- Email: [email protected]
Does the quenched reduced U(∞) chiral model break spontaneously in weak coupling?
Volume
120B, number
4,5,6
PHYSICS
LETTERS
13 January
1983
DOES THE QUENCHED REDUCED U(m) CHIRAL MODEL BREAK SPONTANEOUSLY
IN WEAK COUPLING ?
G. BHANOT CERN, Geneva, Switzerland Received
17 September
1982
The U(N) chiral model, when quenched using Parisi’s rule, has a [U(l) X U(l)]N/U(l) global invariance. To determine whether this symmetry breaks spontaneously in weak coupling for N = -, a one-loop calculation of the distribution of eiaenvalues of the single U(N) matrix of the model is performed. This distribution is shown to be uniform on the unit circle and hence, no symmetry breaking occurs. Further, the order parameter I tr U12/N2, which should be zero at N = - in the absence of spontaneous symmetry breaking, is evaluated in the weak coupling phase for one, two and three dimensions for A; varying from 2 to 50 by Monte Carlo simulation of the quenched model. The data indicate that this parameter indeed goes to zero as N + - implying that the symmetry does not break.
Recently, there has been renewed interest in large N matrix theories. It was initiated by the observation of Eguchi and Kawai [ 1 ] that the loop equations of the N = “,d dimensional lattice gauge theory are identical to those of a reduced model with d links and periodic boundary conditions. However, this model was shown by Bhanot et al. [2] to be an incomplete description of N = w QCD because of spontaneous symmetry breaking in the weak coupling (continuum) limit of the theory. The recipe suggested to cure this problem and obtain the correct weak coupling limit was to “quench” the eigenvalues of the d link matrices [2]. This means that the eigenvalues are not to be treated as dynamical variables to be integrated over in the partition function. Instead. one must average over them after computing Green’s functions in a fixed distribution of these eigenvalues. To illustrate using a rather obvious though partly symbolic notation, consider the free energy F=lnZ=ln
_I dh&exp[S@,n)],
(1)
where the h’s are the eigenvalues and “a” stands symbolically Under quenching, this is replaced by
for the “rest” of the matrix.
(2) Similarly, a Green’s function (3) is replaced under quenching (@Q
by
=jdAzkjdn9 exp[S@,a)l.
Parisi [3] suggested and developed [4] another approach to quenching valid for all N= 00 matrix theories. Using Parisi’s approach, Gross and Kitazawa [S] and Das and Wadia [6] showed that the reduced, quenched gauge model reproduces the planar graphs of weak coupling perturbation theory for N = m QCD. 0 031-9163/83/0000-OOOO/$
03.00 0 1983 North-Holland
371
Volume
120B, number
PHYSICS
4,5,6
13 January
LETTERS
1983
It is now of interest to apply these ideas to N = 00 chiral models. Using the quenching prescription of ref. [3], Bars et al. [7] have succeeded in performing the group integrals in the N = 03 continuum quenched chiral model. This is potentially a very important development. However, it has been pointed out by Heller and Neuberger [8] that there is an additional [u( 1) X U(l)] N/U( 1) sy mmetry left over in the chiral model after quenching. In Monte Carlo studies, this symmetry seems to break spontaneously in weak coupling. This would suggest that the naively quenched chiral model ends up with the wrong vacuum state in the continuum limit. The authors of ref. [S] suggest an alternate model which is quartic and therefore difficult to integrate in the spirit of ref. [7]. In the present paper, I study the question of the spontaneous symmetry breaking of the quenched lattice chiral model in weak coupling within a one-loop approximation. To lowest order, one finds that there is no spontaneous symmetry breaking because the distribution of eigenvalues of the single matrix U of the model is uniform on the unit circle. In the one-loop approximation also, this picture persists as the distribution remains uniform. This strongly indicates that there is no spontaneous symmetry breaking. To confirm this issue, a Monte Carlo simulation was performed with the order parameter L = 1/N2(ltr U12) for d = 1,2,3 dimensions and N ranging from 2 to 50. By factorization, L should vanish as N --, 00. This indeed happens, both for weak and strong coupling in the quenched model. The conclusion is that there is no spontaneous symmetry breaking in this model and the vacuum state in the continuum limit is the correct one. The lattice chiral model is defined by the action SL = flNx$
Re ]tr(W)
u’(x
+ ~111,
(5)
where x, ~1label the sites and directions of a hypercubic lattice and the D”s are elements of the group U(N) The reduced model a la Parisi [3,7] equivalent to this one at N = m is given by the action S = PN F
Re [tr(UDr U'D:)]
,
(6)
where D, is a diagonal matrix of momenta D, =Diag{DL,DL=exp(ipl),
i= I,2 ,..., N, -n<~~
and the p:s are distributed uniformly between --K and rr. The action of eq. (6) obviously has a [U( 1) X U( l)lN/U( The partition function is given by Z =
s
(7) 1) invariance.
