Instanton contributions and the cluster property of the vacuum

Volume 104B, number 6

PHYSICS LETTERS

17 September 1981

INSTANTON CONTRIBUTIONS AND THE CLUSTER PROPERTY OF THE VACUUM Y. IWASAKI

Instituteof Physics,Universityof Tsukuba,Ibaraki305,Japan Received 15 May 1981

Analyzing carefuUy instanton contributions to the correlation functions in the two-dimensional 0(3) non-linear o-model and imposing the cluster property upon the vacuum, we conclude that the vacua are two-fold degenerate and that a new kind of topological symmetry breaking occurs. We also obtain the correlation length, which coincides remarkably with ihe result by Shenker and Tobochnik.

The two-dimensional 0(3) non-linear o-model has asymptotic freedom, n-instanton solutions and no intrinsic scale parameters. In this respect it resembles four-dimensional non-abelian gauge models. To study non-perturbative effects one has to formulate the theories in a non-perturbative way. An 0(3) invariant regularization of the o-model gives the classical 0(3) Heisenberg model. Recent results of Monte Carlo simulations by Shenker and Tobochnik [ 1] show that the two-point correlation function decays exponentially for any finite temperature. Now the problem is whether it is in principle possible to take account of this kind of non-perturbative effect starting from continuum theories. Although continuum theories are well defined by perturbation theory order-by-order, it is not clear how they are defined in non-perturbative ways. Therefore the problem is also how the non-perturbative effects can be taken account of. We will show that at least in the 2d non-linear 0(3) o-model, non-perturbative effects can be exactly taken account of by instanton [2] configurations (not by instanton-anti-instanton configurations) and that the correlation length calculated by this method coincides with the result by Monte Carlo simulations. In the course of deriving these results we will show that the vacua are two-fold degenerate and that a new kind of spontaneous symmetry breaking occurs. With this symmetry breaking the cluster property of the vacuum is satisfied. The non-linear 0(3) o-model is defined by the 458

lagrangian 2

3

L =l__~2fu=l i=1 ~

(8"°i)(Su°i)'

~°2

= 1.

(1)

We assume that the two-dimensional space is euclidean. The correlation functions are given by

(F(o))=l f DoF(o)exp(- f L d2x)

(2)

where Z is the partition function given by

z=f Doexp(-fLd2x).

(3)

W h e n f ~ l, the path integrals may be evaluated by means of the steepest descent method. The solutions of the equation of motion are known as instantons [2]. There are no other solutions with finite action other then instantons and anti-instantons [3]. The contributions of instantons to the path integrals are evaluated in the very interesting papers by Fateev et al. [4] and Berg and Liischer [5], including one-loop quantum corrections around instantons. Let us introduce for later convenience a new field defined by co = (o 1 + io2)/(o 1 + o3)

(4)

and a new space variable z = x I + ix 2.

(5)

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company

Volume 104B, number 6

The instanton contributions to the correlation functions of the o-field are rather complicated. Instead, the contributions to the correlation functions of the co-field can be simply expressed by Schwinger functions of fermion fields. After some calculations we obtain * 1

(CO(Z1) "'" ¢°(Zn)C°- l(Zn+ 1)... 60- l(Z2n ) ) instanton <~2(Zl) ... ¢2(Zn)t~(Zn+l) ... t~(Z2n)) m

(6)

(~2(Zl) ... ¢2(Zn)¢~(Zn+l ) ... ~(Z2n))m= 0 and
. (7)

