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fiWCezMo*! COMMISSARIAT A L'ENERGIE ATOMIQUE CENTRE D'ETUDES NUCLEAIRES DE SACLAY CEA-CONF — 8513 Service de Documentation F91I91 GIF SUR YVETTE CEDEX L1 CONFORMAL INVARIANCE AND FINITE SIZE EFFECTS IN CRITICAL TWO DIMENSIONAL STATISTICAL MODELS Claude Itzykson Service de Physique Théorique, CEN-Saclay, 9H91 Gif-sur-Yvette Cedex, France Communication présentée à : 9. Sitges conference Sitges (Spain) 26-30 May 1986 1 CONFORMAI, INVARIANCE AND FINITE SIZE EFFECTS IN CRITICAL TWO DIMENSIONAL STATISTICAL MODELS Claude Itzykson Service de Physique Théorique, CEN-Saclay, 91191 Gif-sur-Yvette Cedex. France I. PRELIMINARIES 1. A systematic review of the recent achievements in conformai invariant two-di­ mensional field theories would already require a volume size. Here we shall discuss some finite size effects, because they reveal important features as noted by Cardy. The computation of the partition function in a finite geometry -a torus- enables one to understand the spectrum of the theory. The later is a compendium of the critical dimensions of the various operators. Two simple examples, namely bosonic or fermionic free fields (respectively the Gaussian and the Ising model) will serve as a pedagogic introduction, although the corresponding computations are not without some subtleties. We then proceed to considerations related to modular invariance. The forthcoming analysis is of course based on the fundamental work by Belavin, Polyakov and Zamolodchikov (quoted as BPZ) and its subsequent development by a number of authors including Dotsenko, Fateev, Friedan Qui and Shenker, Cardy and many others. It also relies on the mathematics of infinite Lie algebras (also known as current algebras) elaborated by Kac, Feigin and Fuchs, Rocha-Caridi... The whole subject is to a large extent similar to the one of string theories. The following presentation is based on a joint paper with J.B. Zuber and a letter written in collaboration with H. Saleur. Let both of them receive here my warmest thanks. A more thorough exposition is to be found in notes for lectures delivered in Marseille. 2. It will be assumed that the reader has some idea of the essentials of two dimensional conformai invariance presented in detail in the beautiful BPZ article. At variance with the renormalization group analysis this work emphasizes directly the massless critical theory and its associated energy-momentum tensor. The later enables one to study tho effect of a local change of coordinates. This is a familiar construct in the context of quantum field theory, but not until now in statistical mechanics. Changing the coordinate system can be interpreted in a passive way -a mere repa- rametrisation- and therefore without effect on the physical content, if not on its mathematical description. It also admits of an active interpretation as a map from 2 domains to domains (with correlative boundary conditions) in which case it is cf objective significance. The energy-momentum tensor T„.v is the generator of such infi­ nitesimal changes. Given the corresponding local infinitesimal variations SAlx) of the observables or fields Alxj, the basic (Ward) identity for correlations reads in twc dimensions v S A & <A,(x1)...T^v(x)> d»t>e U) = 0 (1 ( I(ÎI)- S(ÏP)-)O*|D^ and for the total free energy In ZJJ 0 'to^lpMl))^™- (2: given the change of coordinates x^" -* x4* • 6£.*(x.). The factor 2ir is included for convenience. If !D is the full Euclidean plane, requiring invariance under translations, rota­ tions and dilatations imposes the conditions £ (A1|x1j...avfcAp|xpj...) T*v(;) = V(x) (3* Z* T£U) = 0 Brackets without subscripts refer to the infinite plane and coordinates have beer, assumed Euclidean x2 * x2 • x2,. Under a general coordinate transformation x -* x + S € |x|, a scalar field behaves as A(X) -* AJx) • 6A(x) LA 6A(x| ) » &«(x).ô • Y Ê'6^*1 AU) (4, where \ is the scaling dimension of A, since the local dilatation factor is 3 DU*6fc) i / z i D(x) oi 1 + - â.&C If &€ decreases fast enough, tnen integrating by parts in (1) results in t This form of the identity states that correlations of 6 *T^v(xj with observables, of arguments x , vanish except at the points x . This is what is meant when one says that T^v is conserved 3%v(x) =0 (6) One can of course write similar expressions in arbitrary dimension d. Invariance under euclidean and scale transformations entails in this framework invariance under the full global conformai group. In two dimensions it can be identi­ fied with the isochronous Lorentz S0(l,3) or equivalently SL(2,C)/Z2 realized as rational (or Môbius) transformations group on the complex variable z = Xj+ix2 az+b z -» z' = f(z) = , ad-bc = 1 (7) cz*d 3. Specific to two-dimensions is the existence of a much larger set of conformai