Calculation of anomalous dimensions for the nonlinear sigma model

Nuclear Physics B275 [FS17] (1986) 561-579 North-Holland, Amsterdam

C A L C U L A T I O N OF A N O M A L O U S D I M E N S I O N S FOR THE N O N L I N E A R S I G M A M O D E L Daniel HOF* and Franz WEGNER lnstitut fffr Theoretische Physik, Rupreeht-Karls Universitiit, D-6900 Heidelberg, Federal Republic" of Germat9'

Received 28 April 1986

The relevant scaling operators without derivativesfor the orthogonal, unitary and symplectic nonlinear o-model are classified and their anomalous dimensions are calculated up to three-loop order. The exponents for the participation ratio and for higher averaged moments of the wave functions for Anderson localization are obtained. For nonmagnetic scattering the two-loop and three-loop term vanishes.

I. Introduction Some time ago a field-theoretic model of localization was proposed in the literature [1]. The Goldstone modes of this model belong to a noncompact symmetry group which is broken by the frequency to its maximal compact subgroup. The emerging nonlinear o - m o d e l is known to be renormalisable in 2 + e dimensions by the analysis of Br6zin et al. [2]. The phase transition was later analysed [3] in terms of the (inverse) participation ratio and related averaged moments of wave functions. These quantities, which besides the conductance distinguish between the regime of localized and extended states turn out to be connected with the relevant scaling operators of the field theory. Their critical exponents were obtained in first order for the model with pure potential scattering [3]. The anomalous dimensions of some of them are greater than the spatial one which is very unusual for critical phenomena. Later Kunz and Souillard [4] argued, that the vanishing of the (one-loop result for the) exponent of the inverse participation ratio in 4 dimensions favoured their conjecture that this is indeed the upper critical dimension of the model. In 1984, Pruisken [5] calculated the exponent of the inverse participation ratio for the case of spin-flip scattering in second order. This exponent if extrapolated also vanishes at d=4. In the present paper we have calculated the ~'-function and the scaling functions of all relevant scaling operators without spatial derivatives in third order for the * Supported in part by GFG-scholarship of Baden-W/irttemberg. 0619-6823/86/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

562

D. H6f, F. Wegner / Anomalous dimensions

potential, spin-flip and spin-orbit scattering. These functions have the form

~s(O, t) = a~t(1 + 3(2mlm 2 - rn)t 2) + (_9(t4) for the O(ml,

mz)/(O(ml) × O(m2) ) model and ~'s(U, t) =

a~t(l + 32(mxrn 2 + 1)t 2) + 0 ( t 4)

for the U(ml, m 2 ) / ( U ( m l ) X U ( m 2 ) ) model. Here t is the coupling constant and a 2 the lowest expansion coefficient of the scaling operator S in the transversal components. It is remarkable that the functions factorize in this way in an operator dependent part a 2 and a model dependent part. One sees directly that for the Anderson transition (m 1 = m 2 ----0) one does not have a correction in two- and three-loop order in the orthogonal case. In the unitary case if the conductivity exponent does not change in the next order then the critical temperature has to stay the same and therefore the participation ratio exponent is changed in three-loop order. In addition to the anomalous dimensions ~s one also obtains relations between the expansion parameters a 2, a 4 . . . by the condition that the divergent terms in the ~-function have to vanish. The paper is organized as follows. In sect. 2 we briefly introduce the models and the concept of inverse participation ratio. In sect. 3 we calculate the ~'-function in three-loop order using a method proposed in [6] and generalize the result for the grassmannian o-models to a wider class of symmetric spaces. Sect. 4 is a discussion of the classification of the relevant scaling operators along the lines of ref. [3] by their Young frames. In sect. 5 all scaling functions are calculated in third order and the exponents for the averaged moments of wave functions given, In sect. 6 we discuss the expansion parameter a 2 for various models and express it in terms of the row lengths of the Young frames in the general case and in terms of the degree of a polynomial for the O(n), CP n and HP" model. We also give the exponent for the participation ratio in the potential and spin-flip case. 2. Definition of the model

The participation ratio p, for an eigenstate li) of a tight binding model is given by 1

t"

where N is the total number of sites r and ~ i ( r ) the normalized wave function at site r. The inverse participation ratio is then introduced as P, =

