Size Scaling for Infinitely Coordinated Systems

Abstract

The finite-size scaling of Fisher and Barber is extended to infinitely coordinated systems. Near ${T}_{c}$ and for a large number of elements $N$, a critical quantity $A$ behaves as ${|T\ensuremath{-}{T}_{c}|}^{a}f(\frac{N}{{N}_{c}})$ with ${N}_{c}\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{\ensuremath{-}{\ensuremath{\nu}}^{*}}$. An argument gives ${\ensuremath{\nu}}^{*}={\ensuremath{\nu}}_{\mathrm{MF}}{d}_{c}$, where ${\ensuremath{\nu}}_{\mathrm{MF}}$ is the mean-field coherence-length exponent and ${d}_{c}$ the upper critical dimensionality of the corresponding short-range system. This is checked on spin systems at $T\ensuremath{\ne}0$ and on the Ising-$\mathrm{XY}$ quantum spin system in a transverse field at $T=0$ for which calculations are reported.