Conformal Bootstrap in two Dimensions

Abstract

From a theoretical point of view, the characteristic feature of the second order phase transitions is the infinite growth of the correlation length in the nearest vicinity of the critical point. This fact, which makes the theoretical investigation of the critical region so difficult, allows one nevertheless to forget about the detailed microscopic structure of the system and to introduce the continuous fields of the order parameters to describe the essentially fluctuating variables. Thus, the problems of the second order phase transitions can be translated into the general language of the quantum field theory. At the critical point the correlation length is infinite and there is no physical scale in the theory; the corresponding quantum field theory is massless and possesses the scale invariance. This means that the interaction of the fields is invariant under the scaling transformations

$$$$

(1.1)

provided one considers the distances much greater than the ultraviolet cutoff (which is of the order of the interatomic distances in realistic systems). Here ξ^a are the space coordinates; a=l,2,...,D;λ is the parameter of the transformation. The fluctuations of the local fields Φ_ℓ(ξ) exhibit under the transformation (1.1) the scaling behaviour

$$$$

(1.2)

Here the exponents d_ℓ are the anomalous dimensions of the fields Φ_ℓ. Computation of the spectrum {d_ℓ} of the anomalous dimensions is the most importat problem of the theory as these quantities determine the critical singularities of the thermodynamic functions.