dU exp(S),
(8)
where dU is the Haar measure on U(N). The free energy, according to the rules of quenching F=j(
[2,3,8]
is defined by
ndpl)lnZ. i,p
(9)
Z will now be computed to one-loop approximation First, rewrite the action as S =flN2d -y
L
c
P
in weak coupling (large 0).
tr(F,F,+),
(10)
with
(11)
F/.,= [KD,l. Maximum action corresponds to Fp = 0 or to u’s which are diagonal. Hence, a convenient struct perturbation theory is [2,8,9] 372
parametrization
to con-
Volume 1208, number 4,5,6
PHYSICS LETTERS
13 January 1983
U = exp(ia) X exp(-ia)
(12)
where h is a diagonal matrix of the eigenvalues of U, i= 1,2 ,..., N, hi=exp(iei),
h=Diag{hj,
-a
(13)
and “a” is a hermitean N X N matrix. The Haar measure becomes [8,9] dU= ( ndX’
(144
i
where
(14b, 4
d2hij = d2gii p(a),
and the last equation is to be thought of as a matrix equation. To compute 2 to one-loop in weak coupling, one must expand the non-gaussian parts of exp(S) in a power series in a up to 0(a4) and the density p(a) to O(a2). This is tedious but straightforward. The result is
z = exp(dpN2)J(
c dX’) ( iFj 2
~@)=$ II d2Qijexp(-oNis i>j
I”ii12 Ihi-XI12Dj~
12 Cl&.12 _flP! 3 i*j
2
12
”
c
i#l,l#k,k+j,j#i
T
IA’ -Ai I2 “0))
c g i#k,k#j,j#l,l+i
(15)
)
[Dijaikakjajlali(Xj
[Dijailalkakjaji {(hi-Xi)
[XT-h;
- 2X, + Xi) (h; - 2X; t A:)]
+ 3 (A; - Xi)] + CL)]
)
+ higherordersin ljD,(lQ
where D..= 4
c u
IDi _Dj12. !J
(17)
P
As discussed in ref. [2], the issue of spontaneous symmetry breaking depends on the effective action governing the dynamics of the h’s. If the distribution of these h’s [which is determined by the curly bracket in eq. (15)] is such that the x’s are uniformly distributed on the unit circle, there is no spontaneous symmetry breaking. If the h’s can clump together on the unit circle th_esymmetry is broken. To decide the issue, one must compute Z(X). To zeroth order (no loops),~(X) is given solely by integrating over the exponential term in eq. (16). These integrals are easily done and the result is Z - exp(dpN2)
(1/fl)(NZ-NJ/2
s 7
dh’.
(18)
It is therefore apparent that to this order, the distribution of the A’s is uniform on the unit circle and there is no spontaneous symmetry breaking. To compute the one-loop correction to eq. (18), one introduces in the usual way the generating function
G(Y)=Jipj d2aij exp
(-PNi$ laijI2
Qij + ij z
j %j 7ij)
Y
(194 373
Volume 120B, number 4,5,6
1
Cr..r..
ON i>i
‘1 1’
13 January 1983
PHYSICS LETTERS
Qf’
,
Q,>O,
From this, it is easy to perform the integrations After some algebra, this gives
Z-exp(dpN2)(l/p)(NZ-N)~2~
ViJ,
i#i.
(19b)
over the Uii’s in eq. (16) by differentiating
7 dhi{exp[-fl-lf(D,X)]
with respect to the 7’~.
