# Renormalization-group study of the critical end point in 4-dimensions

## Abstract

The phase diagrams of a variety of physical systems, among them certain antiferromagnets and ternary fluid mixtures, may include a critical line intersecting a line of first-order transitions at a critical end point rather than at the familiar tricritical point. At the critical end point the order parameter has critical fluctuations in the paramagnetic phase and a finite jump. Within Landau theory this phenomenon appears with the introduction of a negative 6 coupling; here we extend that theory to include fluctuations by a momentum-shell renormalization-group analysis close to four dimensions. In previous real-space analyses the critical end point was associated with a new fixed point. In contrast, the Wilson-Fisher fixed point which characterizes the entire critical line describes the critical end point here. We derive the O corrections to the mean-field exponents which, in Landau theory, describe the approach to the critical end point from high temperatures. At the same time, the discontinuity in at the critical end point predicted by Landau theory is essentially unaltered by fluctuations. It emerges from a renormalization-group calculation of the Gibbs free energy for the Ginzburg-Landau-Wilson model with a negative 6 term and a positive 8 term. © 1982 The American Physical Society.