Tricriticality and the failure of scaling in the many-component limit

Abstract

We analyze exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the $n$-component continuous-spin model with a cubic field term ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}}_{3}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}|}^{2}$. Full "four-field" tricritical behavior is exhibited for dimensionalities $d\ensuremath{\ge}3$. However, orthodox tricritical scaling is shown to be impossible for $3<d<4$: to obtain the correct nonclassical spherical-model exponents on the $\ensuremath{\lambda}$ line ($H={H}_{3}=0, T>{T}_{t}$) it is essential to allow for a dangerous irrelevant scaling variable $p\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}$, where ${R}_{0}$ is the range of the pair interactions. The appropriate crossover exponent is ${\ensuremath{\varphi}}_{p}=3\ensuremath{-}d$ so that $p$ is marginal for $d=3$: orthodox scaling is then possible but the scaling functions are nonuniversal. On the disordered symmetry plane ($H={H}_{3}=0$) only the corrections to scaling survive and describe crossover to Gaussian tricritical behavior. For $T>{T}_{t}$ bicritical crossover from spherical to classical critical behavior occurs when ${H}_{3}$ varies but scaling is fully obeyed. Some inferences for systems with finite $n$ are drawn.