+0(1/P2)),
(20)
where j-(D,X)=-l
c ’ 6 i+j ihi -hi
1
c
-~i#k,j#k
L,’ 12 Dij
_____
c-
lXi-2Xk
Dij
32Ni+k,j#kDikDjk
(4X/f -hi
DkjIhi-Ak121hj-Ak12
+
Xjl* ___
Ihi-Xk121~j-_k12
- 3Xj) + C.C.
(21)
’
f(D, h) is the effective action for the h integration. The h’s like to distribute themselves so as to minimizef. If the uniform distribution of X’s remains an absolute minimum off, the symmetry remains unbroken to one-loop order. This issue is most readily decided numerically. This is done as follows. Choose Dt= exp(ipL) with the p;‘s uniformly spaced in the interval [-n, n] [2,5,7]. Choose hi = exp(i0i) with the 8’s uniformly spaced in the interval (-@, 4). Compute
(22)
~(~)=N-*[f(D,h(~>)-f(D,h(n))l,
as N + 00. When @< rr, the h’s are forced closer together on the unit circle. If the uniform distribution of h’s were an unstable saddle point, .$(@)would be negative for 9 < rr. In fig. 1, ,$‘($J)is plotted versus @forN varying from 2 to 50 ford = 2 dimensions. It is clear that .$is always positive for large N. Thus, pushing the h’s together increases the action. Hence the uniform distribution for the h’s remains a stable saddle in the one-loop approximation. Note that for small N (N - 5 or less), the uniform distribution seems unstable. This may explain the results of ref. [8] where it was concluded that the h’s like to clump together. In one dimension, g(d) behaves just as in two dimensions and the conclusion is that the symmetry there also does not break as is well known from the exact solution. To obtain a more concrete proof for the absence of spontaneous symmetry breaking, one must resort to Monte Carlo simulation. Consider the order parameter L =Ne2(ltr
U12),
(23)
Fig. 1. The effective action r;(e) for the X’s when they are restricted to the interval ous values of N.
374
(-a,
@) ford
= 2 dimensions
and vari-
Volume
120B. number
4,5,6
PHYSICS
lp
d=l, xp=o.4,
=2.0
d=2,
13 January
LETTERS
l/3=12
x6=0.4,
d-3,
./3=Q8
* 0.2-
1983
1
-0.2 -
-0.2-
* A N 5 &
I
--
t
*
-&O.I
-
*
I
-O.l-
-Ol-
4
I;
+
+ ++*
* -
* (. 0
* +
. * : ,
I 0.2
0.1
I 0.3
+ .*I 0
I/N Fig. 2. The order parameter strong coupling (crosses).
+ t
*
I cll
I 0.2
I a3
+ 0
I/N
L versus l/N for N varying
from 4 to 50 in d = 1,2,3
1 0.1
I 0.2
I Q3
I/N dimensions
both for weak coupling
(dots)
and
where the average is in the sense of eqs. (l)-(4). If the eigenvalues of U like to cluster together, it is easy to see that L does not vanish, even at N = ~0. Indeed, if the distribution of 0 i’s pi = exp(iei)] is gaussian with width u, then for u < 71,L = exp(-a2/2) at N= 00. If, on the other hand, the distribution of the 6 i’s is uniform on the unit circle, L tends to vanish when N + * in general like l/N2. However, in the present case, it is sufficient that L vanishes like l/N because the quenched model is expected to be correct only to 0(1/N) [2,3,5,7] and not 0(l/N2). A standard Monte Carlo simulation was done fo determine L both in the weak and strong coupling region for N varying from 2 to 50. The quenched momentap: were chosen uniformly in the interval [-r, 711but care was taken to compute L by averaging it over many different random permutations of these momenta. In fig. 2, L is plotted versus l/N ford = 1,2,3 dimensions. In strong coupling, L was evaluated ford = 1,2 for /3 = 0.4 and it is clear that as N + m, L vanishes faster than l/N in this regime. The choice of which j3 value to use to determine the weak coupling behaviour of L was made as follows. Consider the order parameter E=
& &
(c
tr(UDPPDl
+ h.c.))
=(l/N2d)
aE/W.
(24)
P
E has the weak coupling expansion E = 1 - (1/2d/3) (l-
l/N) + O(l/p2).