E[+,(r)14

D. H6f F. Wegner/ Anomalousdimensions

563

Pi describes the proportion of the total number of atoms which contribute to an eigenstate, whereas P, is the inverse number of orbitals contributing to this state. One therefore expects Pi to vanish for localized states and be positive for extended states and Pi vice versa. For our field-theoretic analysis we consider the averaged quantities

El+i(r)12k6(e,- E) e k (E) = i,r

- ( e c - E)

o(e)

in the region of localized states and

1 - = lim p k)(E) .--.o

E(lq~(r)12~'(E-eil)k i,r

--

p(E) k

( E - Ec) "k

for the region of extended states near the mobility edge Ec. Here e i is the energy of the eigenstate ]i} and p(E) the density of states per site and energy and 8. the lorentzian

The participation ratio belongs to k = 2. Now we translate the formulation of p(k) to the field theory of Wegner and SchMer according to ref. [3]. We consider the O(m 1 + m2)/(O(ml) × O(m2) ), the U(m I + m2)/(U(ml) × U(m2) ) and the Sp(m 1 + m2)/(Sp(ml) × Sp(m2) ) matrix models and their noncompact counterparts O(ma, rn2)/O(ml) × O(m2) etc. For the mobility edge problem one uses the replica trick to integrate out the disorder. Therefore m 1 = m 2 = 0 in this case. The participation ratio is connected to a set of two-point functions in field theory. Only the advanced-retarded part of these functions is critical near the mobility edge. One can prove, that one therefore has to take a hyperbolic symmetry group to ensure convergence. Orthogonal symmetry is related to a random potential without magnetic impurities and spin-orbit interaction; unitary symmetry to magnetic impurities without spin-orbit interaction and symplectic symmetry to nonmagnetic impurities with spin-orbit interaction. The variables of the model are (m I + m2) × (m 1 + m2) matrices Q over the field of real, complex and quaternionic numbers respectively. They have the structure

Q=

011

Q12

021

Q22

in the compact,

Qla Q = ip21 in the noncompact case.

iol21 Q22 ]

564

D. Hb'f, F. Wegner / Anomalous dimensions

Q'J is an m~ × mj matrix and we have the relations QXl = ~/~_ Q12. Q21 , Q22 = _~/¼_ Q2,. Q12, in the compact case and Q l l = ~/j + Q,2. Q : I , Q22= _~/~+ Q21. Q12

in the noncompact case (the square roots are defined by their formal power series). Further the relation Q12 _= Q21"~ holds in both cases. The t sign stands for transposition in the orthogonal case; for hermitian adjunction in the unitary case and for the respective adjunction in the symplectic case. The quantities p(k) are in this formulation expectation values of linear combinations of products of k matrix elements of Q which are biinvariant under the broken subgroup. Their critical exponents can therefore be obtained from the dominant relevant scaling operators that are contained in them. The analysis of these relevant scaling operators is done in sect. 4. The action A for our models consists of (i) a kinetic part

1)

--

dax(tr(OgQ(x).O.Q(x)))

(tr the trace in the group space, summation over Lorentz-indices understood); (ii) a symmetry-breaking part

A,b = ( - 1 ) ' a •.

hsf

4t B aRa

daxtr(QXl(x)-Q22(x)+const)

_.caha t ~tB JR d a X t r ( ( G + h ) . O ( x ) ) ,

=(-I)

D. Hb'f, F. Wegner / Anomalous dimensions

565

with G=

(]~ml 0 0 --11,,2 ' m 2 -- m 1 - "din, m

?~ m=ml

+m

2 ,

SO that tr(G + X) --0; (iii) the measure-part Am, which is described in detail in ref. [7] (c.o. the different definition of the field-variables there).

c=

Ot~

0 1

for compact for noncompact symmetry,

i

for orthogonal for unitary for symplectic symmetry.

For the symplectic case the trace is understood as sum over the m quaternionic diagonal matrix elements. If the quaternions are represented by 2 × 2 complex matrices then tr is understood as half the sum of the 2m diagonal matrix elements. One can obtain the scaling functions and renormalisation constants of the perturbation theory of the noncompact model from those of the compact model by the transformation t13 --* _ l B .

