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2 The Unitary Model: Green's Functions and Action

2 The Unitary Model: Green's Functions and Action

Tải bản đầy đủ - 0trang

264



22 Diffusive Model



We consider a tight-binding model on a lattice with N lattice sites. For simplicity,

we assume cubic symmetry of the lattice and of the model. Moreover, we assume a

unit volume per lattice site. Otherwise a number of factors of the volume per lattice

site would have to be carried over. The Hamiltonian of the model reads

X

jri.tr r0 C fr;r0 /hr0 j:

(22.19)

HD

r;r0



We assume a Gaussian distribution of f ,

fr;r0 D 0;

where Mr



r0



D Mr0



r



fr;r0 fr00 ;r000 D Mr



r0 ır;r000 ır0 ;r00 ;



(22.20)



0. Then products of Green’s functions

G.r; r0 ; z/ D hrj



1

z



H



jr0 i



(22.21)



can be written as

m

Y



G.ri ; ri0 ; zi / D



iD1



Z

ŒD S



m

Y

.i/

.i/

.sbi Sb .ri /Sb .ri0 // exp. S1 .S; 0//;



(22.22)



iD1



Z



!

à Z

m Â

. j/

. j/

Y

sbi N Y d
. j/

d Sf .r/d Sf .r/ ;

ŒD S D

sfi

iD1

j;r

(22.23)

Ž



S1 .S; 0/ D strŒS .sz ˝ 1N s ˝ .t C f //S

X

D

s i S.i/ .r/.zi ır;r0 tr r0 fr;r0 /S.i/.r0 /;



(22.24)

(22.25)



i; ;r;r0



where, as before,



assumes the values b D 0 and f D 1. We note



S.r/ 2 M .2; 2; 1; 0/;



S 2 M .2N; 2N; 1; 0/;



s D diag.sb1 ; sb2 ; sf1 ; sf2 / 2 M .2; 2/;



t; f 2 M .N; 0/



(22.26)



z D diag.z1 ; z2 ; z1 ; z2 / 2 M .2; 2/:

(22.27)



We add a source term

S1 .S; A/ D S1 .S; 0/



Xp

. j/

.i/

sbi sbj Aij .r; r0 /Sb .r0 /Sb .r/;



(22.28)



i;j;r;r0



which allows the calculation of the Green’s functions from the partition function

Z

Z.A/ D



ŒD S exp. S1 .S; A//:



(22.29)



22.2 The Unitary Model: Green’s Functions and Action

m

Y



G.ri ; ri0 ; zi / D



iD1



m

Y

iD1



265



@

Z.A/jAD0 :

@Aii .r; r0 /



(22.30)



The average over the random terms f yields

exp. str.fSSŽ s// D exp. 12

D exp. 12

T.r/ D



X



Mr



r0



SŽ .r/sS.r0 / SŽ .r0 /sS.r//



Mr



r0



str.T.r/T.r0 ///;



r;r0



X

r;r0



p

p

sS.r/SŽ .r/ s 2 M .2; 2/:



(22.31)



We use that SŽ .r/sS.r0 / and SŽ .r0 /sS.r/ are in M .1; 0/. The Hubbard-Stratonovich

transformation yields

Z

exp. str.fSSŽ s//



D



ŒD R exp.



1

2



X



str.R.r/wr



r0 R.r



0



// C



X



r;r0



str.R.r/T.r///;



r



R.r/ 2 M .2; 2/;



(22.32)



where w is the inverse of M. We assume

Pthat the matrices M and thus, also w, is

positive. This is surely the case if M0 > r6D0 jMr j.

Thus, we obtain

X

p

p

S2 D 12

str.R.r/wr r0 R.r0 // str.SŽ s.R z C t C A/ sS/: (22.33)

r;r0



R 2 M .2N; 2N/ consists of the block matrices R.r/. s and z are multiplied by 1N

and t by 12;2 in this rearrangement. Integration over S yields

S3 .R; A/ D



1

2



X



str.R.r/wr



r0 R.r



0



// C ln sdet.R



z C t C A/:



(22.34)



r;r0



This action allows us to give expressions for the one- and two-particle Green’s

functions. Since

ln sdet.R



z C t C A/ D ln sdet.R



z C t/ C ln sdet.1



G A/;



G WD



1

;

z R t

(22.35)



we expand in A by use of (10.36),

S3 .R; A/ D S3 .R; 0/ C str ln.1

D S3 .R; 0/



str .G A/



G A/

1

2



str..G A/2 / C : : :