( ~ l ( Z l ) ... t~l(Zn)~(Zn+l)... ~(Z2n))m= 0 Here the Schwinger functions in the numerator are those of a massive fermion, while the Schwinger functions in the denominator are those of a massless fermion ~1 and @2 are the first and the second components of the two-component fermion fields. The mass m of a massive fermion is related to the coupling constant as rn = 21rp exp [-27r/f0~)] [ 8 / f ~ ) ] e - 1,

ance, spectrum (p2 _ p2 ~> 0, P0 ~> 0) and locality, the vacuum is unique if and only if the n-point functions have the cluster decomposition property. Therefore we require that (¢O(Zl) ... ¢O(Zn)Co-l(Zn+l) ... X co-l(Z2n)) have the cluster property, because the cluster property of the n-point functions of the cofield is closely related with that of the o-field. The rhs of eqs. (6) and (7) have cluster properties. Therefore if we consider instanton contributions or anti-instanton contributions, the correlation functions have the cluster property. However, if we take into account both contributions, they do not have it [8]. (This happens for n/> 2. For n = 1, the cluster property is satisfied.) In the path integral approach, it seems at first to be natural to take into account both contributions. But the situation is not so simple. To realize what happens, it is convenient to consider a more familiar case. Let us consider the Ising model in two dimensions, as an example. The partition function and the correlation functions are given by

Z=

~

{o)

( 60n(z l )co- n (z 2 ) ) instanton = (6on(Zl)¢O-n(z2))anti.instanton (9)

where K 0 is the modified Bessel function. In ref. [6], the present author used maps from the Coulomb gas to the sine-Gordon model and from the sine-Gordon model to the massive Thirring model. The present method is more straightforward. It is well known [7] that in a field theory which satisfies the conditions of Lorentz (euclidean) invari-

and

1

(F(o)) = ~ {~} F(o) exp

(-~aio/).

(10)

Now the problem is what {a} means. For/3 ~c, one has to take only those configurations with a given spin direction at 0% because the symmetry is spontaneously broken. A convenient way to get this result is to add a symmetry breaking term +eo, to take the infinite volume limit and finally to take the limit e ~ 0. [We will refer to this method as method (a).] As is apparent from this example, it is not obvious what contributions one has to consider. The second way [to be referred to as method (b)] is to regard the system as a quantum system in one space dimension. [Note that an n-dimensional statistical system is equivalent to a quantum system in ( n - 1) space dimensions.] Then the ground states, for example at T = 0, are degenerate and belong to inequivalent representations. That is, _(0161

~:~ Details will be published elsewhere.

exp(-f3~oia])

(8)

where f ~ ) is the renormalized coupling constant with Pauli-Villars regularization. The rhs of eqs. (6) and (7) are shown to be euclidean invariant. But they are not identical. As special cases, we have

= [_r(d/dr)Ko(mr)] n,

17 September 1981

PHYSICS LETTERS

...

6 n 10)+ = 0

(for any finite n).

(1 1)

Here 10)+ and 10)_ are two degenerate vacua and 6 459

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is an operator which corresponds to the variable o of statistical mechanics. The third way [method (c)] is to examine the cluster property of the correlation functions. For example, if one takes into account all possible configurations at T = 0, one has (o) = 0,

(12)

whereas one has (oo> = 1.

(13) t

Eqs. (12) and (13) are in conflict with the cluster property. Only when one takes into account half of the configurations, the correlation functions have the cluster property. Now let us return to the case under consideration. We have shown that if we take only instanton contributions or anti-instanton contributions, the correlation functions have the cluster property. However, if we take both contributions, they do not have the cluster property. This means that the ground states of the non-linear 0(3) o-model are degenerate and belong to inequivalent representations [method (c)]. The instanton contribution is equivalent to the Coulomb gas at T = 1/2 [4,5], where the Coulomb gas starts to condensate. Therefore the instanton number density is infinity. If we cut such configuration at some fixed x 2 (say, x 2 = 0), we obtain the wavefunction of the vacuum. That is, symbolically [9] ~1(x2 = 0) =

~ e- A . instanton (x 2 ~<0)

(14)

The wavefunction of the other vacuum is ~2(x2 = 0) =

~ e- A . anti-instanton (x 2 ~<0)

(15)

Then (~2Ol ... On~ 1) = 0

(n = finite),

(16)

because the wavefunctions ~1 and ~b2 are topologically different. This is method (b) to see that the vacuum is degenerate. Method (a) is to add a term c

fq(x) d2x,

to the action. Here density 460

(17)

q(x) is the topological number

17 September 1981

q(x) = (1/87r)euueabeoa(oob/axu)3oe/Ox v.

