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 $\ensuremath{\chi}$ (and of the specific heats) down to temperatures at which ${T}_{\ensuremath{\chi}}\ensuremath{\simeq}30{({T}_{\ensuremath{\chi}})}_{T=\ensuremath{\infty}}$. Firm conclusions about possible two-dimensional critical behavior cannot be reached, although the data are consistent with an exponent ${\ensuremath{\gamma}}_{2}=3.0\ifmmode\pm\else\textpm\fi{}0.5$. The shifts of the apparent critical temperature ${T}_{c}(n)$ from the $d=3$ bulk value can be described by a power law ${n}^{\ensuremath{-}\ensuremath{\lambda}}$ with $\ensuremath{\lambda}\ensuremath{\simeq}1.1\ifmmode\pm\else\textpm\fi{}0.2$. The surface susceptibility in the bulk limit diverges with an exponent ${\ensuremath{\gamma}}^{\mathrm{x}}=2.18\ifmmode\pm\else\textpm\fi{}0.02$ which appears to exceed the scaling prediction ${\ensuremath{\gamma}}^{\mathrm{x}}={\ensuremath{\gamma}}_{3}+{\ensuremath{\nu}}_{3}\ensuremath{\simeq}2.08$; this could indicate the existence of a surface correlation length ${\ensuremath{\xi}}^{\mathrm{x}}(T)$ with exponent ${\ensuremath{\nu}}^{\mathrm{x}}>{\ensuremath{\nu}}_{3}$.