(22.36)



266



22 Diffusive Model



Together with (22.30), we obtain the averaged Green’s functions

G1 .r; r0 ; zi / WD G.r; r0 ; zi / D Gibr;ibr0 ;



(22.37)



G2 .r1 ; r10 ; z1 I r2 ; r20 ; z2 / WD G.r1 ; r10 ; z1 /G.r2 ; r20 ; z2 / D G1br1 ;1br10 G2br2 ;2br20

CG1br1 ;2br20 G2br2 ;1br10 ;



(22.38)



One contribution of (22.38) comes from the square of the second term in (22.36),

the other from the third term.



22.3 Saddle Point and First Order

The saddle point of the action S and the corresponding Green’s functions are

determined.

We determine the saddle point R.0/ by setting

R D R.0/ C ıR



(22.39)



and expand up to first order in ıR,

S3 .R; 0/ D



1

2



X



C



str.R.0/ wr



r;r0



X



str.ıR.r/



r0 R



X



.0/



/ C ln sdet.R.0/



wr



r0 R



.0/



z C t/



/



r0



r



C str ıR.R.0/



z C t/



1



C O..ıR/2 /:



(22.40)



Fourier Transform We introduce the Fourier transform, where we usually indicate

the quantities depending on wave-vectors by a hat. We consider a periodic lattice

with N lattice points. The transformation can be easily performed by use of the

matrix elements

p

hrjqi D eiqr = N;



hqjri D e



iqr



p

= N



(22.41)



and the completeness relations

X

q



jqihqj D 1;



X

r



jrihrj D 1;



(22.42)



22.3 Saddle Point and First Order



267



where q is confined to the first Brillouin zone. Then t may be expressed as

tD



X



jritr



r0 hr



0



jD



r;r0



X



jqihqjritr



r0 hr



0



1 X

jqie

N 0 0



jq0 ihq0 j D



r;r0 ;q;q0



iqr



tr



r0 e



iq0 r0



hq0 j:



r;r ;q;q



(22.43)

r0 D rQ ,



Setting r



X



e



iq.r0 CQr/Ciq0 r0



D Nıq;q0 e



iqQr



;



(22.44)



tr ;



(22.45)



r0



yields

tD



X



jqiOtq hqj with Otq D



X



q



e



iqr



r



where rQ has been replaced by r.

The corresponding transformation yields, for M amd w,

O q wO q D 1:

M



(22.46)



On the other hand, (22.39) can be written

R.0/ D



X



jriR.0/ hrj D



X



r



ıR D



X



jqiR.0/ hqj;



(22.47)



q



jriıR.r/hrj D



X



jqihqjriıR.r/hrjq0 ihq0 j D



r;q;q0



r



1 X

0

jqiei.q

N

0



q/r



ıR.r/hq0 j:



r;q;q



(22.48)

Then

ıR D



X



jqiı RO q



q0 hq



q;q0



0



j with ı RO q



q0



D



1 X i.q0

e

N r



q/r



ıR.r/:



(22.49)



Thermodynamic Limit and Continuum Limit The sum over the wave-vectors q

is in the thermodynamic limit replaced by the integral

Z

1 X

D WD .2 /

N!1 N

q

q

lim



Z

d



d dq



(22.50)



268



22 Diffusive Model



In our explicit calculations, we assume the volume per lattice site v to be unity. If

one wishes to perform the continuum limit, then one replaces

Z

X

v

$ d dr

(22.51)

r



and the periodicity volume V D Nv has to be introduced. Then it has to be checked

to which extent the various quantities have to be multiplied by powers of v.

Now we rewrite S3 (22.40), up to first order in ıR,

S3 .R; 0/ D S3 .R.0/ ; 0/ C str



X

r



!

X

1

1

.ıR.r/ wO 0 R.0/ C

/

N q R.0/ z C Otq

(22.52)



Thus, the saddle-point equation reads

w

O 0 R.0/ C



1

1 X

D 0:

.0/

N q R

z C Otq



(22.53)



.0/



Again there are solutions, Ri ;i0 0 .r/ D Rdi ıii0 ı 0 , diagonal in i and i0 0 . The eqs.

for the diagonal solutions have N 1 real solutions between the z Otq for real z. The

other two solutions, which are relevant for the saddle point, are complex conjugate,

if z lies inside the band. Otherwise they are real too. As before (Sects. 4 and 21), the

.0/

imaginary part of Ri has to be opposite to that of zi .

