Size Scaling for Infinitely Coordinated Systems

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 ${|T-T_c|}^{a}f\left(\frac{N}{N_c}\right)$ with $N_c\sim {|T-T_c|}^{-{\nu }^{*}}$. An argument gives ${\nu }^{*}={\nu }_{\mathrm{MF}}{d}_c$, where ${\nu }_{\mathrm{MF}}$ 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 $T\ne 0$ and on the Ising-$\mathrm{XY}$ quantum spin system in a transverse field at $T=0\mathrm{}$ for which calculations are reported.