(25)
by numerical simulations that at /3 = 2.0,0 = 1.2, /3 = 0.8, respectively ford = 1,2,3 dimensions, between 0.75-0.8 and was well within the weak coupling region described by eq. (25). From fig. 2, it is also clear that in weak coupling, L vanishes as N -+ 00 at least as fast as l/N for two and three dimensions and like l/N2 ford = 1. As stated above, this is sufficient in the quenched model because it is expected to be correct only to 0(1/N) [2,3,5,7]. Fig. 2 also shows that if one does simulations in the quenched model with N - 10, one will see a rather large signal for L. This should not be interpreted to mean that the symmetry breaks spontaneously because it is an artifact of the finiteness of N. This observation probably explains why the authors of ref. [8] concluded that this model suffers spontaneous symmetry breaking. In summary, it has been shown in this paper both by a one-loop weak coupling computation of the distribution of eigenvalues of the matrix CJand by Monte Carlo simulation on the complete quenched model that the [u( 1) U( l)“/U(l) invariance of the reduced quenched chiral U(N) model at N = 00 does not break spontaneously. It was determined
E had a value somewhere
375
Volume
1208,
number
4.56
PHYSlCS
13 January
LETTERS
I would like to thank Itzhak Bars, Dietrich Foerster, Murat Giinaydin, Yankielowicz for very useful discussions.
Giorgio Parisi, Spenta Wadia and Shimon
References [l] [2] [3] [4] [5] [6] [7] [8] [9]
T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063. G. Bhanot, U. Heller and H. Neuberger, Phys. Lett. 113B (1982) 47; 11SB (1982) G. Parisi, Phys. Lett. 112B (1982) 463. G. Parisi and Y. Zhang, Phys. Lett. 114B (1982) 319. D. Gross and Y. Kitazawa, Princeton University preprint PRE 2564 (April 1982). S. Das and S. Wadia, University of Chicago preprint EFI 82-15 (April 1982). I. Bars, M. Gunaydin and S. Yankielowicz, CERN preprint TH. 3379 (1982). U. Heller and H. Neuberger, Phys. Rev. Lett. 49 (1982) 621. M.L. Mehta, Random matrices (Academic Press, New York, 1967).
376
1983
237.
120B, number
4,5,6
PHYSICS
LETTERS
13 January
1983
DOES THE QUENCHED REDUCED U(m) CHIRAL MODEL BREAK SPONTANEOUSLY
IN WEAK COUPLING ?
G. BHANOT CERN, Geneva, Switzerland Received
17 September
1982
The U(N) chiral model, when quenched using Parisi’s rule, has a [U(l) X U(l)]N/U(l) global invariance. To determine whether this symmetry breaks spontaneously in weak coupling for N = -, a one-loop calculation of the distribution of eiaenvalues of the single U(N) matrix of the model is performed. This distribution is shown to be uniform on the unit circle and hence, no symmetry breaking occurs. Further, the order parameter I tr U12/N2, which should be zero at N = - in the absence of spontaneous symmetry breaking, is evaluated in the weak coupling phase for one, two and three dimensions for A; varying from 2 to 50 by Monte Carlo simulation of the quenched model. The data indicate that this parameter indeed goes to zero as N + - implying that the symmetry does not break.
Recently, there has been renewed interest in large N matrix theories. It was initiated by the observation of Eguchi and Kawai [ 1 ] that the loop equations of the N = “,d dimensional lattice gauge theory are identical to those of a reduced model with d links and periodic boundary conditions. However, this model was shown by Bhanot et al. [2] to be an incomplete description of N = w QCD because of spontaneous symmetry breaking in the weak coupling (continuum) limit of the theory. The recipe suggested to cure this problem and obtain the correct weak coupling limit was to “quench” the eigenvalues of the d link matrices [2]. This means that the eigenvalues are not to be treated as dynamical variables to be integrated over in the partition function. Instead. one must average over them after computing Green’s functions in a fixed distribution of these eigenvalues. To illustrate using a rather obvious though partly symbolic notation, consider the free energy F=lnZ=ln
_I dh&exp[S@,n)],
(1)
where the h’s are the eigenvalues and “a” stands symbolically Under quenching, this is replaced by
for the “rest” of the matrix.