As was shown by Oppermann and Jfingling [8] and by Wegner [7] there are in addition the relations Z(Sp; ml, m z, ta) = Z(O; = 2 m , , - 2 m 2, = ltB), Z(U; ma, m 2, tB) = Z(U; -rex, - m 2 , - t B ) , between the renormalisation constants of the models for corresponding scaling operators. Because of these relations we consider only the orthogonal and unitary compact case explicitly in the following sections. We obtain the perturbation theory by expanding the action in powers of Q12. Q 2 1 and Fourier transformation to momentum space. We omit the at least linear

D. Hb'f, F. Wegner / Anomaloa~dimensions

566

divergent measure graphs, which have to cancel with other diagrams of higher loop-order. By this procedure we get the Feynman graphs we need for the calculation up to 3-loop order. Decomposition in the harmonic and the anharmonic part yields Ak + Ah

A s . b . -----A h + A a ,

ol = ~t B fRudaxtr( Q12(x)(h"-

oj. dax E

A a = i6tB

d

k>_l

A)Q21(x))+

(')()

E ( - 4 ) k+k' k5 k'21

mira 2 a h B m 2 tB

} k'

×tr((Q12(x)Q21(x))k(h.- A)(Q12(x)Q2'(x))k') + (Q12 ~ Q2,), where we have used

tr(G.Q(x))= ~m-tr((Q(x)-½G)2). The propagators are tB

G ( p ) - p2+h u for both models. A 2n-vertex maps under n rotations into itself. In addition in the orthogonal case it maps into itself under n reflections. Thus the symmetrized (QlZQ 21)n contributions of A a are multiplied by 2n and n respectively to obtain the value of the corresponding vertices

V4 :

~B(2hB-~- (plArp2)2"~ (p2-~-p3)2)~(~i pi),

1(

O6 :

with

tB

6h,+

6 (pi+Pi+l) 2)trt 3 Pi ,

~

i=1

p7=pv 3. Renormalisation and calculation of the ~-function

As was proven in ref. [2] the above models are all renormalisable in 2 + e dimensions with the two renormalisation constants t ~ = Z 1 "t, h B= Z 2 •h R.

D. Hb'f, F. Wegner / Anomalous dimensions

567

F r o m the /3-function of the model (in the MS scheme in which we shall always work)

et

~(/,) =

1 + t" ( d / d t ) l n ( Z , ( t ) ) '

/~(O, t) = at - (m - 2)t 2 - (2mlm 2 - m)t 3 + 0 ( t 4 ) , /3(U, t) = e t - mt 2 - 2(rnam 2 + a)t 3 + O ( t 4 ) , we obtain for Z 1 the formulas t t2 t2 Z I ( O , t) = 1 + (m - 2 ) - + (2rn,m 2 - m ) - + (m - 2)2-3- + 0 ( t 3 ) , g

g

l

t2

E-

t2

Z I ( U , t) = 1 + m - + (rnlmz + 1 ) - - + m2-g + O(t3). E

8

E

The ~-function is defined by d Z2(t ) ~(t) -- ]3(t)~-;ln ZI(/,)

.

We calculate the quotient Z 2 / Z 1 from the finiteness of the vacuum energy F. We obtain for F the formula:

mxm2o F = --2m

tB + ½mlm2ajR daq In 1 +

+mlm2V4tBhl+e+ mlm2 (V 2+

V 6 ) t.B,~B 2 / ~ l + 3 e / 2 + (_0(/,3)

V4, V~ and V6 represent the vacuum diagrams with one and two 4-vertices and one 6-vertex respectively as listed in tables 1 to 3. In these tables the integrals I, the symmetry factors S and the multiplicities M are listed for the unitary (u) and orthogonal (o) model respectively. An overall factor mlm 2 and appropriate factors t B and h B are extracted. The final result for F is F ( O ; tB)

=

mtm2 hB 2m

tB

h1+~/2

+ mlm2--

e(2 + e)

+ mlm2 t2hl+3~/2 mxm2 hB

F ( U , tB) = - rn

tB

+

+mlm2tBhB 2 1+

4

--

m l m z t h 1+~

+ ~

B B

2mlm 2 m

6e 3

- 6 --

2mlm 2

m

12e 2

+

8e

2mlm2h~ +~/2 e(2 + e)

3e/2

(1 1)

(mlm2 + 1) - - -3e - +z

~e

+ (_0(t3B).

m) --

+(9(t3)'

