Gaussian dominance on compact spin manifolds

Volume 149B, number 1,2,3 PHYSICS LETTERS 13 December 1984

GAUSSIAN DOMINANCE ON COMPACT SPIN MANIFOLDS

A. PATRASCIOIU l and J.L. RICHARD Centre de Physique Th~orique 2, CNRS Luminy, Case 907, 13288 Marseille Cedex 9, France

Received 16 July 1984

The low temperature regime of continuous spin models is discussed. The relevance of the weak coupling expansion for the calculation of invaxiant Green's functions is analyzed. Notably it is found that in two dimensions Green's functions of invariant operators cannot be computed perturbatively.

In two previous papers [1 ~.] certain aspects of functional integration on compact spaces were discussed. For the concrete case of a non-relativistic par- ticle free to move on S N - I ( H = L2/2MR 2) it was found that:

(i) The Feynman-Kac formula for the propagator is correctly given by the continuum limit of the one- dimensional non linear O(N) o-model

Qlf, Tin O, 0) = lim ([3L/2rr) L dn i L. -~ oo

Xexp(-[3Li~l l (n i -n i_ l )2 .= , (1)

~=MR2/hT, n L = n f .

(ii) For any concrete parametrization of the sphere S N - l , one cannot interpret the probability measure in eq. (1) (naively) as

exp - dt ~MR2h 2 , (2) 0

with h 2 expressed in the chosen coordinates. This observation, forming the basis of Ito's calculus appears in many papers starting with De Witt's [3]. The naive action in eq. (2) must be modified by the introduc-

1 Permanent address: University of Arizona, Physics Depart- ment, Tucson, AZ 85721, USA.

2 Laboratoire propre du CNRS.

0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

tion of a potential term of O(h2). The generalization of the Ito calculus to field theory and its intimate connection with the O(N) symmetry was discussed by us in a recent paper [4].

(iii) The semi-classical approximation of the functional integral in eq. (1) gives the correct answer for T ~ 0 (/3 ~ 0o), but fails for T ~ o. (fl-~ 0). In particular in this approximation one does not obtain a discrete spectrum.

In this paper we continue our investigation of functional integration on compact spaces by consider- ing the weak coupling perturbation expansion. The concrete questions analyzed are:

(i) As the coupling constant goes to zero, doesthe spin fluctuation diminish so that the assumption of gaussian dominance becomes legitimate to compute the Green's functions?

(ii) Does the naive continuum action used for perturbative computations have to be modified by Ito- like terms [5] 9 .

Our results regarding the applicability of perturbation theory follow from the simple observation that this approximation cannot be used if by lowering the temperature one cannot control the magnitude of the spin fluctuations. In 1 and 2 dimensions the Mermin- Wagner theorem [6] implies unbounded spin fluctuations at any temperature in the infinite volume limit. In 2, 3 and 4 dimensions the fluctuations become unbounded in the continuum limit. We Fred no way of relating the temperature to the volume or the lattice spacing such that a non-trivial limit is reached. We

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would like to emphasize that the criterion discussed above for the applicability of perturbation theory is strictly necessary and in general not sufficient. This point will be illustrated later on with a concrete example.

All our computations were performed in d dimensions. We appealed to d = 1 occasionally to compare the perturbative results to the known exact ones. To distinguish between infrared and ultraviolet difficul- ties the non-linear O(N) o-model was modified by coupling the spin to an external magnetic field. Our Findings are:

(i) For the O(N) o-model (with or without magnetic) field in the continuum limit perturbation theory is inapplicable for d /> 2. For d < 2 it cannot be ruled out either when the magnetic field is non-vanishing or in the small volume limit.

(ii) The same statements apply to any topological sector of the semi-classical approximation.

('lii) For the Yang-Mills lattice models, regarded as spin models, perturbation theory cannot be ruled out as a scheme to compute the Green's functions of the spin for any d in the continuum limit.

(iv) In those cases where applicable, perturbation theory with the naive action (no Ito4ike terms) pro- duces the correct results. In particular the Green's functions so computed possess the desired symmetry.

A similar analysis has been performed recently by Hasenfratz [7]. He points out that perturbation theory cannot be applied even on fmite lattices if the zero modes due to global rotation are not separated out. Such a procedure has always been used in semi¢lassi- cal approximations (collective coordinates for the in- stanton) and our analysis of perturbation theory is done only after all the zero modes have been properly eliminated, for instance by following Hasenfratz's prescription on an L d periodic lattice.

We therefore consider the following L d periodic lattice model

N-1

Z= x~_AI-I f dS(X)AF[S ] nI"I 1= 8 (x~EASn(X))

d

X e x p - 2 3 a e 2 3 . (3) x~-,~ v=l 2a2

Here A C Z d is a finite sublattice of linear size L and a = T/L is the lattice spacing. An inverse length/a is

introduced so as to render g dimensionless. AF(S ) de- notes the Faddeev-Popov determinant. When intro- ducing an external magnetic field H, the partition function Z becomes

Z H = xEA f l fdS(x)exp [ Idd-2 E x~A

X t t 2a 2 •

To keep the discussion simple we consider the 0(2) model and represent

S ( x ) = (sin ~o(x), cos ¢(x)) .

