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Saddle-points in the twisted reduced chiral model
Nuclear Physics B240[FS12] (1984) 270-280 0 North-Holland Publishing Company
SADDLE-POINTS IN THE TWISTED REDUCED CHIRAL MODEL F.R. KLINKHAMER Institute for Theoretical Physics, Universityof California, Santa Barbara, CA 93106, USA Received 5 January 1984 (Revised 27 March 1984)
It is shown that the abnormal behaviour found in numerical simulations of the twisted reduced SU( N * cc) chiral model in two dimensions results from fluctuations around a non-trivial extremum of the action. The possible role of such saddle-points in the crossover to the strong coupling regime is discussed.
1. Introduction In the limit of a large number N of “colors” many interesting field theories in d dimensions are equivalent to a reduced model containing only a few (N x N matrix) variables [l]. Notably this applies to the SU(N) lattice gauge theory in d = 4 space-time dimensions and to the SU( N) chiral model in d = 2 dimensions. Recently this d = 2 reduced chiral model, in the twisted version 121, was studied in extensive Monte Carlo simulations by Das and Kogut [3]. At intermediate coupling strengths they found that the system could be in an “abnormal” state with somewhat lower internal energy and very different behaviour of the correlation function. In the present article I will argue that this abnormal state corresponds to the first non-trivial extremum of the action. This reminds of the alleged role of instantons in gauge field theories [4]. But the typical Boltzmann factor there is exp( - 8n2N/g2N), so that one thinks (naively) that these configurations cannot be very relevant for the N = co theory, where in the N + co limit g2N is held fixed. But ‘t Hooft [5] has shown that there may be other types of extrema that do survive the large-N limit. These were also found [6] in the twisted reduced model of the gauge theory. The extrema of the d = 2 reduced chiral model are similar, but not quite the same. Hence, our results might also shed some light on the precise relation [7] of d = 4 gauge and d = 2 chiral field theories. The contents of this paper are as follows. In sect. 2 I describe the model and its extrema around which in weak coupling the fields fluctuate. In sects. 3 and 4 I show that the (ab)normal numerical results [3] of the internal energy and correlation function can be explained by considering the (first non-) zero action extremum. Finally in sect. 5 I discuss the possible role of these saddle-points in making the 270
F. R. Klinkhamer / Twisted reduced chiral model
271
crossover to the strong coupling regime and also comment briefly on related results in d = 4 gauge theories.
2. Extrema I will specialize immediately to d = 2 dimensions and will use basically the same notation as ref. [3]. The model is given by the partition function and action Z =/dUePas,
S=
i
(2.1)
Tr(1 - Ur,U+rJ
+ h.c.),
(2.2)
p=l
where dU is the normalized Haar measure over W(N) have special commutation relations rlr2
=
and the SU(N) matrices r,
e2~~n~~/Nr2rl.
(2.3)
In order to reproduce the Feynman expansion of the field theory one must choose the twist to be n *i= -ni*=
-1.
(2.4)
Specifically, one may choose r, and r. to be the Q and P matrices of ref. [5]. To calculate in the reduced model the corresponding value of an average (O(U(x))) in the N = cc field theory, where 0 is any combination of the site variables U(x) that is invariant under the global symmetries of the field theory, one uses on 0 the reduction rule
u(x) -+ D(x)UD(x)+,
(2.5)
and then integrates with the Boltzmann weight as given in (2.1). The use in (2.5) of the matrices
(2.6) maps space-time translations into manipulations in group space, which is large enough for N --, co. Consider now the following set of “configurations”:
{UlU=D(n,m)}.
(2.7)
F. R. Klinkhamer /
212
Twisted reduced chiral model
For m = - n these will turn out to be the extrema promised in the introduction. small fluctuations around these configurations U=
I expand the “potential”
D(n, m)e’“.
For
(24
a( = at) as follows
a=
N-l
N-l
91 =o
92’0
c c
(not:
4qMqL
(2.9)
9, =O=q*)
where u(q) E C and (2.10) so that conjugation with rP r,A (q)r{
= eSniqJNA( q)
(2.11)
shows that the q act just as lattice momenta, cf. (2.5). The sum in (2.9) contains N* - 1 terms and covers SU(N). Inserting (2.8) in the action I find to order a0 2N(2 - cos(27rn/N)
(2.12)
- cos(2mm/N))
and an order u* term g(ql
-m) ] +cos [$q*+“)]-
cos%m
- cos$n (2.13)
Note that q-independent factors result from expanding one of the U’s to second order and keeping the other D(n, m) and that the O(u) term vanishes by Tr A(q) = 0. Setting m = -n + d, n > 0, and introducing the short-hand notation i = 27rZ/N, some cossinology allows to write the large square bracket in (2.13) as [ cos ii (cos qF2+ cos q,cos 2) - sin iI (sin q2 + sin qrcos d) + sin d( sin ti cos q1 + cos pi sin &) - cos ti (1 + cos d) - sin n sin d] .
(2.14)
We are interested in the lowest action levels, so that n - 0. Thus (2.14) reduces in the limit N+ca to [ cos q2 + cos qrcos 2 + sin qi sin C?- 1 - cos d] .