568

D. Hb'f, F. Wegner / Anomalous dimensions TABLE 1 Feynman graphs of the free energy with one 4-vertex Graph

I

SO

1 4

0

2

Su

Mu

1 2

DI

m

1

~2

We have rescaled t B by a factor

Mo

2

K(d) and daq

F by

fR h +q2

h /2

K(d) K(a)

1 where

K(d) is defined by

K ( d ) is analytic in d at d = 2. We obtain for the renormalisation constant mt (~)2 (t)3

Z2(o;t)=l---+m Z1

-~m -~ (8-2m+(2mxm2-m)(2e+32e2))

e

+0(t4), --Z2(u;t)=I Z1

mt

3(1 3) -~+--~e

2m(mlm2+l)t

E

+0(t4)

-

TABLE2 Feynman graphs of the free energy with two 4-vertices

0

1 4

D,/2

1

2m

l

4

4

8 2

3

~3 + ~2

16

~2 +

1 16

20 2 383 + 3t,2

1

1

S

2mlm

2 + 2

m+2

1 2

m 2

81

2mira 2 + 2

569

D. H6f, F. Wegner / Anomalous dimensions

TABLE3 Feynmangraphs of the free energywith one 6-vertex

0

4 E3

'6

m~+m~

1 2

m

2 E3

6 ~3

6 E3

4

'~

m~+m~

2mlm 2

2~-

2mira 2

2

1 6

1 4

1 12

2

and for the ~'-function ~'(O; t) = - r o t ( 1 + ¼(2mxrn 2 - m ) t 2) + 0 ( t 4 ) , ~'(U; t) = - r o t ( 1 + ~ ( m l m 2 + 1)t 2) + O(t4). We conclude from [7] ~'(Sp; t ) = - r o t ( 1 + 3(4mlm 2 + m ) t 2) + 0 ( t 4 ) . By the same arguments as in ref. [9] we obtain for the ~'-function of other symmetric spaces

~,~ m,(l+ ~(2mlm2+m(1~)4(1 1)(1 4))t2)+~4~

570

D. Hb'f, F. Wegner / Anomalous dimensions

We check this result with that for the CP "-~ model in ref. [10] and get in accordance ~,(Cpn 1 ; t ) = ~ . ( U ; m l = l , m

= -n,(1 +

2=n-l,t)

2) + O(,4).

For the ~'-function of the O(n) o-model we obtain ~'(O(n); t) =~'(a = n = -(n-

1, m, = 1, m 2 = 1,

¼at)

1)12t(1 + ~ ( n - 2)t 2) + (9(t4),

which differs from that in ref. [11] by a factor of 2 due to the different definition of the Z constants in this model.

4. Discussion of the relevant scaling operators After the calculation of the ~-function we want to discuss the scaling properties of the other relevant operators which do not contain spatial derivatives. We proceed similarly to ref. [12] where the O(n) o-model was treated. Since the Q-field is dimensionless in 2 dimensions any local function without derivatives of it is a relevant operator. Under the action of the renormalisation group a set of such operators which belong to an irreducible representation of the symmetry group will not be mixed with other operators. One can also verify by Frobenius reciprocity that to each irreducible representation belongs only one renormalisation constant, i.e. a 1-dimensional subspace, that transforms trivially under the subgroup, to which the full group is broken. In the case of the O(n) o-model the corresponding operators are given by Gegenbauer polynomials [12], in the general case, we are interested in the zonal spherical functions. Harish-Chandra has given an explicit formula for all zonal spherical functions in terms of functionals from the complexified Lie algebra modulo an operation of the Weyl group [13]. As is shown in the next section we only need certain coefficients of these functions which can be computed from elementary group theory and are already known for the orthogonal model from the one-loop order. As in ref. [14] we realise the representations on the linear combinations E 71" 1 . . . . . i 2 k Q i l , i 2 * ' " I"1 • . . . , 1"2k

Qi2k

l,i2k'

with traceless tensors T (a tensor is called traceless, if any contraction of two indices vanishes) which are symmetric under any permutation of pairs of indices

(i2p 1, i2p)~'--)(i2q l'i2q) "