(2) Similarly, a Green’s function (3) is replaced under quenching (@Q
by
=jdAzkjdn9 exp[S@,a)l.
Parisi [3] suggested and developed [4] another approach to quenching valid for all N= 00 matrix theories. Using Parisi’s approach, Gross and Kitazawa [S] and Das and Wadia [6] showed that the reduced, quenched gauge model reproduces the planar graphs of weak coupling perturbation theory for N = m QCD. 0 031-9163/83/0000-OOOO/$
03.00 0 1983 North-Holland
371
Volume
120B, number
PHYSICS
4,5,6
13 January
LETTERS
1983
It is now of interest to apply these ideas to N = 00 chiral models. Using the quenching prescription of ref. [3], Bars et al. [7] have succeeded in performing the group integrals in the N = 03 continuum quenched chiral model. This is potentially a very important development. However, it has been pointed out by Heller and Neuberger [8] that there is an additional [u( 1) X U(l)] N/U( 1) sy mmetry left over in the chiral model after quenching. In Monte Carlo studies, this symmetry seems to break spontaneously in weak coupling. This would suggest that the naively quenched chiral model ends up with the wrong vacuum state in the continuum limit. The authors of ref. [S] suggest an alternate model which is quartic and therefore difficult to integrate in the spirit of ref. [7]. In the present paper, I study the question of the spontaneous symmetry breaking of the quenched lattice chiral model in weak coupling within a one-loop approximation. To lowest order, one finds that there is no spontaneous symmetry breaking because the distribution of eigenvalues of the single matrix U of the model is uniform on the unit circle. In the one-loop approximation also, this picture persists as the distribution remains uniform. This strongly indicates that there is no spontaneous symmetry breaking. To confirm this issue, a Monte Carlo simulation was performed with the order parameter L = 1/N2(ltr U12) for d = 1,2,3 dimensions and N ranging from 2 to 50. By factorization, L should vanish as N --, 00. This indeed happens, both for weak and strong coupling in the quenched model. The conclusion is that there is no spontaneous symmetry breaking in this model and the vacuum state in the continuum limit is the correct one. The lattice chiral model is defined by the action SL = flNx$
Re ]tr(W)
u’(x
+ ~111,
(5)
where x, ~1label the sites and directions of a hypercubic lattice and the D”s are elements of the group U(N) The reduced model a la Parisi [3,7] equivalent to this one at N = m is given by the action S = PN F
Re [tr(UDr U'D:)]
,
(6)
where D, is a diagonal matrix of momenta D, =Diag{DL,DL=exp(ipl),
i= I,2 ,..., N, -n<~~
and the p:s are distributed uniformly between --K and rr. The action of eq. (6) obviously has a [U( 1) X U( l)lN/U( The partition function is given by Z =
s
(7) 1) invariance.
dU exp(S),
(8)
where dU is the Haar measure on U(N). The free energy, according to the rules of quenching F=j(
[2,3,8]
is defined by
ndpl)lnZ. i,p
(9)
Z will now be computed to one-loop approximation First, rewrite the action as S =flN2d -y
L
c
P
in weak coupling (large 0).
tr(F,F,+),
(10)
with
(11)
F/.,= [KD,l. Maximum action corresponds to Fp = 0 or to u’s which are diagonal. Hence, a convenient struct perturbation theory is [2,8,9] 372
parametrization
to con-
Volume 1208, number 4,5,6
PHYSICS LETTERS
13 January 1983
U = exp(ia) X exp(-ia)
(12)
where h is a diagonal matrix of the eigenvalues of U, i= 1,2 ,..., N, hi=exp(iei),
h=Diag{hj,
-a
(13)
and “a” is a hermitean N X N matrix. The Haar measure becomes [8,9] dU= ( ndX’
(144
i
where
(14b, 4
d2hij = d2gii p(a),
and the last equation is to be thought of as a matrix equation. To compute 2 to one-loop in weak coupling, one must expand the non-gaussian parts of exp(S) in a power series in a up to 0(a4) and the density p(a) to O(a2). This is tedious but straightforward. The result is
z = exp(dpN2)J(
c dX’) ( iFj 2
~@)=$ II d2Qijexp(-oNis i>j
I”ii12 Ihi-XI12Dj~
12 Cl&.12 _flP! 3 i*j
2
12
”
c
i#l,l#k,k+j,j#i
T
IA’ -Ai I2 “0))
c g i#k,k#j,j#l,l+i
(15)
)
[Dijaikakjajlali(Xj
[Dijailalkakjaji {(hi-Xi)
[XT-h;
- 2X, + Xi) (h; - 2X; t A:)]
+ 3 (A; - Xi)] + CL)]
)
+ higherordersin ljD,(lQ
where D..= 4
c u
IDi _Dj12. !J
(17)
P
As discussed in ref. [2], the issue of spontaneous symmetry breaking depends on the effective action governing the dynamics of the h’s. If the distribution of these h’s [which is determined by the curly bracket in eq. (15)] is such that the x’s are uniformly distributed on the unit circle, there is no spontaneous symmetry breaking. If the h’s can clump together on the unit circle th_esymmetry is broken. To decide the issue, one must compute Z(X). To zeroth order (no loops),~(X) is given solely by integrating over the exponential term in eq. (16). These integrals are easily done and the result is Z - exp(dpN2)
(1/fl)(NZ-NJ/2
s 7
dh’.
