Condensed Matter Physics, 2000, vol. 3, No. 3(23), p. 559-575, English

Author(s): C.Bagnuls (Service de Physique de l'Etat Condense, CE Saclay, F91191 Gif-sur-Yvette Cedex, France), C.Bervillier (Service de Physique Theorique, CE Saclay, F91191 Gif-sur-Yvette Cedex, France)

Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the $O(1)$-symmetric scalar Hamiltonian. In three dimensions, in addition to the usual critical surface $S_{\rm c}$ (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain $S_{\rm f}$ separated from $S_{\rm c}$ by the tricritical surface $S_{\rm t}$ (attraction domain of the Gaussian fixed point). $S_{\rm f}$ and $S_{\rm c}$ are two distinct domains of repulsion for the Gaussian fixed point, but $S_{\rm f}$ is not the basin of attraction of a fixed point. $S_{\rm f}$ is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the $\varphi^{4}$-coupling. This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the $\varphi_{4}^{4}$ renormalized field theory asymptotically free but with a wrong sign of the perfect action). We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions). This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.

Comments: Figs. 6, Refs. 28, Tabs. 0.

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