(2.15)
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273
Assume that the 1a(q) 1 distribution is smooth and peaked at q - 0. For small q, one sees that the third term of (2.15) destabilizes the extremum. Also if present (2.15) would not correspond to the inverse propagator of the field theory, moreover, the land 2-directions would be treated differently. Thus I set d = 0 or m=
-n,
(2.16)
where the possibility of having d = )N is not interesting, since it would lead to a large action (2.12). In conclusion one has for 0 < n +z $N a saddle-point approximation to the action given by
2N 2-2cos.T i
-cos[+(q,+n)]
+ C[2cosF 4 -cos
[
%(q*+n)
[u(q)j” . 11 1
(2.17)
Of course, the levels I considered above (T/T;“, 0 -Cn -C +N) are degenerate with those that switch the role of r, and r,, which can be achieved by replacing n by -n or, equivalently, n by N - n. This degeneracy factor of 2 is important in solving the model, for example when calculating the partition function (2.1). But in the present article I will only consider contributions from a given extremum. Actually the extrema given by (2.16) are not entirely stable at finite N, see the appendix at the end of this paper. Thus for small n one has, expanding the cosine in (2.17) extrema with a (classical) weight exp( - 8m2n2~/N),
(2.18)
which survives the N --+ cc limit where P/N is held constant. In sect. 5 I will compare this with the situation in d = 4 SU( N + cc) gauge theories, but first I turn to the calculation of two observables using the saddle-point approximation established in this section.
3. Internal energy Defining the mean internal energy as
(3.1)
F. R. Klinkhamer /
214
Twisted reduced chiral model
it becomes around the extremum U = D( n, - n ) (E)‘“) = 2 cos-
2mn N
+ c cos 27f
Y$(ql+n)
4
t
[
[
1 [
(n) 1 )
1
2m
+cos
-2cos7 27rn 14qV
$qz+n)
)
gaussian
(3.2) where in the expectation value on the right-hand side the action (2.17) must be used. The large square bracket for n -=x$N is approximately equal to
cos% +
(3.3)
N
which is proportional
to what one would have around the normal n = 0 minimum.
1.5
W >-
X0 - l -Y&V+=-
00
l
-
l
0
l
a
00 l
l
1.4-
X ,_-2L*-x_x,-_ X X X
xx
xxx
0
0
= 0
XXxX
x
.-SC-xx
-
L--x--A-XaG--u x
x2-
z < z
CL E z-
1.3-
2-1.2I 0
2000 I I I
I
6000 I I I 10000
I I I
1
10000 I I I 20000
I
ITERATION NUMBER (:) Fig. 1. Two sets of Monte Carlo data [3] on the internal energy are shown as dots (N = 36) and crosses (N = 24). The energy drops at iteration - 5000 (N = 24) and - 6000 (N = 36) agree with the interpretation of a change of saddle-point around which the field fluctuates, viz. from no) = 0 to ]t~(~)]= 1. The energy bands between the levels ]n 1 = 0, 1,2 and 3 were obtained from the formulae (3.4) and (3.5).
F. R. Klinkhamer /
215
Twisted reduced chiral model
Also the gaussian weight used in (3.2) is nearly equal to that for n = 0. So I estimate (E)‘“’
=
COST {2+ (E)&ian}>
(3.4)
with
(E)&,,i,= -i(P/N)-‘*
(3.5)
Now compare this in fig. 1 with the numerical results [3] for N = 24 and 36 at nearly equal couplings P/N = 0.50 and 0.52. Because /3/N is not large at all I will use (3.4) and (3.5) to calculate the energy differences and choose (E)(O) to fit the data best. Both sets of data show that the energy difference observed is consistent with that expected between the n = 0 and n = 1 extrema. The scatter is quite large, but the nearest other level expected (n = 2) is clearly separated. As a final remark I think it is unlikely that during the fluctuations around the lower level the state n might change to the degenerate one with n’ = - n, but if the third before last N = 24 configuration in fig. 1 is indeed a brief return to the nC3)= 0 extremum as claimed in ref. [3] (and not just a large fluctuation), then the last two configurations belonging to the extremum with lnC4’1= 1 may or may not have, with equal probability, nC4) equal to the n@) of the configurations before the jump.
4. Correlation function Define the two-point correlation function between 0 and x = (x1, x2) as, cf. (2.5) G(x) = (&Tr( Again developing around the I’;r;”
Uo(x)Uto(x)t
+ h.c.))
.
(4.1)
extremum gives
cos[ %(x1+4]-( ~[c+h+nx*)]
G(x)'"'=
-cos
$$lx,+nx,+q.x)1a(q)1’ [
(n)
11 )
gaussian
.
