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5 Isotropic ϕ2σ-Theories with Negative Number of Components

5 Isotropic ϕ2σ-Theories with Negative Number of Components

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222



20 Dimensional Reduction



The exponent Á carries a factor .n=2 C

d D 2 =.

1/

[256, 266, 267].



1/Š.n=2/Š at order



2



with dimension



Problems

20.1 Set

2



r D



n

X

iD1



x2i



Cc



r

X



Âi Âi ;



iD1



n

r

X

X

@2

@2

0

4ss D

C

c

@Âi @Âi

@x2i

iD1

iD1



Which condition have c; c0 to obey such that 4ss f .r2 / is a function of r2 only?

20.2 Derive Eq. (20.79) from (20.78).



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Chapter 21



Random Matrix Theory



Abstract In this chapter the Gaussian random matrix ensembles are investigated.

We determine their Green’s functions and show that for small energy differences a

soft mode appears. As a consequence, the non-linear sigma-model is introduced and

the level correlations are determined.



21.1 Green’s Functions

Green’s functions and their products are introduced.

The Green’s function between states j˛i and jˇi is defined as

G.˛; ˇ; z/ D h˛j



1

z



H



jˇi



(21.1)



It is obtained from the time-integrals (<Á > 0)

Z



0



i



d te.i!CÁ



iH/t



D



1



Z



C1



i



d te.i!



Á iH/t



0



D



1





H



1

! C iÁ



H



!



;



(21.2)



:



(21.3)



The upper Green’s function is called advanced (=z < 0) and the lower one retarded

(=z > 0). The density of states per orbital is obtained from the difference of both

Green’s functions

.˛; E/ D lim .G.˛; ˛; E

Á!C0



iÁ/



G.˛; ˛; E C iÁ//=.2 i/:



(21.4)



Q

In the following, we consider averaged products m

iD1 G.˛i ; ˇi ; zi / of Green’s

functions of the random matrix model of Sect. 4.4. Since the distribution of matrixelements (4.13) is invariant under unitary transformations of the matrix-elements

P.f / D P.U Ž fU/, only averaged products of Green’s functions differ from zero,

if ˛i and ˇj agree pairwise. Thus, the only averaged one-particle Green’s function

different from 0 is G.˛; ˛; z/ and the only two-particle Green’s functions different

© Springer-Verlag Berlin Heidelberg 2016

F. Wegner, Supermathematics and its Applications in Statistical Physics,

Lecture Notes in Physics 920, DOI 10.1007/978-3-662-49170-6_21



227



228



21 Random Matrix Theory



from zero are the averages of G.˛; ˛; z/G.ˇ; ˇ; z0 / and G.˛; ˇ; z/G.ˇ; ˛; z0 /. Since

the distribution of the matrix-elements is invariant under permutation of the ˛s, the

averaged one- and two-particle Green’s functions can be expressed in terms of the

one-particle Green’s function G and two-particle Green’s functions K and K 0

G.˛; ˇ; z/ D ı˛;ˇ G.z/;



(21.5)



G.˛; ˇ; z1 /G. ; ı; z2 / D ı˛ˇ ı ı K.z1 ; z2 / C ı˛ı ı ˇ K 0 .z1 ; z2 /:



(21.6)



The correlations between the various levels of the eigenstates is described by

Q 1 ; z2 / WD

n2 K.z



X



G.˛; ˛; z1 /G.ˇ; ˇ; z2 / D n2 K.z1 ; z2 / C nK 0 .z1 ; z2 /



(21.7)



˛ˇ



with ˛; ˇ D 1 : : : n. The combination

nKQ 0 .z1 ; z2 / WD



X



G.˛; ˇ; z1 /G.ˇ; ˛; z2 /



˛ˇ



D



X

h˛j

˛;ˇ



D



X

h˛j

˛



D



1

z1

.z1



1

z1



H



z2



jˇihˇj



1

H/.z2



X

.h˛j

˛



1

z2

H/



1

z2



H



H



j˛i



(21.8)



j˛i



j˛i



h˛j



1

z1



H



j˛i/



yields

G.z2 /

KQ 0 .z1 ; z2 / D K.z1 ; z2 / C nK 0 .z1 ; z2 / D

z1



G.z1 /

z2



(21.9)



and is thus determined by the one-particle Green’s function.



21.2 Reduction of the Gaussian Unitary Ensemble

to a Matrix Model

The determination of the averaged product of m Green’s functions in the Gaussian

unitary ensemble is reduced to the determination of correlations in a model of a

2m 2m super-matrix.



21.2 Reduction of the Gaussian Unitary Ensemble to a Matrix Model



229



Table 21.1 Gaussian random ensembles

Gaussian

Abbreviation

Matrix elements

Ensemble invariant under



Unitary

GUE

Complex

Unitary



Orthogonal

GOE

Real

Orthogonal



Symplectic

GSE

Quaternion

Symplectic unitary



Ensemble

Numbers

Transformations



Gaussian Random Matrix Ensembles We consider three types of Gaussian

ensembles listed in Table 21.1.