(18)

Take the limit e --> 0 after the thermodynamical limit. In our case the topological number density (q(x))/V is infinite. This means that the configurations with large Iql are the dominant contributions. Therefore the topological symmetry breaking term eq. (17) will choose one of two possible configurations, instanton configuration or anti-instanton configuration, depending on the sign of e. From these considerations we conclude that the vacuum of the non-linear 0(3) o-model is degenerate and that a new kind of symmetry breaking happens. We call it topological symmetry breaking. Because the two representations are inequivalent, the 2n (n >/2) point functions are different in general. An order parameter to distinguish two representations is (sign

q(x)) = -+1.

(19)

Because of instanton or anti-instanton contributions, we also have

(4rrq(x)/½ [Duo(x)] 2)

=

-+1.

(20)

The vacuum is not parity invariant. This results in the fact that, for example, four-point functions are not parity invariant. This topological symmetry breaking is not in conflict with Coleman's theorem [ 10], because the topological charge is invariant under the continuous 0(3) transformation. As stated already, an 0(3) invariant regularization of the non-linear 0(3) model gives the classical 0(3) Heisenberg spin model. Then from the analysis above, we may conclude that at low temperature T( = f) ~< 1/1.3 the ground state of the classical 0(3)Heisenberg model is degenerate. The topological number density defined by eq. (18) for the continuum theory has a topological meaning. There is no unique way to define a topological number for the lattice theory; one way is to replace the differentiation by the difference in eq. (18) and another way is to define it-to have some topological meaning [ 11 ]. It seems that both are good order parameters, since in the scaling limit they reduce to the same quantity. At any rate a non-zero value of this order parameter is not in conflict with the Merwin-Wagner [ 12] theorem which forbids a

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PHYSICS LETTERS

non-zero value for the expectation value of the spin operator. Let us denote the topological number density at site i as q(i). [Here we do not care about its detailed definition.] Then we have (sign

q(i)) = +-1

(21)

(22)

where (E) = (-S i" Si+ 1 + 1) and (q) = (q(i)). At high temperature the system is disordered by Bloch walls with finite width. The system has no net topological number density: (sign

q(i)) = 0.

(23)

This can be checked for example by a high-temperature expansion. Thus the system does have a kind of phase transition at finite temperature depending on the topological number density, although the system does not have a phase transition at finite temperature in the sense that the two-point function decays exponentially for any finite temperature. We guess that the phase transition occurs at f ~ 1/1.3 (which we call fT), because f o r f ~ < f T the W i l s o n - K a d a n o f f renormalization group works well and the renormalization group coincides with that of the continuum theory. Whether the picture we have concluded is really correct can be checked, in principle, by Monte Carlo simulations. Since an essential ingredient of the model at low temperature is a topological object, one must be careful of finite-size effects and boundary condition effects [13].. Let us consider, for example, the classical 0 ( 3 ) Heisenberg model in three dimensions. The system has a spontaneous magnetization for T < T c. However, this happens only for an infinite system. For a finite system we have (M) = 0 even for T < T c. In order to see the effect of the spontaneous magnetization by a finite-size system, one must measure [13]

[(-~-~iSi)2] 1/2 = ((M2)) 1/2.

I((~q(i))2)]l/2=((q2))

1/2.

(26)

In eqs. (22) and (26), (E) is the nonperturbative contribution to the energy density. We have to subtract the perturbative part from the total: (E) = (E)to t - (E)pert ,

(27)

where [ 11 ] (E}pert = 1/13 + 1/8/32 + 0.078//33 + ....