(4.2)
The square bracket with the two cosines in (4.2) can be rewritten as cos [
[
3x,+
nx*) ](I-cos[~(q.x)])+sin[~(xr+x,)]sin[~(q.x)]]. (4.3)
F. R. Klinkhumer /
216
Twisted reduced chirul model
so that, assuming the ]a( q) 1 to be dominant at small momenta q - 0, one has for N+Cc
Just as for (3.2) this gaussian part at n < +N is approximately the same as that calculated around n = 0. Remark that the multiplicative factor in (4.4) is not in general equal to cos(27rn ]x]/N). For N = 36 fig. 2 shows the Monte Carlo data [3] of G(x) with the normal and abnormal behaviour, which clearly fits the expected behaviour (4.4) for n = 0 and n = 1 with the { }-factor equal to 0.40. But for a lower value of N, viz. 24, the fit is not so good: (i) the factors { }(‘) appear to differ for n=O and n=l; and (ii) with { }(l) set to 0.22 the agreement with the cosine curve (dashed) is less close than for N = 36, but still the qualitative behaviour is born out correctly by (4.4). Apparently for N = 24 one should try to calculate (4.2) better than I did above.
I
g
I
1
I
I
I
I
I
I
0.8 8
zc
-
go.4I? 6 2 ii! S 8
:
u &*~tx--)4tY*-.rreX0*
c
0
-0.4 I
0
I
I
I
I
I
6 DISTANCE
12
I
I
I_
18
x
Fig. 2. Monte Carlo values [3] for the N = 36 two-point correlation function G(x) are shown (dots). The (lower) upper branch corresponds to (ab)normaJ behaviour, which is related to the existence of a (non-)trivial extremum; the full curves result from perturbation theory (4.4) for (n = 1) n = 0. Also shown are the data (crosses) and perturbation theory curves (dashed) for N = 24.
F. R. Klinkhamer /
Twisted reduced chiral model
211
Note that fluctuations around an arbitrary (n, m) extremum give in zeroth order G(x)=cos[~(nx,-mx,)].
(4.5)
Hence the G(x,,O) a cos(277x,/N) behaviour in fig. 2 could also arise from the n = 0, Irn1 = 1 level. Only simultaneous measurements of E can discriminate between the levels that operate (see also the appendix).
5. Discussion In the preceding I have argued that the abnormal behaviour found at intermediate couplings by numerical simulations [3] is a signal of having fluctuations around a non-trivial extremum of the action. Both the internal energy (sect. 3) and the correlation function (sect. 4) indicate that this is the n = 1 level, which has lowest non-zero action. Even longer Monte Carlo runs at /3/N - 0.50 and N = 36, say, might show higher action levels. Of course, these levels are also present at weaker couplings, but to see them would require runs of impossibly long lengths. Also the value of N to be used is a compromise; consider in fig. 1 the energy spacings and widths for the different N values. We may be grateful that the chiral model allowed these compromises, so that the numerical results [3] could show for the first time the validity of the semiclassical (or saddle-point) approximation. Naively one would expect this approximation to be valid as long as /3/N > l/877’, compare (2.18). The crossover (see below) at /3/N - 0.30 [3] shows that this is too naive indeed. For the instantons of QCD the folklore [4] is that towards the crossover quantum effects give something more like a liquid than like the ideal gas of instantons of the semiclassical approximation. Back to the chiral model, the Monte Carlo data suggest that for increasing couplings (or lattice spacings) the levels get more and more filled, but also the fluctuations grow larger up to the point (P/N - 0.2550.30?) that there are no more clear sectors labelled by n in configuration space*. Let me discuss this briefly. In fig. 3 the dots are N = 24 Monte Carlo averages [3] for the mean action (S) = 2N(2 - (E)), together with the curves (dashed) of the strong coupling expansion (S) = 4N(land of the standard (n = 0) perturbation
P/N),
(5.1)
theory
(s) = +N(P/N)-'.
*The referee has brought to my attention ref. [9], which discusses the rBle of stable first-order phase transitions for lattice gauge theories with extended Wilson actions.
(5.2) extrema
in
F. R. Klinkhamer /
218
Twisted reduced chiral model
Note that the value 4N of (S) at infinitely strong coupling nicely interpolates between the minimum (0) and maximum (8N) values of S, as reached by U = 1 and U = D(iN, - +N) for example. Comparing at /3/N - 0.60 the data and (5.2) I estimate the O(N*/p*) contribution, which extrapolates towards stronger coupling as shown by the full (n = 0) curve in fig. 3. I expect this to give a reasonably good approximation to (S)(‘) for /3/N >, 0.30. Also are presented the curves for the higher saddle-points, which were obtained by assuming (3.4) could be extended to (E)(“)
cos% {
2
(E)L~toquarticorder}
.
(5.3)
But the difference between (3.4) and (5.3) is not large, so that the general behaviour of the curves (at least for n < 3) in fig. 3 may be trusted. In the coupling interval indicated by the wavy line the fluctuations of S in the Monte Carlo run (fig. 1) are not too large and two distinct levels are seen: crosses and dots in fig. 3, the true value of (S) at these couplings is some average of them, roughly $(S( X) + S(O)). For smaller /3/N one may safely assume that the Monte Carlo procedure gives a good sampling for (S), while at larger P/N the runs used [3] were too short to show the higher levels, which in turn implies that these are not very important for the true
I
I
I
5N-
I
1
I
u
N=24
$4N\
6-
5 3NI= 8
~2: 4
“’ l
0
-
I
0.25
I
0.375
I
I
0.50
Fig. 3. The dots arc Monte Carlo results [3] for the mean action (S); at two values of P/N near 0.50 one should take the average of dot and cross values. The dashed curves are from the strong coupling expansion (5.1) and standard (n = 0) perturbation theory (5.2). The lowest full curve (n = 0) includes the 0( N*/p*) contribution as estimated from the data points at P/N 2 0.60. Higher saddle-point curves from (5.3) are drawn where they contribute significantly. For n > 3 our approximations in the calculation of (S)(“) become inaccurate: the drawn curves (n = 3,4,5) based on eq. (5.3) are expected to be too small. Also shown are the zero-coupling action levels (2.17) for n = 1,. ,6.