(18)
It is therefore apparent that to this order, the distribution of the A’s is uniform on the unit circle and there is no spontaneous symmetry breaking. To compute the one-loop correction to eq. (18), one introduces in the usual way the generating function
G(Y)=Jipj d2aij exp
(-PNi$ laijI2
Qij + ij z
j %j 7ij)
Y
(194 373
Volume 120B, number 4,5,6
1
Cr..r..
ON i>i
‘1 1’
13 January 1983
PHYSICS LETTERS
Qf’
,
Q,>O,
From this, it is easy to perform the integrations After some algebra, this gives
Z-exp(dpN2)(l/p)(NZ-N)~2~
ViJ,
i#i.
(19b)
over the Uii’s in eq. (16) by differentiating
7 dhi{exp[-fl-lf(D,X)]
with respect to the 7’~.
+0(1/P2)),
(20)
where j-(D,X)=-l
c ’ 6 i+j ihi -hi
1
c
-~i#k,j#k
L,’ 12 Dij
_____
c-
lXi-2Xk
Dij
32Ni+k,j#kDikDjk
(4X/f -hi
DkjIhi-Ak121hj-Ak12
+
Xjl* ___
Ihi-Xk121~j-_k12
- 3Xj) + C.C.
(21)
’
f(D, h) is the effective action for the h integration. The h’s like to distribute themselves so as to minimizef. If the uniform distribution of X’s remains an absolute minimum off, the symmetry remains unbroken to one-loop order. This issue is most readily decided numerically. This is done as follows. Choose Dt= exp(ipL) with the p;‘s uniformly spaced in the interval [-n, n] [2,5,7]. Choose hi = exp(i0i) with the 8’s uniformly spaced in the interval (-@, 4). Compute
(22)
~(~)=N-*[f(D,h(~>)-f(D,h(n))l,
as N + 00. When @< rr, the h’s are forced closer together on the unit circle. If the uniform distribution of h’s were an unstable saddle point, .$(@)would be negative for 9 < rr. In fig. 1, ,$‘($J)is plotted versus @forN varying from 2 to 50 ford = 2 dimensions. It is clear that .$is always positive for large N. Thus, pushing the h’s together increases the action. Hence the uniform distribution for the h’s remains a stable saddle in the one-loop approximation. Note that for small N (N - 5 or less), the uniform distribution seems unstable. This may explain the results of ref. [8] where it was concluded that the h’s like to clump together. In one dimension, g(d) behaves just as in two dimensions and the conclusion is that the symmetry there also does not break as is well known from the exact solution. To obtain a more concrete proof for the absence of spontaneous symmetry breaking, one must resort to Monte Carlo simulation. Consider the order parameter L =Ne2(ltr
U12),
(23)
Fig. 1. The effective action r;(e) for the X’s when they are restricted to the interval ous values of N.