Then the gaussian approximations of Z and Z H read simply

x exp a a E + g x~A v=l 2a 2 '

(5) and

[ a-2 Z~ )= r I d~o(x)exp . . . . . E a d

x~-.A -** g xEA

X ÷H = 2a2 '

(6)

choo~ng H = (0, H). Our discussion is based on the observation that the

gaussian approximations (5) and (6) are manifestly in- correct i f a s g ~ 0 , the average value of ~(x) 2 becomes larger than lr 2. This quantity can easily be estimated in the same approximation (eqs. (5) or (6)). It is

= (zl -2) (az )-d

d 7fly ~ -1 X ~ 4a -2 G sin" 2-~---] , (7)

I=__.A v=l !,#o

and, respectively,

= 0Uua -2) (aL)-d d

X ~ ( 4 a - 2 ~ s i n 2 ffl v

I~-A x v=l - L ' - + H t " (8)

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Obviously for t'mite L, asg ~ 0 the spin fluctuation (~p(x)2) ~ 0, hence the approximation is self consistent. For L ~ oo we distinguish two cases depending upon the value ofaL:

(a) Infinite volume limit (a fixed, L ~ oo). As g ~ 0, (¢ (x)2)H ~ 0 in any dimension d(H :/: 0). I f H = 0, (~o(x)2) ~ oo for any g > 0 if d ~< 2 (Mermin-Wagner theorem [6] ) but goes to zero as g ~ 0 i fd > 2.

(b) Continuum limit (aL fixed,L -~ ~o). I f d = 1,as g ~ 0 for any H the spin fluctuation goes to zero. But i fd /> 2, the spin fluctuation diverges for anyg andH (ultraviolet divergence).

We have therefore classified the cases in which an a priori perturbation theory is inapplicable. To complete the discussion we must verify if it is possible to make g a function of L such that as L ~ oo, g -~ 0 and (~(x)2) < oo. Unfortunately this is not possible in the non-linear o-models without rendering them trivial. Indeed a simple computation [8] gives for 0(2)

(S(x) • SO')) = 1 + (~(x) ~ 0 ' ) ) - (~(x) 2) + .... (9)

It is straightforward to verify that (i) For d /> 1 the difference in eq. (9) is infrared

finite even when (~(x) 2) is infrared divergent. (ii) For d t> 2 in the continuum limit

(~O(X)~OO'))/(tp(X)2)~O asL ~ o * .

Therefore it is not possible to make (~0(x) 2) < by using g =g(L) and obtain non-trivial correlations in % (7) or (8).

Next we discuss lattice gauge theories. They can be regarded as lattice spin models, with the spins attach- ed to the links. The action is the sum over all the pla- quette actions, the latter being:

(a/.t)d-4f[inv. (Si, Sj, Sk, Sl) ] . (10)

Besides the different spin interactions, the notable difference with the o-models in the power ofa/a changed from d - 2 to d - 4. (This is done in an attempt to det'me a theory in which in the continuum limit the usual gauge observables exist). I f we wish to compute the Green's functions of the spins S i, perturbation theory cannot be ruled out in the continuum limit for d < 4. In the infinite volume limit the same discussion as for the o-models applies.

Finally we would like to address the question of the form of the continuum action which is to be used for perturbative calculations in those cases in which

this approximation is applicable. Specifically does one have to use an effective action containing Ito-like terms [5] to obtain the correct answer? We have veri- fied that computing perturbatively up to two loops with the naive action:

(a) For d = 1 , H = 0, ~T/MR 2 ~ 0 the result agrees with the exact answer.

(b) For d arbitrary,H:~0, the Green's functionsof O(N) invariant operators depend only on IHI.

Thus Ito calculus is not necessary in perturbation theory. We interpret this result as being due to the fact that in perturbation theory the fluctuations vanish as g -+ 0. Hence the spin never probes the full manifold on which a priori it would be allowed to fluctuate.

In closing we would like to emphasize that our paper addresses an important question of physics: are the invariant Green's functions calculated perturbatively in d = 2 for the O(N) o-model [9], known to be infrared i~mite [10], the Green's functions of the continuum limit of the O(N) lattice o-model? Our answer is no, although they have all the desired properties. This answer is a corollary of a criterion which while necessary, is not sufficient. An interesting example of the subtleties involved is provided by the following spin model:

Z= x~_,x_l-I fd~o(x)exp(--[Jx~ea(alz)d-2

d X ~ [~o(x + lv) - ~p(x)] 2 ) .

v=l (11)

Fro d > 2 in the infinite volume limit, as 3 ~ oo one may think that the gaussian approximation is correct. In fact our criterion (with the zero mode properly treated) would not rule it out. Yet it has been rigo- rously shown by McBryan and Spencer [ 11 ] that the true correlation (~px~_0) decays exponentially, while its gaussian approximant as a power of 1/Ixl. The reason for which our criterion is not sufficient is that it does not estimate the error, which can grow with L, as the above example shows.

A. Patrascioiu would like to thank C-P.T. - Marseille for its hospitality.

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References

[ 1 ] A. Patrascioiu and M. Visinescu, The path integral for motions on compact spaces, preprint Central Institute of Physics (Magnrele, Romania, 1984).

[2] A. Patrascioiu and J.L. Richard, Remarks concerning functional integration on compact spaces, Marseille preprint CPT-84/P. 1599.

[3] B.S. DeWitt, Rev. Mod. Phys. 29 (1957) 377. [4] A. Patrascioiu and J.L. Richard, Ito calculus for a-

models and Yang-MiUs theories, Marseille prepnnt CPT (1984).

[5 ] D.W. McLaughlin and L.S. Schulman, J. Math. Phys. 12 (1971) 2520.

[6] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133.

[7] P. Hasenfratz, Perturbation theory and zero modes m O(N) lattice o-models, preprint TH 3846-CERN.

[8] S. Elitzur, Nucl. Phys. B212 (1983) 501. [9] E. Brezin and J. Zlrm-Justin, Phys. Rev. B14 (1976)

3110. [10] F. David,Phys. Lett. 96B (1980) 371. [11] O.A. McBryan and T. Spencer, Commun. Math. Phys.

53 (1977) 99.

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Gaussian dominance on compact spin manifolds