E R. Klinkhamer /
Twisted reduced chiral model
279
value of (S). I should have drawn these curves of (S)(“’ for all /3/N larger than 0.30, say, but I have drawn them only when they contribute significantly to (S)@Ota’), which is roughly given by the Monte Carlo data (dots). Of course, for P/N 5 0.48 the fluctuations are equal to or larger than the level separations. The quantitiue conclusion is that for p - 0.50 and - 0.375 only a few levels suffice [0 < n < M with M = 1 and 3, respectively, although they are getting fuzzy at /3 - 0.3751, but that at “the” crossover point /3/N - 0.30 their number is of the order of N. Also note that (S) appears smooth for all couplings, whereas the d = 4 gauge model has a transition at P/N - 0.36 [2]. Let me continue the comparison with large-N gauge theories in 4 dimensions, but this may also be of some relevance to the N = 3 theory. For the reduced d = 4 gauge model with symmetric twist (i.e. all N, equal Np = L, where N* = np=l,,_dNp and (rp)“: = 1; the twist (2.4) is also symmetric with L = N) has extrema [6] with
,&S=8n2k(P/N),
k~Z+/{6,11,...
},
(5.4)
which has some structure, just as (2.18) for the d = 2 chiral model. For the asymmetric twists ( Np f- Iv,) of ref. [8] it is known that although the levels remain the same as (5.4) the gap pattern can change. Still one expects the N = 00 physics to be twist independent as long as ‘VP+ CG for all p when N + co. So the meaning of having structure as in (2.18) and (5.4) is uncertain. Numerical simulations, such as the one [3] analyzed in this article, may be of some help in understanding these problems. I thank S.R. Das for sending me the preprint [3] that triggered off the present paper and that provided all the data points in its figures. Also I acknowledge support of the Netherlands Organization for the Advancement of Pure Research (ZWO). The Institute for Theoretical Physics is funded by the National Science Foundation under grant no. PHY 77-27084 and by the National Aeronautics and Space Administration.
Appendix The derivation leading to the preferred extrema (2.16) buried some fine points, which I will resurface in this appendix*. The crucial simplifications occurred near (2.15) in the main text: (i) the n/N + 0 limit taken and (ii) the assumed dominance of q values near zero. Actually one can see from (2.13) that for ally (n, m) # (0,O) pair there are unstable modes, for example qI = m and q2 = N - n if n, m > 0. Here only the lowest action levels with n and ]m] a N will be considered. * I thank P. van Baal for some useful comments.
280
F. R. Klinkhamer /
Twisted reduced chval model
The negative of the large square bracket part of (2.13) can be written as, cf. (2.14),
-sind{sinEcosLj,+costisincf,}].
(A.1)
For the extrema with d = 0 there still are some destabilizing modes because of the sin ii part in (A.l). For these modes at least one of the q’s lies in the interval [C,N-l]withCoforderN-n.HenceforN + CQand n fixed the relative number of unstable modes decreases. Now consider extrema with d # 0, and let us specialize to the case of n and Jdl of the same order, but both < N. Recall that n is taken to be positive. For d > n there are other unstable modes than those mentioned above, for example q2 = 0 and q1 small enough. Namely (A.l) reduces then to 1 -cosq,
+(E--J)sinq,,
(A-2)
which is negative for q1 E [l, D], with D of order n - d. Hence levels with n, m > 0 are unstable already at smallmomenta. For the small n values considered in the text this implies that besides the d = 0 levels (0,O) and (1, - 1) also the (1,O) level could occur and even the (2,0) one. Indeed the N = 24 data in fig. 1 seem to show some preference for E values midway between the (0,O) and (1, - 1) levels drawn, which would correspond to an extremum of type (1,O). It is a pity that the scatter in the data is so large.
References [l] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063 [2] A. Gonzalez-Arroyo and M. Okawa, Phys. Rev D27 (1983) 2397 [3] S.R. Das and J. Kogut, Nucl. Phys. B235[FSll] (1984) 521 [4] C.G. Calan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717; E.V. Shuryak, Nucl. Phys. B203 (1982) 93,116, 140 [5] G. ‘t Hooft, Comm. Math. Phys. 81 (1981) 267 [6] P. van Baal, Comm. Math. Phys. 92 (1983) 1 [7] A.A. Migdal, Sov. Phys. JETP 42 (1975) 413, 743; L. Kadanoff, Ann. of Phys. 100 (1975) 359 [8] F.R. Klinkhamer and P. van Baal, Nucl. Phys. B237 (1984) 274 [9] C.P. Bachas and R.F. Dashen, Nucl. Phys. B210[FS6] (1982) 583
SADDLE-POINTS IN THE TWISTED REDUCED CHIRAL MODEL F.R. KLINKHAMER Institute for Theoretical Physics, Universityof California, Santa Barbara, CA 93106, USA Received 5 January 1984 (Revised 27 March 1984)
It is shown that the abnormal behaviour found in numerical simulations of the twisted reduced SU( N * cc) chiral model in two dimensions results from fluctuations around a non-trivial extremum of the action. The possible role of such saddle-points in the crossover to the strong coupling regime is discussed.