D. Hb'f, F. Wegner / Anomalous dimensions

571

As mentioned in sect. 2 the tensors of rank 2k belong to the kth order inverse participation ratio of the mobility edge problem. One can characterize the irreducible representations of our groups by the symmetry under the permutation group of the indices of the tensors [14]. The irreducible representation with highest weight l = (/a . . . . . lr) is equivalent to the tensor representation with Young symmetry of the frame )~ = l. For the orthogonal model we have in addition symmetry under permutations of the indices within a pair as the variables Q are symmetric matrices. This means that not all representations are contained in this invariant space of tensors. It was conjectured in ref. [3] that those representations are contained, for which all l s are even. This can be proven in two steps if m > k. One first shows, that all these representations are indeed contained in this tensor space. Finally by dimension arguments it is easy to calculate that the dimension of the whole space - ( 2 k - 1)!! - is already the sum of the dimensions of these representations. In ref. [3] the one loop order of the critical exponents for this model was also calculated. For the unitary model we get all irreducible representations by permuting only the advanced or retarded indices as we don't have a real symmetry between them. By dimension arguments one can again show, that all Young frames of the advanced or retarded indices are contained precisely once. The symplectic model is in a sense dual to the orthogonal. If one chooses the coordination of ref. [7] then the matrices o~,.q~,~ are antisymmetric. So we conclude, we get all irreducible representations by taking those Young frames precisely once which have an even column length. An easy argument for the above classification of the irreducible representations is obtained from looking at the expectation value

(Qil,i2...Qizk

1,i2k) -

For the orthogonal model one has to link the indices ir by diagrams to r closed cycles of length (Xi) where }2"~Xi = 2k, i=l

and as by one link always two indices are connected the number of indices in one cycle is always even. Then one sums over all possibilities to permute within the cycles, antisymmetrizes between the cycles respectively and projects on the space of traceless tensors. So the eigenoperators are characterized by the even partitions of the number 2k and these partitions belong precisely to the above mentioned Young frames. For the unitary model one can only link the advanced and retarded indices separately which again corresponds to Young frames of

i=1

D. Hb'f, F. Wegner / Anomalous dimensions

572

For the symplectic model the individual indices are free as in the orthogonal model but now one has to take care of the antisymmetry relation above and so one has to take the "dual" Young frames.

5. The scaling function of a general scaling operator We now consider the scaling behaviour of a general scaling operator S. We expand S in powers of Q12. Q21 and define $2 =

1

tr(Q12. Q21),

mlm 2

1 $2, 2 = - -

(tr(Q12.

Q21))2

mlm 2

$4 =

1

tr((Q12. Q21)2),

mlm 2

$6=

1

tr((Q12. Q21)3),

mlm 2

$2,4 =

1

tr(al2.a21)tr((Ol2.O21)2),

mlm 2

1 $2,2,2 = - -

(tr(Q12 • Q2X)) 3 .

mlm 2

We now write S as S = 1 + a2S 2 + a4S , + a2,2S2, 2 + a6S 6 + a2,4S2, 4 + a2,2,2S2,2, 2 + 0 ( Q 8 ) . It can be proven that any relevant scaling operator S has a nonzero constant part in the proposed expansion and therefore can always be normed to begin with 1. Operators, which contain derivatives, cannot be written in this way but they are by one or more dimensions down. We now want to calculate the scaling function ~'s(t) by the following procedure. We first form the expectation value ( S ) = SB= Z s ( t ) -I" SR, where S R is the finite part of S B. The function ~s(t) is then given by d ~s(t) = ~ ( t ) - ~ l n ( Z s ( t ) ) .

D. H6f F. Wegner/ Anomalousdimensions

573

TABLE4 Vacuumgraphs of highermomentsof the field variables Graph

Value

2

~2

©

3e3

In order to evaluate the expectation values of S with the exception of S 2 we use the values of the graphs in table 4. The expectation value ($2) is obtained from the free energy by first calculating the order parameter from the free energy (tr((G + ~ ) Q ) )

2mamz(1-m(S2)-m(S4)-2m(S6))+(.O(Q8 ) m

and then subtracting the terms proportional to ($4~ and ($6). The divergent parts of the expectation values (Sz) to ($2,2,2) are calculated:

tle

( 8 2 ) ( O ' tl)

-g

(m+2+e)(~) 2

(4mlm2+14m+32+e(-2mlm2+lOm+30)+9e2),

~S4)(O, tl) = ( t-~l)2(m e + l) + 2( t~se) 3 × ( ( m + 1)(m + 2 + e) + 4(re,m2 + m) + 4),

~

~.