(28)

Because (E) is a small difference between two large quantities, we need precise measurements on (E)to t and higher-order calculations on (E)pert. In ref. [ 11], Berg and Liischer have already measured (E)to t and (q2) in other contexts. To compare their results with eq. (26) we need higher-order terms of eq. (28), at least, up to/3-4. It may be suitable to measure the size dependence of ((q2))1/2 by Monte Carlo simulations in order to conclude that (q) has a finite value. We hope that we can report on it in future publications. We have explicitly obtained the correlation functions of co and co- 1. To rewrite these formulae into those of the a-field is not straightforward. Here we only discuss the two-point functions. We have obtained

(wn(zl)oo-n(z2)) = [_r(d/dr)Ko(mr)] n

(29)

Now we can expand the lhs in terms of 0 ( 3 ) invariant correlation functions as follows: * 2

(con(zl)w-n(z2)) = ~

anm(Sm),

(30)

m=n

where

S m is Pm(O(z 1)" a(z2)) and

anm -

47r(2m + 1)(m - n)!(m q- n)! [m(m 1)(n - 1)] 2

(24)

Then we have limN.+=((M2))l/2 = ICM)I. For the topological symmetry breaking discussed above we have a similar situation. Instead of measuring q one has to measure

(25)

In the limit N - + % ((q2))1/2 = I(q)l. Then we have from eq. (22) (E) = 47r((q2)) 1/2.

and corresponding to eq. (4.14) we have (E) = -+4n(q),

17 September 1981

,2 I would like to thank Dr. W. Riihl for his kind correspondence with the suggestion of the method to obtain the correlation function. 461

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We conjecture that in SU(N) non-abelian (lattice) gauge theories in four dimensions similar phenomena occur as in the 0 ( 3 ) model in two dimensions. We will discuss it in a separate paper.

Solving inversely eq. (30), we obtain (O(Zl).O(z2)) = ~ m=l

=~

m=l

blm(6om(zl)t-o-m(z2)) blm[-r(d/dr)Ko(mr)]m.

(31)

Note that the blm can be solved recursively. From eq. (31) we have (O(Zl) .O(Z2)) oc e-mr

+ O(e-2mr)

(32)

and m = 167r/a e-2~/f(u) [ 1/f(/.t)] e - 1 .

(33)

Therefore we obtain ~=gel

e2~r/f(u)[f(l~)/27r]/Ia .

(34)

To compare this with the result of Shenker and Tobochinik [1], we have to rewrite eq. (34) in terms of the coupling constant defined in lattice regularization, f(a). The transformation factor was obtained by Parisi [14]. Thus we obtain [6] = exp(1 -

lr/2)/32x/~e 27r/f(a)[f(a)/2rr] a,

(3 5)

where a is the lattice spacing. The numerical coefficient is given by C = exp(1 - 7r/2)/32x/~ -= 0.0126.

(36)

This number coincides remarkably with the numerical value 0.01 obtained by Monte Carlo simulations [ 1]. This result supports our conclusion that instantons or anti-instantons dominate the path integrals.

462

17 September 1981

I would like to thank M. Kobayashi, B. Sakita, S. Takada and T. Yoneya for valuable discussions.

References [1] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462. [2] A.A. Belavin and A.M. Polyakov, Zh. Eksp. Teor. Fiz. Pis'ma 22 (1975) 503. [3] W. Garber, S. Ruijsennaars and E. Seiler, Princeton preprint (1978). [4] V.A. Fateev, I.V. Frolov and A.S. Schwarz, NucL Phys. B 154 (1979) 1. [5] B. Berg and M. Liischer, Commun. Math. Phys. 69 (1979) 57. [6] Y. lwasaki, preprint UTHEP-73. [7] See, e.g.R. Haag, Nuovo Cimento 25 (1962) 287. [8] See also: B. Berg, DESY preprint 79/58; G. Lazaxides, Nucl. Phys. B167 (1980) 327. [9] See, e.g.G. Mack and V.B. Petkova, Ann. Phys. (NY) 123 (1979) 447 [see, especially, eq. (1.15)]. [10] S. Coleman, Commun. Math. Phys. 31 (1973) 259. [11] B. Berg and M. Liischer, Bern preprint BUTP-1/81 (1981). [12] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [13] See, e.g.K. Binder, in: Phase transitions and critical phenomena, Vol. 56, eds. C. Domb and M.S. Green (Academic Press, London).