Finite-Size and Surface Effects in Heisenberg Films

Abstract

High-temperature expansions for the zero-field susceptibility and specific heat to seventh order in $K=\frac{J}{k_{B}T}$ are reported for ferromagnetic Heisenberg-model simple-cubic-lattice "films" of $n=1, 2, \dots , 6$ layers. Extrapolation of the series yields reliable estimates of the susceptibility $\chi$ (and of the specific heats) down to temperatures at which $T_{\chi }\simeq 30{\left(T_{\chi }\right)}_{T=\infty }$. Firm conclusions about possible two-dimensional critical behavior cannot be reached, although the data are consistent with an exponent ${\gamma }_2=3.0\mathrm{}\pm 0.5$. The shifts of the apparent critical temperature $T_c\mathrm{}\left(n\right)\mathrm{}$ from the $d=3\mathrm{}$ bulk value can be described by a power law $n^{-\lambda }$ with $\lambda \simeq 1.1\pm 0.2$. The surface susceptibility in the bulk limit diverges with an exponent ${\gamma }^{\mathrm{x}}=2.18\mathrm{}\pm 0.02$ which appears to exceed the scaling prediction ${\gamma }^{\mathrm{x}}={\gamma }_3+{\nu }_3\simeq 2.08$; this could indicate the existence of a surface correlation length ${\xi }^{\mathrm{x}}\mathrm{}\left(T\right)\mathrm{}$ with exponent ${\nu }^{\mathrm{x}}>{\nu }_3$.