8 - Two-dimensional conformal field theories  pp. 272-307

Two-dimensional conformal field theories

By Cisar Gómez, Martm Ruiz-Altaba and German Sierra

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Introduction: critical phenomena

The main method of discovering the various phases in which a quantum system may exist is the study of the system's behavior under scale transformations, which are described by the renormalization group equations. In particular, the fixed points of the scale transformation correspond to the critical points where, since the correlation length becomes infinite, the system enjoys invariance under dilatations.

The renormalization group approach to critical phenomena allows for the explicit computation of critical exponents. These were originally introduced as empirical fitting parameters to describe the scaling behavior of physical magnitudes in the neighborhood of the critical point. Universality means that the critical exponents depend only on the dimensionality and the symmetries of the system. From the point of view of the renormalization group, the critical exponents are just the weights of representations of the dilatation group. Associated with each universality class, characterized by a set of critical exponents, is a set of representations of the dilatation group, whose weights are precisely the above critical exponents. Two systems are in the same universality class if they are related by a renormalization group transformation.

For quantum systems that are invariant under translations and rotations (i.e. homogeneous and isotropic), and with short-range interactions, dilatation invariance implies, in general, invariance under the bigger group of conformal transformations. Therefore, its critical behavior will be described by a quantum field theory invariant under the whole conformal group. This observation, originally due to Polyakov, has dramatic consequences if the dimensionality of the system is two: in this case, the conformal group is infinite dimensional.