In all cases the matrices are hermitian. The matrix elements are independently distributed apart from the condition of hermiticity. Denoting the matrices

by f , the probability distribution is proportional to exp. c tr .f 2 //. The average

.UfU 1 /˛ˇ .UfU 1 / ı does not depend on the transformation matrix U.

We return to the Gaussian unitary ensemble (GUE) whose density of states,

or equivalently one-particle Green’s functions, we have considered in Sect. 4.4.

Here, we will mainly consider the two-particle Green’s function, which yields the

correlations between the energy levels.

We start from (4.12), but denote the complex components x and the Grassmann

components as components of the array S

.i/



x˛.i/ D Sb˛ 2 A0 ;



.i/

˛



.i/



D Sf˛ 2 A1 :



Instead of the indices b and f we will sometimes use the Z2 -degree

product of Green’s functions is expressed by



(21.10)

D 0; 1. A



à Z

m Â

.i/

.i/

Y

sbi n Y d
.i/

d Sf˛ d Sf˛ /

G.˛i ; ˇi ; zi / D

.

sfi

iD1

iD1

i;˛

Y

.i/ .i/

.sbi Sb˛i Sbˇi / exp. S1 .S; 0//;



m

Y



(21.11)



i



p p Ž

S1 .S; A/ D str..sz

sA s/S S sSŽ fS/

X

.i/

. / s ;i S.i/;˛ .zi ı˛;ˇ f˛;ˇ /S ;ˇ

D



(21.12)



i; ;˛;ˇ



Xp

p .j/ .i/

sbi Ai;j sbj Sb˛ Sb˛ :

i;j



with sbi D isign.=zi /, (4.11), where we use (3.35) or (13.33). For the moment we

do not decide on the value of sfi . The last term is a source term, which allows one

to determine the Green’s functions. We introduce the source term only for the even

components Sb . This is sufficient to determine the Green’s functions.



230



21 Random Matrix Theory



The field S and the diagonal matrices s and z are given by

0

.1/

S D @ Sb˛



1

:::

:::

.m/ .1/

.m/

: : : Sb˛ Sf˛ : : : Sf˛ A 2 M .n; 0; m; m/;

:::

:::



z D diag.z1 ; : : : zm ; z1 ; : : : zm /;



s D diag.sb1 ; : : : sbm ; sf1 ; : : : sfm /:



(21.13)

(21.14)



.i/



The indices i and of S ˛ number the columns, and the subscript ˛ the rows.

As in Sect. 4.4, we perform the average over the matrix elements f , which are

Gaussian distributed with

f˛ˇ D 0;



f˛ˇ f



ı



D



1

ı˛ı ıˇ ;

ng



(21.15)



which yields

p p Ž

1

str..SsSŽ /2 //:

sA s/S S/ C

2gn



exp. S1 .S; A// D exp. str..sz



(21.16)



Next the Hubbard-Stratonovich transformation has to be performed which, formally (i.e. withoutpconsideration

of convergence requirements) reads (we use

p

str..SsSŽ /2 / D str.. sSŽ S s/2 /),

p

p

1

str.. sSŽ S s/2 // D

2gn



Z



p

p

ng

str.R2 / C str.R sSŽ S s//

2

(21.17)

with a super-matrix R 2 M .m; m/. Its precise form will be determined in Sect. 21.4.

The Hubbard-Stratonovich transformation allows the reduction of the biquadratic

interaction in S to a bilinear expression in S at the expense of a coupling between

such a bilinear term in S and the new degree of freedom R. Now the averaged

m-particle Green’s function reads

exp.



m

Y



Z

G.˛i ; ˇi ; zi / D



ŒD SŒD R



ŒD R exp.



Y

.i/ .i/

.sbi Sb˛i Sbˇi / exp. S2 .S; R; 0//



(21.18)



i



iD1



with

S2 .S; R; A/ D



ng

str.R2 /

2



p

str. s.R



p

z C A/ sSŽ S/:



(21.19)



The Green’s functions can be determined from the ‘partition function’

Z.A/ D



à Z

m Â

Y

sbi n

iD1



sfi



ŒD SŒD R exp. S2 .S; R; A//



(21.20)



21.3 Saddle Point



231



by taking the derivatives with respect to A,

@Z

D nG.z1 /;

@A11



(21.21)



@2 Z

Q 1 ; z2 / D n2 K.z1 ; z2 / C nK 0 .z1 ; z2 /;

D n2 K.z

@A11 @A22



(21.22)



@2 Z

D nKQ 0 .z1 ; z2 / D nK.z1 ; z2 / C n2 K 0 .z1 ; z2 /:

@A12 @A21



(21.23)



Next, the integration over the fields S is performed. All components, from ˛ D 1 to

˛ D n, yield the same contribution. Thus,

Z

Z.A/ D



ŒD R exp. nS3 .R; A//;



S3 .R; A/ D



g

str.R2 / C ln sdet.R

2



z C A/:



(21.24)

We have thus reduced the calculation of the averaged Green’s functions to the

determination of the expectation values of a model of matrix R. However, we have

not yet determined the manifold over which R varies. In order to obtain convergence

for the vector field S, it was necessary to choose sbi D isign=zi . sfi is not

determined up to now.