1. Introduction In the limit of a large number N of “colors” many interesting field theories in d dimensions are equivalent to a reduced model containing only a few (N x N matrix) variables [l]. Notably this applies to the SU(N) lattice gauge theory in d = 4 space-time dimensions and to the SU( N) chiral model in d = 2 dimensions. Recently this d = 2 reduced chiral model, in the twisted version 121, was studied in extensive Monte Carlo simulations by Das and Kogut [3]. At intermediate coupling strengths they found that the system could be in an “abnormal” state with somewhat lower internal energy and very different behaviour of the correlation function. In the present article I will argue that this abnormal state corresponds to the first non-trivial extremum of the action. This reminds of the alleged role of instantons in gauge field theories [4]. But the typical Boltzmann factor there is exp( - 8n2N/g2N), so that one thinks (naively) that these configurations cannot be very relevant for the N = co theory, where in the N + co limit g2N is held fixed. But ‘t Hooft [5] has shown that there may be other types of extrema that do survive the large-N limit. These were also found [6] in the twisted reduced model of the gauge theory. The extrema of the d = 2 reduced chiral model are similar, but not quite the same. Hence, our results might also shed some light on the precise relation [7] of d = 4 gauge and d = 2 chiral field theories. The contents of this paper are as follows. In sect. 2 I describe the model and its extrema around which in weak coupling the fields fluctuate. In sects. 3 and 4 I show that the (ab)normal numerical results [3] of the internal energy and correlation function can be explained by considering the (first non-) zero action extremum. Finally in sect. 5 I discuss the possible role of these saddle-points in making the 270
F. R. Klinkhamer / Twisted reduced chiral model
271
crossover to the strong coupling regime and also comment briefly on related results in d = 4 gauge theories.
2. Extrema I will specialize immediately to d = 2 dimensions and will use basically the same notation as ref. [3]. The model is given by the partition function and action Z =/dUePas,
S=
i
(2.1)
Tr(1 - Ur,U+rJ
+ h.c.),
(2.2)
p=l
where dU is the normalized Haar measure over W(N) have special commutation relations rlr2
=
and the SU(N) matrices r,
e2~~n~~/Nr2rl.
(2.3)
In order to reproduce the Feynman expansion of the field theory one must choose the twist to be n *i= -ni*=
-1.
(2.4)
Specifically, one may choose r, and r. to be the Q and P matrices of ref. [5]. To calculate in the reduced model the corresponding value of an average (O(U(x))) in the N = cc field theory, where 0 is any combination of the site variables U(x) that is invariant under the global symmetries of the field theory, one uses on 0 the reduction rule
u(x) -+ D(x)UD(x)+,
(2.5)
and then integrates with the Boltzmann weight as given in (2.1). The use in (2.5) of the matrices
(2.6) maps space-time translations into manipulations in group space, which is large enough for N --, co. Consider now the following set of “configurations”:
{UlU=D(n,m)}.
(2.7)
F. R. Klinkhamer /
212
Twisted reduced chiral model
For m = - n these will turn out to be the extrema promised in the introduction. small fluctuations around these configurations U=
I expand the “potential”
D(n, m)e’“.
For
(24
a( = at) as follows
a=
N-l
N-l
91 =o
92’0
c c
(not:
4qMqL
(2.9)
9, =O=q*)
where u(q) E C and (2.10) so that conjugation with rP r,A (q)r{
= eSniqJNA( q)
(2.11)
shows that the q act just as lattice momenta, cf. (2.5). The sum in (2.9) contains N* - 1 terms and covers SU(N). Inserting (2.8) in the action I find to order a0 2N(2 - cos(27rn/N)
(2.12)
- cos(2mm/N))
and an order u* term g(ql
-m) ] +cos [$q*+“)]-
cos%m
- cos$n (2.13)
Note that q-independent factors result from expanding one of the U’s to second order and keeping the other D(n, m) and that the O(u) term vanishes by Tr A(q) = 0. Setting m = -n + d, n > 0, and introducing the short-hand notation i = 27rZ/N, some cossinology allows to write the large square bracket in (2.13) as [ cos ii (cos qF2+ cos q,cos 2) - sin iI (sin q2 + sin qrcos d) + sin d( sin ti cos q1 + cos pi sin &) - cos ti (1 + cos d) - sin n sin d] .
(2.14)
We are interested in the lowest action levels, so that n - 0. Thus (2.14) reduces in the limit N+ca to [ cos q2 + cos qrcos 2 + sin qi sin C?- 1 - cos d] .