One-Particle Green’s Function The one-particle Green’s function (22.37) is given

within the saddle point approximation, R D R.0/ , by

.0/



G1 .r; r0 ; zi / D hrj.zi



t/ 1 jr0 i;



Rdi



(22.54)



which reads

.0/



G1 .r; r0 ; zi / D



1 X iq.r

e

N q



r0 /



O .0/ .q; zi /;

G



O .0/ .q; zi / D

G



zi



1

Rdi



Otq



;:

(22.55)



and one obtains from (22.53)

.0/



G1 .r; r; zi / D wO .0/ Rdi ;



.E/ D



wO 0



=Rdi with zi D E



i0



(22.56)



with the averaged density of states per energy and lattice site. .E/ is normalized

so that

Z



C1

1



d E .E/ D 1:



(22.57)



22.4 Second Order and Fluctuations



269



The exact one-particle Green’s function, as well as the approximation given here,

obeys the symmetry relations

G.r; r0 ; z/ D G .r0 ; r; z /;



O

O .q; z /:

G.q;

z/ D G



(22.58)



Two-Particle Green’s Function The two-particle Green’s function (22.38) factorizes within this approximation,

G2 .r1 ; r10 ; z1 I r2 ; r20 ; z2 / D G.0/ .r1 ; r10 ; z1 /G.0/ .r2 ; r20 ; z2 /:



(22.59)



Whereas the result for the one-particle Green’s function is quite reasonable, we will

see in the next section that fluctuations yield an important contribution to the twoparticle Green’s function.



22.4 Second Order and Fluctuations

Fluctuations around the saddle-point manifold are included in second order.

Leading contributions to the diffusion constant and the conductivity are obtained.

Expand G (22.35), around R.0/ ,

G D G.0/ C G.0/ ıRG.0/ C G.0/ ıRG.0/ ıRG.0/ C : : : ;



(22.60)



O .0/ .q0 /

O .0/ .q/ıq;q0 C G

O .0/ .q/ı RO q q0 G

hqjG jq0 i D G

X

O .0/ .q/ı RO q q00 G

O .0/ .q00 /ı RO q00 q0 G

O .0/ .q0 / C : : :

C

G



(22.61)



i.e.



q00



Then we obtain, up to second order in ıR,

O .0/ .q; zi / C .G

O .0/ .q; zi //2

O 1 .q; zi / D G

G



X



O .0/ .q0 ; zj /ı RO q

G



q0 ;ib;j



ı RO q0



q;j ;ib :



q0 j



(22.62)

With

G2 .r1 ; r10 ; z1 I r2 ; r20 ; z2 / D



1

N2



X

q1 q01 ;q2 q02



ei.q1 r1



q01 r10 Cq2 r2 q02 r20 /



O 2 .q1 ; q01 ; z1 I q2 ; q02 ; z2 /;

G

(22.63)



270



22 Diffusive Model



we obtain, starting from (22.38),

O 2 .q1 ; q01 ; z1 I q2 ; q02 ; z2 / D hq1 1bjG jq0 1bihq22bjG jq0 2bi

G

1

2



(22.64)



Chq1 1bjG jq02 2bihq22bjG jq01 1bi

O 1 .q1 ; z1 /ıq ;q0 G

O 1 .q2 ; z2 /

D ıq1 ;q01 G

2 2



(22.65)



O .0/ .q01 ; z1 /G

O .0/ .q2 ; z2 /G

O .0/ .q02 ; z2 /

O .0/ .q1 ; z1 /G

CG

.ı RO q1



O



q01 ;1b;1b ı Rq2 q02 ;2b;2b



C ı RO q1



O



q02 ;1b;2b ı Rq2 q01 ;2b;1b /:



We have omitted the fluctuation terms in first order, since they vanish at the saddlepoint. The action yields, up to second order in ıR,

S3 .R; 0/ D S3 .R.0/ ; 0/ C

D 12 N



X

q;i ;j



1

2



X



str.ıR.r/wr



r0 ıR.r



˘O q .zj ; zi //ı RO



q;i ;j



0



1

2



ı RO q;j



0 ;i



r;r0



. / .w

Oq



0



str.G.0/ ıRG.0/ ıR/



//



;



(22.66)



0



with

1 X O .0/

O .0/ .q0 ; zi /:

