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
iÁ
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 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
Ã 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
S1 .S; A/ D str..sz
sA s/S S sS 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
1
str..SsS /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..SsS /2 / D str.. sS S s/2 /),
p
p
1
str.. sS S s/2 // D
2gn
Z
p
p
ng
str.R2 / C str.R sS 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 SŒD 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/ sS S/:
(21.19)
The Green’s functions can be determined from the ‘partition function’
Z.A/ D
Ã Z
m Â
Y
sbi n
iD1
sfi
ŒD SŒD 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
Dı
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)