(2.15)
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Twisted reduced chiral model
273
Assume that the 1a(q) 1 distribution is smooth and peaked at q - 0. For small q, one sees that the third term of (2.15) destabilizes the extremum. Also if present (2.15) would not correspond to the inverse propagator of the field theory, moreover, the land 2-directions would be treated differently. Thus I set d = 0 or m=
-n,
(2.16)
where the possibility of having d = )N is not interesting, since it would lead to a large action (2.12). In conclusion one has for 0 < n +z $N a saddle-point approximation to the action given by
2N 2-2cos.T i
-cos[+(q,+n)]
+ C[2cosF 4 -cos
[
%(q*+n)
[u(q)j” . 11 1
(2.17)
Of course, the levels I considered above (T/T;“, 0 -Cn -C +N) are degenerate with those that switch the role of r, and r,, which can be achieved by replacing n by -n or, equivalently, n by N - n. This degeneracy factor of 2 is important in solving the model, for example when calculating the partition function (2.1). But in the present article I will only consider contributions from a given extremum. Actually the extrema given by (2.16) are not entirely stable at finite N, see the appendix at the end of this paper. Thus for small n one has, expanding the cosine in (2.17) extrema with a (classical) weight exp( - 8m2n2~/N),
(2.18)
which survives the N --+ cc limit where P/N is held constant. In sect. 5 I will compare this with the situation in d = 4 SU( N + cc) gauge theories, but first I turn to the calculation of two observables using the saddle-point approximation established in this section.
3. Internal energy Defining the mean internal energy as
(3.1)
F. R. Klinkhamer /
214
Twisted reduced chiral model
it becomes around the extremum U = D( n, - n ) (E)‘“) = 2 cos-
2mn N
+ c cos 27f
Y$(ql+n)
4
t
[
[
1 [
(n) 1 )
1
2m
+cos
-2cos7 27rn 14qV
$qz+n)
)
gaussian
(3.2) where in the expectation value on the right-hand side the action (2.17) must be used. The large square bracket for n -=x$N is approximately equal to
cos% +
(3.3)
N
which is proportional
to what one would have around the normal n = 0 minimum.
1.5
W >-
X0 - l -Y&V+=-
00
l
-
l
0
l
a
00 l
l
1.4-
X ,_-2L*-x_x,-_ X X X
xx
xxx
0
0
= 0
XXxX
x
.-SC-xx
-
L--x--A-XaG--u x
x2-
z < z
CL E z-
1.3-
2-1.2I 0
2000 I I I
I
6000 I I I 10000
I I I
1
10000 I I I 20000
I
ITERATION NUMBER (:) Fig. 1. Two sets of Monte Carlo data [3] on the internal energy are shown as dots (N = 36) and crosses (N = 24). The energy drops at iteration - 5000 (N = 24) and - 6000 (N = 36) agree with the interpretation of a change of saddle-point around which the field fluctuates, viz. from no) = 0 to ]t~(~)]= 1. The energy bands between the levels ]n 1 = 0, 1,2 and 3 were obtained from the formulae (3.4) and (3.5).
F. R. Klinkhamer /
215
Twisted reduced chiral model
Also the gaussian weight used in (3.2) is nearly equal to that for n = 0. So I estimate (E)‘“’
=
COST {2+ (E)&ian}>
(3.4)
with
(E)&,,i,= -i(P/N)-‘*
(3.5)
Now compare this in fig. 1 with the numerical results [3] for N = 24 and 36 at nearly equal couplings P/N = 0.50 and 0.52. Because /3/N is not large at all I will use (3.4) and (3.5) to calculate the energy differences and choose (E)(O) to fit the data best. Both sets of data show that the energy difference observed is consistent with that expected between the n = 0 and n = 1 extrema. The scatter is quite large, but the nearest other level expected (n = 2) is clearly separated. As a final remark I think it is unlikely that during the fluctuations around the lower level the state n might change to the degenerate one with n’ = - n, but if the third before last N = 24 configuration in fig. 1 is indeed a brief return to the nC3)= 0 extremum as claimed in ref. [3] (and not just a large fluctuation), then the last two configurations belonging to the extremum with lnC4’1= 1 may or may not have, with equal probability, nC4) equal to the n@) of the configurations before the jump.
4. Correlation function Define the two-point correlation function between 0 and x = (x1, x2) as, cf. (2.5) G(x) = (&Tr( Again developing around the I’;r;”
Uo(x)Uto(x)t
+ h.c.))
.
(4.1)
extremum gives
cos[ %(x1+4]-( ~[c+h+nx*)]
G(x)'"'=
-cos
$$lx,+nx,+q.x)1a(q)1’ [
(n)
11 )
gaussian
.
(4.2)
The square bracket with the two cosines in (4.2) can be rewritten as cos [
[
3x,+
nx*) ](I-cos[~(q.x)])+sin[~(xr+x,)]sin[~(q.x)]]. (4.3)
F. R. Klinkhumer /
216
Twisted reduced chirul model
so that, assuming the ]a( q) 1 to be dominant at small momenta q - 0, one has for N+Cc
Just as for (3.2) this gaussian part at n < +N is approximately the same as that calculated around n = 0. Remark that the multiplicative factor in (4.4) is not in general equal to cos(27rn ]x]/N). For N = 36 fig. 2 shows the Monte Carlo data [3] of G(x) with the normal and abnormal behaviour, which clearly fits the expected behaviour (4.4) for n = 0 and n = 1 with the { }-factor equal to 0.40. But for a lower value of N, viz. 24, the fit is not so good: (i) the factors { }(‘) appear to differ for n=O and n=l; and (ii) with { }(l) set to 0.22 the agreement with the cosine curve (dashed) is less close than for N = 36, but still the qualitative behaviour is born out correctly by (4.4). Apparently for N = 24 one should try to calculate (4.2) better than I did above.