~

~

~

o+

o

+

+

0

o

©

0

~

~

~.

v

II

~

"~

A

.

~.

v

~

~

II

~1~

~

--a

~

+

+

~

3

~

II

~

II

v

3

v

+

II

+

~-

~ 1 ~ ~"

v

II

~.----.

~

v

q

~--

II

I

II

©

+

"~

v

l,J

A

+

I

II

0

v

~

I

II

©

v

÷

Ix~

~ 1 .'~

+

+

-t-

II

0

bo

-F

---F

X

v

A

~.

?n

.1~

D. Hb'f, F. Wegner / Anomalous dimensions

575

with the 2 coefficients A o = a~ + (3m + 2)a 2 - 2(m + 1 ) & - 2(mlm 2 + 2)a2, 2 , B o = a3~ + (5m + 2)a22 - 3(m + 1)a4a 2 - 3(mlm 2 + 2)a2a2, 2 + ( 5 m 2 + 3m + 2mira 2 + 4)a 2 - (10m 2 + 22m + 8mlm 2 + 28)a 4 - 2 ( 5 m l m 2 m + 18m + 2mam 2 + 12)a2, 2

+3(m2 + 3m+mtm2+a)a6+

3(mlm2+4)(m+

1)a2,4

+3((m2m22 + 5mlm 2 + 8)a2.2,2). When we claim, that the divergent parts of the scaling function vanish, we get the formula in the introduction and the two equations between the coefficients A o = Bo = 0. For the unitary case we get l2

t3

~'s(U, t) = a2/(1 + 32(mlm 2 + 1)t 2) + --~-Av + t T B u + 0 ( t 4) and the two coefficients A U = a 2 + 3ma 2 - 2ma 4 - 2(mlm 2 + 1)a2, 2= 0, B U = a32 + 5 m a ~ - 3(mlm 2 + 1)a2,2a 2

-3ma2a 4+

(5m 2 + 6 ( m l m 2 + 1))a 2 - (5m e + 12(mlm 2 + l ) ) a 4

- 2 m ( 5 m l m 2 + l l ) a 2 , 2 + 3(m 2 + m l m 2 + 1)a 6 + 1 2 m ( m l m 2 + 2)a 4

+ 3 ( m l m 2 + 2 ) ( m l m 2 + 1)a2,2, 2 = 0. By the symmetry relations in [7] we obtain for the symplectic model (Szk ...... 2k,)(Sp; m 1, m2, t) = 4 ( - 2) - ' (S2k ' ..... 2 k ; ) ( O ; - 2 m 1 , - Z m 2 - ½ t ) (k~ the lengths of the closed cycles of Q's). Thus: ( S ( a 2 , a4, a2, 2 . . . . ))(Sp; m,, m 2, t)

--- ( S ( - 2 a 2, - 2 a 4, a2, 2.... ) ) ( O ; - 2 m r , - 2 m 2 , - - I ~ I ), which yields ~'s(SP, t) = a2t(1 + ~ ( 4 m l m 2 + m ) t 2) + O(t4).

576

D. H6f, E Wegner / Anomalous dimensions

6. The coefficient a 2 One sees that for the three models the scaling functions are determined by the coefficient a 2 alone. It changes sign, as one turns from the compact to the noncompact model and so one obtains the same scaling function in both cases. It is known from the one-loop order for the orthogonal model in the case m 1 = m 2 = 0 [3] and can be obtained similarly for the other models. One had for the orthogonal model in the notation of sect. 5:

~s(O, t) - k~(O, t) = -2at + d)(t2), a= i=l

where the Xi are the lengths of the rows of the Young frame. Thus we obtain:

a~-- ~ ½X~(2i- X~- m ) . i=l The corresponding one-loop calculation for the unitary model yields:

a2U= ~ )~i(2i_)~i_m_ l), i=1

and for the symplectic model

a~p= i ¼ X / ( 2 i - X

i-2m-2).

i=1

Obviously the duality relations

a~(Y,m)= - a ~ ( Y * , - m ) , a~zp(y,m)= _L_.otv*2~2k , -2m) between the Young frame Y and its dual Y* hold, which imply ~'(U, Y, rex, m 2, t) = ~'(U, Y*, - m 1, - m 2 , - t ) , ~'(Sp, Y, m,, m2, t) = ~'(0, Y*, - 2 m , , - 2 m 2 , - ~t).

