Renormalization group domains of the scalar Hamiltonian

TL;DR

This paper uses the renormalization group approach to map the parameter domains of the scalar Hamiltonian, revealing critical and first-order transition regions, and discusses implications for phase behavior and asymptotic freedom.

Contribution

It explicitly characterizes the domains of the scalar Hamiltonian using the local potential approximation, including the first-order transition domain and its relation to fixed points.

Findings

Identification of the critical surface $S_c$ and first-order transition domain $S_f$

Existence of a renormalized trajectory in $S_f$ with negative $^4$-coupling

Potential explanation for classical critical behavior in ionic systems

Abstract

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