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Critical region of D-dimensional spins: Extension and analysis of the hierarchical reference theory

The hierarchical reference theory (HRT) is generalized to spins of dimensionality $D$. Then its properties are investigated by both analytical and numerical evaluations for supercritical temperatures. The HRT is closely related to the self-consistent Ornstein-Zernike approximation (SCOZA) that was developed earlier for arbitrary $D$. Like the $D=1$ case we studied earlier, our investigation is facilitated by a situation where both HRT and SCOZA give identical results with a mean spherical model (MSM) behavior (i.e. $D=\infty$). However, for the more general situation we find that an additional intermediate term appears. With an interplay between leading and subleading contributions, simple rational numbers, independent of $D$ ($<\infty$), are found for the critical indices.