D. Hb'f, F. Wegner / Anomalousdimensions

577

Along the lines of ref. [3] we get the exponents:

~r~=(k-1)(2+ l - k ) +O(e3), #°k = k ( k -

1) + O(e3).

For the unitary model one principally needs the four-loop result of the fl-function. In the parametrisation of Hikami [9] we get: u=(k-a) k(k-

I't~

(1

k 1 ( l _ a (3B _2C+ 5)e)_ ~L ( B 1 2

2 C + ~ )) + O(e),

1)

27'27 ( 1 - ~ ( B 1 - 2 C + ~ ) e ) + O ( e ) '

where B 1 and C are constants from ref. [9] and B 1 is not known. We don't give the exponents for the symplectic model here because they maybe do not correspond to the mobility edge problem as the calculated fixed point is negative in this case. We also remark that the scaling functions for the O(n) o-model as given in ref. [15] can, like the ~-function, be evaluated to

~,(,)

=

-

½1(l+

n

2),(1 + 3(n - 2), 2) + d~(t4).

-

In the general case of a group G~(n)/G~(n parametrize the matrix Q by a vector

-

1) in the notation of Hikami [9] we

ep~ S ~" = G~(n)/G.(n -

1),

in the form QiJ

=

q~,q~7-

1

~3i, j

,

where the star operation * is the identity for a = 1 and complex and higher conjugation for a = 2,4. The zonal spherical functions can be constructed as polynomials of q~lq~T as all other invariant combinations of the ~'s can be reduced to this one. We write such a function fk as k

k

k = E

'= E

i=1

i=1

+

k i=1 k

E di(tr(Q 11) + ½ ) i ( t r ( Q 22) + ½ ( n - 1)) k - i , i=1

578

D. Hb'f, F. Wegner / Anomalous dimensions

where ~r is the vector of the last n - 1 components of ¢. In the last equation fk is in the tensor form of sect. 4 and one can prove that the tracelessness of the tensor which belongs to fk is equivalent to the equation A L = O,

where 02

z=2

)2.

In this way one obtains for the zonal spherical functions the Jacobi polynomials ]X,P

Pk

(2¢Iqh - I),

with /~= ~ a ( n - 1 ) - 1, and

u = S1 a - 1 and for the coefficient a [

a~ = -k~n

2

so

+o(t'). In the formula a = 1 corresponds to the already known case of the O(n) o-model, a = 2 to the CP n-1 model and a = 4 to the HP n-1 model.

References [1] F. Wegner and L. Schiller, Z. Phys. B38 (1980) 113 [2] E. Br6zin and J. Zinn-Justin, Phys. Rev. B14 (1976) 3110 E. Br6zin, S. Hikami and J. Zinn-Justin, Nucl. Phys. B165 (1980) 528 [3] F. Wegner, Z. Phys. B36 (1980) 209 [4] H. Kunz and B. Souillard, J. Phys. Lett. 44 (1983) 503 [5] A.M.M. Pruisken, Phys. Rev. B31 (1985) 416 [6] A. McKane and M. Stone, Nucl. Phys. B163 (1980) 169 [7] F. Wegner, Nucl. Phys. B180 [FS2] (1981) 77 [8] K. J~ingling and R. Oppermann, Z. Phys. B37 (1980) 369

D. Hb'f, F. Wegner / Anomalous dimensions

579

R. Oppermann and K. Jfingling, Phys. Lett. 76A (1980) 449 S. Hikami, Nucl. Phys. B215 [FS7] (1983) 555 D.J. Amit and G.B. Kotliar, Nucl. Phys. B170 [FS1] (1980) 187 S. Hikami and E. Brfzin, J. Phys. A l l (1978) 1141 E. Br6zin, J. Zinn-Justin and J.L. le Guillou, Phys. Rev. D14 (1976) 2615 S. Helgason, Groups and geometric analysis (Academic Press, 1984) A.O. Barut and R. Raczka, Theory of group representations and applications, 2nd edition, (PWN-Polish Scientific Publishers, Warschau, 1980) [15] E. Br6zin, J. Zinn-Justin and J.L. le Guillou, Phys. Rev. B14 (1976) 4976

[9] [10] [11] [12] [13] [14]