Quasi-One-Dimensional and Quasi-Two-Dimensional Magnetic Systems: Determination of Crossover Temperature and Scaling with Anisotropy Parameters

Magnetic systems of lattice dimensionality $\overline{d}$ are considered, with interaction strength $J$ in $d$ lattice directions and $RJ$ in the remaining ($\overline{d}-d$) directions; recent experimental work on quasi-one-dimensional and quasi-two-dimensional systems are germane to the cases $d=1,2\mathrm{}$, respectively (and $\overline{d}=3\mathrm{}$). Rigorous relations are established for the first few derivatives with respect to $R$ of the susceptibility $\chi \mathrm{}\left(R\right)\mathrm{}$, the second moment of the correlation function ${\mu }_2\mathrm{}\left(R\right)\mathrm{}$, and the specific heat $C_H\mathrm{}\left(R\right)\mathrm{}$. These results permit detailed statements about the coefficients in the expansions of $\chi \mathrm{}\left(R\right)\mathrm{}$, ${\mu }_2\mathrm{}\left(R\right)\mathrm{}$, and $C_H\mathrm{}\left(R\right)\mathrm{}$ in Taylor expansions in $R$ about the $d$-dimensional limit ($R=0\mathrm{}$), and thus permit estimates concerning the temperatures at which each of these functions exhibits significant departures as $T\to T_c^{+}$ from its high-temperature $d$-dimensional behavior. These are the first estimates of the "crossover temperature" at which the system crosses over to $\overline{d}$-dimensional behavior from its limiting high-temperature behavior (at which the weak interactions $RJ$ are ineffective compared to the strong interactions $J$ in the $d$ lattice directions). In particular, they emphasize the fact that the crossover temperature depends upon the function being considered as well as upon the system and the value of $R$. These results also give strong support to the generalized scaling hypothesis in which $R$ is scaled.