transition particle singlesimply are

seigenvalue of absence In . elements diagonal on/off and diagonal the

on energies transition particle single withequation eigenvalueMatrix

*

**

,*

,

K

KK

0c K

,

,

Look for zeros of -1 equivalent to poles of -1= o

-1 - K* = 0 an eigenvalue equation

• Bethe-Salpeter Equation: Expansion of functions of 2 or 4 variables

• Need all 4 arguments of o


lkjiijkl

i*k

*jiijkllkjilkji

lklkjiji

jiij

*jiijjijijjii

φφGφφG

)(φ)(φ)(φ)(φGφφφφφφGφφ

φφφφGφφφφ),,,G(

φFφF

)(φ)(φFφφφFφφφFφφ),F(

4321

4321

2121

1,t1 2,t2(1,2)

1,t1

2,t2

(1,2,3,4)

3,t3

4,t4

• Bethe-Salpeter Equation: Solution as an eigenvalue problem• o and o

-1 are diagonal in the basis of single particle states


)-(

...-)()ψ(ψ)-(

)()ψ()ψ()ψ(ψ)()ψ(ψdddd)(

...)-(

)()ψ()ψ()ψ(ψ ),,,,(

...)-(

)()ψ()ψ()ψ(ψ),,(

,

mn

mnmn

*

mn

*nn

*mm*R

,o

mn

*nn

*mm

unoccocc,

nm,

Ro

mn

*nn

*mm

unoccocc,

nm,

Ro

ii

i

i

i

411324

23 4321

13244321

122121

)()ψ(ψ * 23

1,t1 2,t2(1,2)

1,t1

2,t2

(1,2,3,4)

3,t3

4,t4

• Bethe-Salpeter Equation: Solution as an eigenvalue problem• K* in the basis of single particle states


)()ψ(ψ),()()ψ(ψdd )()ψ(ψ),,()()ψ(ψdd

)()ψ(ψ),(),(),()()ψ(ψdddd

)()ψ(ψ),,(),(),()()ψ(ψdddd

)()ψ()ψ,,,,()()ψ(ψdddd)(

),(),(),(),,(),(),( ),,,,(

****

**

**

***,

*

*

112122 21212121 21

4121324123 4321

4121423123 4321

41432123 4321

2132412142314321

vW

v

W

KK

vWK

)()ψ(ψ * 23

Direct term -W(1,2,)

1,t1 3,t3

2,t24,t4

1,t1 3,t3

2,t24,t4

Exchange term (singlet excitons only) v(1,2)

• Bethe-Salpeter Equation: Solution as an eigenvalue problem


stransition single to ingcorrespond

indices (compound) single are and

energy transition particle single a is

mn

,1-R

,o

,R,o

-

...)(

...- )(

i

Ne

)(ψ* 1

)(ψ 2

)(ψ* 2

)(ψ 1

v(q)

W(q)

v(q)

• Bethe-Salpeter Equation: numerical calculation of matrix elements


Direct term -W(1,2,)

1,t1 3,t3

2,t24,t4

1,t1 3,t3

2,t24,t4

Exchange term (singlet excitons only) v(1,2)

)()(),()()( c'cvv' 222111 k'kkk' W )()(),(v)()( c'v'vc 222111 k'k'kk

kkqr)G'qrG)q

GG,q,

GG,kk kkkk

GqGq

q',

.i(-.i(

'

1'

2

''' 'c'ecve'v''

)0,(e4

cvvcK

k'k'kk

TrGrGkk

kpkkpkqq kkkk

v'c'vc

.i-.i

0G2

2

'c'v',vc EE

'c''v

EE

vcˆˆ'v'e'c'vec

G

1e4 x 2

K

Excitons in solid Ne

Expt. Runne and Zimmerer Nucl. Instrum. Methods Phys. Res. B (1995). DFT/GW Galamic-Mulaomerovic and Patterson Phys. Rev. B (2005).

Singlet Ne energy levels, band gaps, binding energies (eV)

n En BSE En EXPT EB BSE EB EXPT

1 17.25 17.36 4.44 4.22

2 19.90 20.25 1.79 1.33

3 20.55 20.94 1.14 0.64

4 20.95 21.19 0.74 0.39

5 21.15 21.32 0.54 0.26

Eg

21.69

Eg

21.58

LT

0.30

LT

0.25

Excitons in solid Ar

Expt. Runne and Zimmerer Nucl. Instrum. Methods Phys. Res. B (1995). GW/BSE Galamic-Mulaomerovic and Patterson Phys. Rev. B (2005).

Singlet Ar energy levels, band gaps, binding energies (eV)

n En BSE En EXPT EB BSE EB EXPT

1 11.60 12.10 2.09 2.06

2 13.05 13.58 0.64 0.58

3 13.45 13.90 0.24 0.26

Eg

13.69

Eg

14.25

LT

0.36

LT

0.15

numerical aspects of many-body theory

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