I
g
I
1
I
I
I
I
I
I
0.8 8
zc
-
go.4I? 6 2 ii! S 8
:
u &*~tx--)4tY*-.rreX0*
c
0
-0.4 I
0
I
I
I
I
I
6 DISTANCE
12
I
I
I_
18
x
Fig. 2. Monte Carlo values [3] for the N = 36 two-point correlation function G(x) are shown (dots). The (lower) upper branch corresponds to (ab)normaJ behaviour, which is related to the existence of a (non-)trivial extremum; the full curves result from perturbation theory (4.4) for (n = 1) n = 0. Also shown are the data (crosses) and perturbation theory curves (dashed) for N = 24.
F. R. Klinkhamer /
Twisted reduced chiral model
211
Note that fluctuations around an arbitrary (n, m) extremum give in zeroth order G(x)=cos[~(nx,-mx,)].
(4.5)
Hence the G(x,,O) a cos(277x,/N) behaviour in fig. 2 could also arise from the n = 0, Irn1 = 1 level. Only simultaneous measurements of E can discriminate between the levels that operate (see also the appendix).
5. Discussion In the preceding I have argued that the abnormal behaviour found at intermediate couplings by numerical simulations [3] is a signal of having fluctuations around a non-trivial extremum of the action. Both the internal energy (sect. 3) and the correlation function (sect. 4) indicate that this is the n = 1 level, which has lowest non-zero action. Even longer Monte Carlo runs at /3/N - 0.50 and N = 36, say, might show higher action levels. Of course, these levels are also present at weaker couplings, but to see them would require runs of impossibly long lengths. Also the value of N to be used is a compromise; consider in fig. 1 the energy spacings and widths for the different N values. We may be grateful that the chiral model allowed these compromises, so that the numerical results [3] could show for the first time the validity of the semiclassical (or saddle-point) approximation. Naively one would expect this approximation to be valid as long as /3/N > l/877’, compare (2.18). The crossover (see below) at /3/N - 0.30 [3] shows that this is too naive indeed. For the instantons of QCD the folklore [4] is that towards the crossover quantum effects give something more like a liquid than like the ideal gas of instantons of the semiclassical approximation. Back to the chiral model, the Monte Carlo data suggest that for increasing couplings (or lattice spacings) the levels get more and more filled, but also the fluctuations grow larger up to the point (P/N - 0.2550.30?) that there are no more clear sectors labelled by n in configuration space*. Let me discuss this briefly. In fig. 3 the dots are N = 24 Monte Carlo averages [3] for the mean action (S) = 2N(2 - (E)), together with the curves (dashed) of the strong coupling expansion (S) = 4N(land of the standard (n = 0) perturbation
P/N),
(5.1)
theory
(s) = +N(P/N)-'.
*The referee has brought to my attention ref. [9], which discusses the rBle of stable first-order phase transitions for lattice gauge theories with extended Wilson actions.
(5.2) extrema
in
F. R. Klinkhamer /
218
Twisted reduced chiral model
Note that the value 4N of (S) at infinitely strong coupling nicely interpolates between the minimum (0) and maximum (8N) values of S, as reached by U = 1 and U = D(iN, - +N) for example. Comparing at /3/N - 0.60 the data and (5.2) I estimate the O(N*/p*) contribution, which extrapolates towards stronger coupling as shown by the full (n = 0) curve in fig. 3. I expect this to give a reasonably good approximation to (S)(‘) for /3/N >, 0.30. Also are presented the curves for the higher saddle-points, which were obtained by assuming (3.4) could be extended to (E)(“)
cos% {
2
(E)L~toquarticorder}
.
(5.3)
But the difference between (3.4) and (5.3) is not large, so that the general behaviour of the curves (at least for n < 3) in fig. 3 may be trusted. In the coupling interval indicated by the wavy line the fluctuations of S in the Monte Carlo run (fig. 1) are not too large and two distinct levels are seen: crosses and dots in fig. 3, the true value of (S) at these couplings is some average of them, roughly $(S( X) + S(O)). For smaller /3/N one may safely assume that the Monte Carlo procedure gives a good sampling for (S), while at larger P/N the runs used [3] were too short to show the higher levels, which in turn implies that these are not very important for the true
I
I
I
5N-
I
1
I
u
N=24
$4N\
6-
5 3NI= 8
~2: 4
“’ l
0
-
I
0.25
I
0.375
I
I
0.50
Fig. 3. The dots arc Monte Carlo results [3] for the mean action (S); at two values of P/N near 0.50 one should take the average of dot and cross values. The dashed curves are from the strong coupling expansion (5.1) and standard (n = 0) perturbation theory (5.2). The lowest full curve (n = 0) includes the 0( N*/p*) contribution as estimated from the data points at P/N 2 0.60. Higher saddle-point curves from (5.3) are drawn where they contribute significantly. For n > 3 our approximations in the calculation of (S)(“) become inaccurate: the drawn curves (n = 3,4,5) based on eq. (5.3) are expected to be too small. Also shown are the zero-coupling action levels (2.17) for n = 1,. ,6.
