Monte Carlo study of three-dimensional—to—one-dimensional crossover in the Ising model

Abstract

Monte Carlo calculations have been used to study the behavior of a simple cubic spin-½ Ising model with spatially anisotropic nearest-neighbor coupling. The interaction in the $x$ and $y$ directions, which is equal to ($J_{\perp }$), was systematically decreased compared to that in the $z$ direction ($J_{\parallel }$), and results were obtained for values of $\Delta =\frac{J_{\perp }}{J_{\parallel }}$ as small as 0.003. For small $\Delta$ the system behaves like a one-dimensional Ising chain at high temperature before undergoing a low-temperature transition to a three-dimensionally ordered state. We find that the asymptotic ($\Delta \to 0$) variation of $T_c$ is accurate over a surprisingly wide range of $\Delta$.