21.3 Saddle Point

The saddle point of the matrix R is determined. The averaged one- and two-particle

Green’s functions are calculated. If the difference of the energies of the retarded and

advanced Green’s functions are small, then there is a saddle point manifold for the

matrix R.

For large n, the integration starts by calculating the saddle point of S3 at A D 0.

The saddle point equation reads

gR.0/



1

R.0/



z



D0



(21.25)



with a diagonal solution

.0/



Rki;k0 i0 D ıkk0 ıii0 Rdi



(21.26)



given by

p



1



gRdi D p

g.zi



s

p

gzi

˙i 1

D

2

Rdi /



gz2i

D e˙i i :

4



(21.27)



232



21 Random Matrix Theory



Including the diagonal source terms in the bosonic sector, one obtains

1

sdet.Rd



m

Y



D



z C A/



Rdi



Rd

iD1 i



zi

:

zi C Aii



(21.28)



Thus,

d sdet.Rd1



zCA/



d Ajj



jAD0 D



1

zj



Rdj



1

sdet.Rd



z/



:



(21.29)



the saddle point solution for the one-particle Green’s function is then

G.zj / D



1

Rdj



zj



D



p sj j

ge :



(21.30)



The assignment of sj D i=zj is due to the discussion in Sect. 4.4. The last equality

of (21.27) constitutes a mapping z ! e˙i with the assignment ˙i D s. The real

p

p

interval 2= g Ä z Ä C2= g is mapped onto the unit circle ( is real): Real

p

p

z > 2= g is mapped onto the real interval .0; C1/; real z > 2= g onto . 1; 0/.

z with =z > 0 are mapped into the lower half unit circle and z with =z < 0 into the

upper half unit circle.

The saddle point solution (21.30) yields, with (21.4), (21.27),

G.E



si 0/ D gRdi D



gE

C si

2



.0/



.E/;



(21.31)



which gives Wigner’s semi-circle law in accordance with (4.24), (4.25).

Now expand S3 .R; A/ around the saddle point with

R



i; 0 j







0



1

ıij Rdi C p X

g



i; 0 j :



(21.32)



Then

g

1

str.R2 / D str.e2s C 2es X C X 2 /;

2

2

sdet.R



z C A/ D sdet.Rd



z/sdet.1



es .X C



(21.33)

p

gA//;



(21.34)



with the matrix es D diag.esi i /. The superdeterminant can be rewritten

sdet.1



O D sdet.exp ln.1

X/



O

X//



1

1

D sdet.exp. .XO C XO 2 /// D exp. str.XO C XO 2 //;

2

2



(21.35)



21.3 Saddle Point



233



where we have used (10.35) and expanded up to XO 2 . This yields

S3 .R; A/ D



p



g str.es A/ C



1

str.X 2

2



p

gA//2 /:



.es .X



(21.36)



Completing the square for X, we obtain

0



nX

exp. nS3 .R; 0// D Z.A/ exp @

˘

2 0



1

Q i; 0 j XQ 0 j; i A

i; 0 j X



(21.37)



i; j



with

XQ



i; 0 j



˘



i; 0 j



DX



i; 0 j



C ı 0ı



D . / .1

0



es i



00



esi i Csj j

Aij ;

˘ i; 0 j



i Csj j



(21.38)



/;



X

p

Z.A/ D exp @n g str.es A/ C

ij



1

ng

2.e



si



i



sj



j



1/



Aij Aji A :



(21.39)

(21.40)



The integral over XQ yields unity, since the contributions from the bosonic and the

fermionic components cancel. Thus, the ‘partion function’ Z.A/ is given by the

terms left after completing the square.

The expression for ˘ vanishes for s1 D s2 at the band edges. This means, close

to the band edge, one has to go beyond second order in X. It turns out that there are

tails to the band. We will not pursue this further. If s1 D s2 , then ˘ vanishes for

1 D 2 , i.e. for z1 D z2 . This will lead to special behavior for the correlations of

states energetically nearly degenerate.

Let us determine the Green’s functions using the saddle point approximation

from (21.40),

@Z

p

D n gesi i D nG.zi /;

@Aii

@2 Z

D n2 ges1

@A11 @A22

@2 Z

D

@A12 @A21

e



1 Cs2 2



Q 1 ; z2 / D n2 G.z1 /G.z2 /;

D n2 K.z



ng

s1



1



s2



2



(21.41)



1



D nKQ 0 .z1 ; z2 /:



(21.42)

(21.43)



KQ factorizes in leading order, n0 . In later sections, the corrections to K for ! D

O.1=n/ will be calculated. They yield the level correlations. Setting

z1 D E C 12 !;



z2 D E



1

!;

2



E 2 R;



=! > 0;



(21.44)



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