E R. Klinkhamer /
Twisted reduced chiral model
279
value of (S). I should have drawn these curves of (S)(“’ for all /3/N larger than 0.30, say, but I have drawn them only when they contribute significantly to (S)@Ota’), which is roughly given by the Monte Carlo data (dots). Of course, for P/N 5 0.48 the fluctuations are equal to or larger than the level separations. The quantitiue conclusion is that for p - 0.50 and - 0.375 only a few levels suffice [0 < n < M with M = 1 and 3, respectively, although they are getting fuzzy at /3 - 0.3751, but that at “the” crossover point /3/N - 0.30 their number is of the order of N. Also note that (S) appears smooth for all couplings, whereas the d = 4 gauge model has a transition at P/N - 0.36 [2]. Let me continue the comparison with large-N gauge theories in 4 dimensions, but this may also be of some relevance to the N = 3 theory. For the reduced d = 4 gauge model with symmetric twist (i.e. all N, equal Np = L, where N* = np=l,,_dNp and (rp)“: = 1; the twist (2.4) is also symmetric with L = N) has extrema [6] with
,&S=8n2k(P/N),
k~Z+/{6,11,...
},
(5.4)
which has some structure, just as (2.18) for the d = 2 chiral model. For the asymmetric twists ( Np f- Iv,) of ref. [8] it is known that although the levels remain the same as (5.4) the gap pattern can change. Still one expects the N = 00 physics to be twist independent as long as ‘VP+ CG for all p when N + co. So the meaning of having structure as in (2.18) and (5.4) is uncertain. Numerical simulations, such as the one [3] analyzed in this article, may be of some help in understanding these problems. I thank S.R. Das for sending me the preprint [3] that triggered off the present paper and that provided all the data points in its figures. Also I acknowledge support of the Netherlands Organization for the Advancement of Pure Research (ZWO). The Institute for Theoretical Physics is funded by the National Science Foundation under grant no. PHY 77-27084 and by the National Aeronautics and Space Administration.
Appendix The derivation leading to the preferred extrema (2.16) buried some fine points, which I will resurface in this appendix*. The crucial simplifications occurred near (2.15) in the main text: (i) the n/N + 0 limit taken and (ii) the assumed dominance of q values near zero. Actually one can see from (2.13) that for ally (n, m) # (0,O) pair there are unstable modes, for example qI = m and q2 = N - n if n, m > 0. Here only the lowest action levels with n and ]m] a N will be considered. * I thank P. van Baal for some useful comments.
280
F. R. Klinkhamer /
Twisted reduced chval model
The negative of the large square bracket part of (2.13) can be written as, cf. (2.14),
-sind{sinEcosLj,+costisincf,}].
(A.1)
For the extrema with d = 0 there still are some destabilizing modes because of the sin ii part in (A.l). For these modes at least one of the q’s lies in the interval [C,N-l]withCoforderN-n.HenceforN + CQand n fixed the relative number of unstable modes decreases. Now consider extrema with d # 0, and let us specialize to the case of n and Jdl of the same order, but both < N. Recall that n is taken to be positive. For d > n there are other unstable modes than those mentioned above, for example q2 = 0 and q1 small enough. Namely (A.l) reduces then to 1 -cosq,
+(E--J)sinq,,
(A-2)
which is negative for q1 E [l, D], with D of order n - d. Hence levels with n, m > 0 are unstable already at smallmomenta. For the small n values considered in the text this implies that besides the d = 0 levels (0,O) and (1, - 1) also the (1,O) level could occur and even the (2,0) one. Indeed the N = 24 data in fig. 1 seem to show some preference for E values midway between the (0,O) and (1, - 1) levels drawn, which would correspond to an extremum of type (1,O). It is a pity that the scatter in the data is so large.
References [l] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063 [2] A. Gonzalez-Arroyo and M. Okawa, Phys. Rev D27 (1983) 2397 [3] S.R. Das and J. Kogut, Nucl. Phys. B235[FSll] (1984) 521 [4] C.G. Calan, R. Dashen and D.J. Gross, Phys. Rev. D17 (1978) 2717; E.V. Shuryak, Nucl. Phys. B203 (1982) 93,116, 140 [5] G. ‘t Hooft, Comm. Math. Phys. 81 (1981) 267 [6] P. van Baal, Comm. Math. Phys. 92 (1983) 1 [7] A.A. Migdal, Sov. Phys. JETP 42 (1975) 413, 743; L. Kadanoff, Ann. of Phys. 100 (1975) 359 [8] F.R. Klinkhamer and P. van Baal, Nucl. Phys. B237 (1984) 274 [9] C.P. Bachas and R.F. Dashen, Nucl. Phys. B210[FS6] (1982) 583