˘O q .zi ; zj / D

G .q C q0 ; zj /G

N 0



(22.67)



q



Since Rq;i

ı RO



;j



0



DR



q1 ;i1 1 ;j1 10



q;j 0 ;i



ı RO q2 ;j2



, we obtain, by means of (13.33),



0

2 ;i2 2



D ıq1 ;q2 ıi1 ;i2 ıj1 ;j2 ı



1;







0 0

1; 2



. 1/

N



0

1



Oq1 .zj1 ; zi1 /



(22.68)



with

Oq .zj ; zi / WD



wO q



1

:

O

˘q .zj ; zi /



(22.69)



Thus, the two contributions for D 0; 1 in the one-particle Green’s function (22.62)

O RO in (22.65) contributes to the twocancel. Only the second fluctuation term ı Rı

particle Green’s function

O 1 .q1 ; z1 /ıq ;q0 G

O .q2 ; z2 / C

O 2 .q1 ; q01 ; z1 I q2 ; q02 ; z2 / D ıq ;q0 G

G

1 1

2 2 1

Oq



1



O .0/



q02 .z1 ; z2 /G



O .0/



.q1 ; z1 /G



O .0/ .q02 ; z2 /:

O .0/ .q2 ; z2 /G

G



ıq1 Cq2



q01 q02 ;0



N



.q01 ; z1 /

(22.70)



22.4 Second Order and Fluctuations



271



This may be rewritten as

G2 .r1 ; r10 ; z1 I r2 ; r20 ; z2 / D G1 .r1 ; r10 ; z1 /G1 .r2 ; r20 ; z2 / C



X



G1 .r1 ; rN ; z1 /



rN;Nr0



G1 .Nr ; r20 ; z2 /G1 .r2 ; rN0 ; z2 /G1 .Nr0 ; r10 ; z1 / .Nr0



rN ; z1 ; z1 /

(22.71)



with

.Nr0



rN ; z1 ; z2 / WD



1 X iq.Nr0

e

N q



rN/



Oq .z1 ; z2 /:



(22.72)



For q D 0, one can perform a partial fraction decomposition and obtain by means

of (22.53),

1X

˘O 0 .zj ; zi / D

N 0 Rdi

q



1

zi C Otq0 Rdj



wO 0 .Rdj Rdi /

1

D d

zj C Otq0

Rj zj Rdi C zi



(22.73)



and thus,

wO 0



˘O 0 .zj ; zi / D



Rdj



wO 0 .zi zj /

:

zj Rdi C zi



(22.74)



Expression (22.70) for the two-particle Green’s function, with O and ˘O given

by (22.69) and (22.67), is equivalent to the coherent-potential approximation [258]

and the n D 1-limit of the n-orbital model. [269].



22.4.1 Diffusion

In this subsection, we consider the massive and the diffusion modes. From

Eq. (22.74), it is obvious that for q D 0 and ! D z1 z2 , =! > 0, as in (21.44), one

obtains, as ! ! 0,

w

O0



˘O 0 .E C 12 !; E



1

!/

2



D



i! wO 20

;

2 .E/



(22.75)



where we have used (22.53), (22.56). Thus, wO 0 ˘O 0 vanishes in this limit and the

modes for s2 D s1 are soft. In contrast, if the imaginary parts of z1 and z2 have

the same sign, s1 D s2 , then Rd1 Rd2 vanishes proportional to !, so that wO 0 ˘O 0

remains finite and the modes are massive. Let us consider the soft modes further.



272



22 Diffusive Model



From Eqs. (22.55), (22.67), one deduces

X

1 X O .0/ 0

O .0/ .q C q0 ; zj / D

˘O q .zj ; zi / D

eiqr G.0/ .r; 0; zi /G.0/ .0; r; zj /:

G .q ; zi /G

N 0

r

q



(22.76)

If the hopping matrix elements are real tr D tr , then Otq D Ot q . Then the linear term

in an expansion of (22.76) in powers of q vanishes. We expand

Oq D

M



X



e



iqr



Mr D



r



X



1

2



Mr



X

O 0 .1

.qr/2 Mr C : : : D M



r



r



q2 2

r / C :::;

2d M

(22.77)



where we assume cubic symmetry and define the average range rM of M

2

rM



P 2

r Mr

WD Pr

:

r Mr



(22.78)



Then

O q 1 D wO 0 .1 C

wO q D M



q2 2

r / C :::;