transformations beyond the global ones. While it remains true that the above Mobius transformations are the only global ones of the completed plane (i.e. the Riemann sphere), it is nevertheless true that an analytic function mapping one-to-one a domain D on a domain f (3)) = D' induces a conformai map B -» D'. Let z -» z' = f(z) f-analytic in D (1) be this invertible map. One assumes the existence of primary tensor-like fields which transform according to AJz.z) dzh dzh = A'(z'.z') dz'hdz'h (2) The pair (h,hj defines the conformai dimensions or weights associated to A. This con­ tains both, the tensor character of A (i.e. its behavior under rotations), thus h-h may be called the spin of A and has to be an integer if A is unaffected by a 2ir rota­ tion, and the scaling dimension k \ = h • h (3 In particular a real scalar field will be such that h = h while in general h * h (and the bar over h does not imply complex conjugation). It is convenient to use z and z as a substitute for Xj and x2, remembering that z and z have to be treated as independent variables. The metric takes the form ds2 = dzdz g =g = 0 g - = g- = - ZZ 2 2 ZZ ZZ 2 CO 9z 2Ux* * dx2) bz 2\bxx dx2) From the constraints on T v, it follows that it has only two independent non vanishing components T = T iT T T ZZ |< ll- 12> -z-z " 5( 11*«12) (5) With the proviso explained above, the conservation law (2.6) implies that T (T } is zz \ z z; only a function of z (of z). Henceforth we write T = T(z) T- - = T(Z) (6) zz z z > / Since there is a constant parallel between properties in z and z, only the z part of the formula (or the z-dependence of fields) will generally be explicited. It is understood that the z miror equation should be added. The Ward identity can be transformed to read 2 (g(zp) ~* hp g«up)J <A,(«,)...> . kg_g<«) CTUJVZ,) (7) where the contour "t encircles the points z} Equivalently by Cauchy's theorem this is also an expression for the short distance expansion ( h-p 1 d^ <T(z)A,( ) ..>.! <A1(z1)...> (8) Zl ( - )2 U"Z ) 9z J P 2 z p p 5 When g(z) in (7) is a polynomial in z of degree at most 2 it corresponds to an infi­ nitesimal generator of the global conformai group, in which case both sides must vanish. This is equivalent to requiring that T(z) behaves as z'1 at infinity and leads to the following invariance constraints on correlation functions translations L-> -— <A,(z,)...> = 0 P 6ZP 1 scale transformations 2-, zp -— • h <Ax(z,)...> = 0 (9 and rotations p \ °zp ) special conformai Z- \zl -— • 2h z 1 <A, (z, )...> = 0 p p p l x transformations p { dzp J In particular this determines the 2~ and 3~point correlations in the plane. 4. For the purpose of normalization we assume that the partition function is the plane is unity. This amounts to say that the free energy per unit area vanishes. Thus <T> = <T> = 0 (1) On the other hand the correlation function for two T's (or two T's) has no reason to vanish. Its non vanishing will be the signal of some anomaly as we shall soon see. Contrarily to what is implied in the title of these lectures and other works on massless two -dimensional field theories, the fact is that they are not invariant under general conformai transformations, but, should one say, covariant in a definite way, which we are now going to characterize. As T(z) behaves as z'u for large z, the two-point function takes the form <T(z,) T(z2)> =- \ . (2} 2 4 (z,-z2) The constant c will be the same for (T T) and real, under mild reality assumptions or. the statistical model. It plays a major role in the theory and is called the central charge for reasons which will appear below. Identifying c will, in a large number of cases, be the key to understanding the critical properties. Let us insert (2) in the Ward identity stating the behavior T -» T*6T of T under an "infinitesimal" conformai transformation z -» z' = z*&€ g(z). As a result <6T(z)> = 6€ <|> —- g(z') <T(z)T(z')> (3) = &€ ^ g"'{2) b Corresponding to (3«7) we would expect the transformation law for T to be homogeneous. with hT = 2 6T(z) = Seig(z) ^ • 2g'(z)j T(z) false (V! which would have required <&T{z)> = 0. This is not the case if c * 0, and is therefore the signal mentioned above: T does not behave as a primary field, but rather picks an inhomogeneous part in its transformation law 6r(2) = ScfgU) j^ + 2g,(z)j T(z) + ^ 6€ g-(z) (5) For a finite transformation, the integrated form of (5) reads T(z) dz2 = T'(z*) dz'2 • — {z'.z} dz2 (6) where {z',z} is the Schwarzian derivative of z' = f(z) with respect to z, namely d3f/dz3 d2f/dz2' 3 {z'.z} (7) df/dz 2 df/dz This derivative vanishes when z' is related to z by a Mobius transformation, hence the added term does not modify the behavior of T under global conformai transformations, but it does so in any other case.

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