Lattice two-point functions and conformal invariance

A new realization of the conformal algebra is studied which mimics the behaviour of a statistical system on a discrete albeit infinite lattice. The twopoint function is found from the requirement that it transforms covariantly under this realization. The result is in agreement with explicit lattice calculations of the (1 + 1)D Ising model and the d−dimensional spherical model. A hard core is found which is not present in the continuum. For a semi-infinite lattice, profiles are also obtained. Since the pioneering work of Polyakov (1970), conformal invariance has become a powerful tool in the description and understanding of critical phenomena, in particular for two-dimensional systems (Belavin et al 1984). For example critical exponents and the n−point correlation functions at criticality can be exactly determined and a classification of the universality classes is obtained, see e.g. Christe and Henkel (1993) and di Francesco et al (1997) for an introduction. The basis of this theory is the assertion that there is a certain class of scaling operators, called (quasi)primary, which transform covariantly under (global) conformal transformations. For notational simplicity, we shall work in a two-dimensional setting throughout, but the generalization to higher spatial dimensions will be immediate. For the same reason, we restrict attention to scalar scaling operators. The generators of the conformal algebra may be written down in the form ln = −X̂n+1P̂ , where X̂ and P̂ are operators which satisfy the commutation relation [P̂ , X̂] = 1 (we neglect here the terms involving the central charge). Usually, an underlying continuum theory is assumed and then the choice of realization X̂ = z , P̂ = ∂ ∂z (1) is natural. However, other realizations are perfectly possible. For example, in the context of 2D turbulence, new realizations of the conformal algebra which yield logarithmic two-point functions, are needed (Rahimi Tabar et al 1997, Flohr 1996). Here, we shall consider yet another realization which mimics the behaviour of a system defined on a discrete lattice. Field theories on a discrete (space-time) lattice are presently under active study, e.g. Kauffman and Noyes (1996), Winitzki (1997) and † Unité Mixte de Recherche CNRS No 7556