Crossover scaling functions for two-point correlations near tricritical points

Abstract

The expectations of phenomenological crossover scaling theory are worked out for two-point correlation functions in zero field above the primary critical point. Renormalization-group recursion relations are then used to construct the two-point correlation function for the changeover from tricritical to critical-line behavior, above the tricritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, for $3\ensuremath{\le}d<4$. An explicit expression is obtained for the normalized double-scaling function ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}({q}^{2}{\ensuremath{\xi}}^{2}, \frac{g}{{t}^{\ensuremath{\phi}}})$, for the general scaling variable ${q}^{2}{\ensuremath{\xi}}^{2}$, and the crossover variable $w=\frac{g}{{t}^{\ensuremath{\phi}}}$, $q\ensuremath{\ll}{a}^{\ensuremath{-}1}$ being the wave vector; $\ensuremath{\xi}\ensuremath{\gg}a$ is a second-moment correlation length and $a$ the lattice spacing. In the tricritical regime, $w\ensuremath{\simeq}0$, ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}$ (${x}^{2}, w$) has the Ornstein-Zernicke form and the changeover to the critical regime $w\ensuremath{\simeq}\ensuremath{\infty}$ is explicitly discussed. The limitation of the results, within our calculations to first order in $\ensuremath{\epsilon}$, are pointed out.