Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications

TL;DR

This paper presents a method to understand logarithmic conformal field theories as limits of ordinary CFTs with varying central charge, revealing their structure and applications in statistical models and critical phenomena.

Contribution

It introduces a new approach to logarithmic CFTs as limits of ordinary CFTs and applies it to various physical models, clarifying the origin of logarithmic behavior.

Findings

Identified logarithmic operators in models like percolation and self-avoiding walks.

Resolved the c->0 paradox and evaluated the b-parameter in these theories.

Derived sum rules for the effective central charge and b-parameter in 2D.

Abstract

We describe an approach to logarithmic conformal field theories as limits of sequences of ordinary conformal field theories with varying central charge c. Logarithmic behaviour arises from degeneracies in the spectrum of scaling dimensions at certain values of c. The theories we consider are all invariant under some internal symmetry group, and logarithmic behaviour occurs when the decomposition of the physical observables into irreducible operators becomes singular. Examples considered are quenched random magnets using the replica formalism, self-avoiding walks as the n->0 of the O(n) model, and percolation as the limit Q->1 of the Potts model. In these cases we identify logarithmic operators and pay particular attention to how the c->0 paradox is resolved and how the b-parameter is evaluated. We also show how this approach gives information on logarithmic behaviour in the extended Ising model, uniform spanning trees and the O(-2) model. Most of our results apply to general dimensionality. We also consider massive logarithmic theories and, in two dimensions, derive sum rules for the effective central charge and the b-parameter.