2d M



X



r2 wr D



2

wO 0 rM

:



(22.79)



r



One obtains, for small ! and q,

wO q



˘O q .E C 12 !; E



1

2 !/



D



2

i! wO 20

wO 0 q2 rM

q2 X 2 .0/

C

C

r G .r; 0; E C i0/

2 .E/

2d

2d r



G.0/ .0; r; E



i0/:



(22.80)



Thus, in the hydrodynamic limit, one obtains the diffusion pole

Oq .E C 1 !; E

2



1

!/

2



D



.0/



D



D



1

wO q

dw

O0



˘O q .E C !2 ; E

2

rM



!

/

2



D



2 =wO 20

;

i! C D.0/ q2



1 X 2 .0/

C

r G .r; 0; E C i0/G.0/ .0; r; E

wO 0 r



(22.81)

!

i0/ :



(22.82)



22.4 Second Order and Fluctuations



273



22.4.2 Conductivity

Due to Kubo [156] and Greenwood [100], the frequency dependent conductivity at

temperature T can be expressed as

T .!/ D



1

!



Z



1

2 !/



d E.fT .E



fT .E C 12 !// .!; E/



(22.83)



with

.!; E/ D



X

e2

!2

r2 S2 .0; r; E C 12 !; E

2d

r



1

!/

2



(22.84)



E2 /jri;



(22.85)



and the averaged two-particle spectral function

S2 .r; r0 ; E1 ; E2 / D hrjı.H



E1 /jr0 ihr0 jı.H



where d is the dimension of the lattice and fT .E/ the Fermi function. In the zerotemperature limit, T D 0, one obtains

0 .!/ D



1

!



Z



C!=2

!=2



d E .!; E/;



(22.86)



which in the dc-limit, ! D 0, yields 0 .0/ D .0; /, with the chemical potential .

We evaluate the two-particle spectral function S2 by means of

hrjı.H



E/jr0 i D



1

G.r; r0 ; E

2 i



i0/



G.r; r0 ; E C i0/ :



(22.87)



From (22.63), we obtain

G2 .r; r0 ; z1 I r0 ; r; z2 / D



1 X iq.r

e

N q



r0 /



KO q .z1 ; z2 /;



1 X O

KO q .z1 ; z2 / D

G2 .q1 ; q2 C q; z1 I q2 ; q1

N q ;q

1



(22.88)

q; z2 /



2



D ˘O q .z1 ; z2 /.1 C ˘O q .z1 ; z2 / Oq .z1 ; z2 //

D wO q ˘O q .z1 ; z2 / Oq .z1 ; z2 /:



(22.89)



274



22 Diffusive Model



For small !, only the averages of products of Green’s functions with opposite

imaginary parts contribute. Then one obtains

SO 2 .q/ D w

O q Œ.˘O q Oq /.E C 12 ! Ci0; E



i0/C.˘O q Oq /.E



1

2!



1

1

2 ! Ci0; E C 2 !



i0/;

(22.90)



where we may replace wO q ˘O q D w

O 20 and obtain

2 =w

O 20

2 =w

O 20

4 D.0/ q2

C

/D 2

:

.0/

2

.0/

2

i! C D q

Ci! C D q

! C .D.0/ q2 /2

X

X

SO 2 .q/ D 4 2

eiqr S2 .0; r; : : :/ D 4 2

S2 .0; r; : : :/



SO 2 .q/ D wO 20 .



r

2



(22.91)



r



4

q2

2d



X



r2 S2 .0; r; : : :/ C : : :



(22.92)



r



Hence

X



r2 S2 .0; r; : : :/ D



r



2d D.0/

:

!2



(22.93)



Thus, we obtain the dc-conductivity

.0; E/ D e2 D.0/ :



(22.94)



22.5 Nonlinear -Model

The model is reduced to the saddle-point manifold. This yields the nonlinear model. Expressions for the averaged Green’s functions are given.

As for the random matrix model of Chap. 21, one may introduce the same parametrization for the matrices R and integrate over the massive modes. Then one is

left with a model of interacting matrices Q.r/. Thus, in generalization of (21.57)–

(21.59), we choose

R.r/ D T.r/P.r/T



1



T.r/sT Ž .r/ D s



.r/;



(22.95)



and the saddle-point P.0/ D R.0/ of R.r/,

P.0/ D


i



.0/



wO 0



:



(22.96)



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