374
(-a,
@) ford
= 2 dimensions
and vari-
Volume
120B. number
4,5,6
PHYSICS
lp
d=l, xp=o.4,
=2.0
d=2,
13 January
LETTERS
l/3=12
x6=0.4,
d-3,
./3=Q8
* 0.2-
1983
1
-0.2 -
-0.2-
* A N 5 &
I
--
t
*
-&O.I
-
*
I
-O.l-
-Ol-
4
I;
+
+ ++*
* -
* (. 0
* +
. * : ,
I 0.2
0.1
I 0.3
+ .*I 0
I/N Fig. 2. The order parameter strong coupling (crosses).
+ t
*
I cll
I 0.2
I a3
+ 0
I/N
L versus l/N for N varying
from 4 to 50 in d = 1,2,3
1 0.1
I 0.2
I Q3
I/N dimensions
both for weak coupling
(dots)
and
where the average is in the sense of eqs. (l)-(4). If the eigenvalues of U like to cluster together, it is easy to see that L does not vanish, even at N = ~0. Indeed, if the distribution of 0 i’s pi = exp(iei)] is gaussian with width u, then for u < 71,L = exp(-a2/2) at N= 00. If, on the other hand, the distribution of the 6 i’s is uniform on the unit circle, L tends to vanish when N + * in general like l/N2. However, in the present case, it is sufficient that L vanishes like l/N because the quenched model is expected to be correct only to 0(1/N) [2,3,5,7] and not 0(l/N2). A standard Monte Carlo simulation was done fo determine L both in the weak and strong coupling region for N varying from 2 to 50. The quenched momentap: were chosen uniformly in the interval [-r, 711but care was taken to compute L by averaging it over many different random permutations of these momenta. In fig. 2, L is plotted versus l/N ford = 1,2,3 dimensions. In strong coupling, L was evaluated ford = 1,2 for /3 = 0.4 and it is clear that as N + m, L vanishes faster than l/N in this regime. The choice of which j3 value to use to determine the weak coupling behaviour of L was made as follows. Consider the order parameter E=
& &
(c
tr(UDPPDl
+ h.c.))
=(l/N2d)
aE/W.
(24)
P
E has the weak coupling expansion E = 1 - (1/2d/3) (l-
l/N) + O(l/p2).
(25)
by numerical simulations that at /3 = 2.0,0 = 1.2, /3 = 0.8, respectively ford = 1,2,3 dimensions, between 0.75-0.8 and was well within the weak coupling region described by eq. (25). From fig. 2, it is also clear that in weak coupling, L vanishes as N -+ 00 at least as fast as l/N for two and three dimensions and like l/N2 ford = 1. As stated above, this is sufficient in the quenched model because it is expected to be correct only to 0(1/N) [2,3,5,7]. Fig. 2 also shows that if one does simulations in the quenched model with N - 10, one will see a rather large signal for L. This should not be interpreted to mean that the symmetry breaks spontaneously because it is an artifact of the finiteness of N. This observation probably explains why the authors of ref. [8] concluded that this model suffers spontaneous symmetry breaking. In summary, it has been shown in this paper both by a one-loop weak coupling computation of the distribution of eigenvalues of the matrix CJand by Monte Carlo simulation on the complete quenched model that the [u( 1) U( l)“/U(l) invariance of the reduced quenched chiral U(N) model at N = 00 does not break spontaneously. It was determined
E had a value somewhere
375
Volume
1208,
number
4.56
PHYSlCS
13 January
LETTERS
I would like to thank Itzhak Bars, Dietrich Foerster, Murat Giinaydin, Yankielowicz for very useful discussions.
Giorgio Parisi, Spenta Wadia and Shimon
References [l] [2] [3] [4] [5] [6] [7] [8] [9]
T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063. G. Bhanot, U. Heller and H. Neuberger, Phys. Lett. 113B (1982) 47; 11SB (1982) G. Parisi, Phys. Lett. 112B (1982) 463. G. Parisi and Y. Zhang, Phys. Lett. 114B (1982) 319. D. Gross and Y. Kitazawa, Princeton University preprint PRE 2564 (April 1982). S. Das and S. Wadia, University of Chicago preprint EFI 82-15 (April 1982). I. Bars, M. Gunaydin and S. Yankielowicz, CERN preprint TH. 3379 (1982). U. Heller and H. Neuberger, Phys. Rev. Lett. 49 (1982) 621. M.L. Mehta, Random matrices (Academic Press, New York, 1967).
376
1983
237.