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Abstract |
The purpose of renormalisation group and quantum field theory approaches to critical phenomena is to
diagonalise the dilatation operator. Its eigenvalues are the critical exponents that determine the power law
decay of correlation functions. However, in many realistic situations the dilatation operator is, in fact, not
diagonalisable. Examples include geometrical critical phenomena, such as percolation, in which the
correlation functions describe fluctuating random interfaces. These situations are described instead by
logarithmic (conformal) field theories, in which the power-law behavior of correlation functions is modified by
logarithms. Such theories can be obtained as limits of ordinary quantum field theories, and the logarithms
originate from a resonance phenomenon between two or more operators whose critical exponents collide in
the limit. We illustrate this phenomenon on the geometrical Q-state Potts model (Fortuin-Kasteleyn random
cluster model), where logarithmic correlation functions arise in any dimension. The amplitudes of the
logarithmic terms are universal and can be computed exactly in two dimensions, in fine agreement with
numerical checks. In passing we provide a combinatorial classification of bulk operators in the Potts model in
any dimension. |
