Phenomenological Weak-Scaling Theory

Abstract

Arguments are presented which allow one to characterize weak-scaling correlations in terms of thermodynamic homogeneity. Using as a guide the present indications on the spin-spin correlation function of the three-dimensional Ising model, it is argued that the thermodynamic effects of the smooth deviations from strong scaling in dimension $d>2$ can be accounted for most easily by looking at the correlations at a second effective length $\ensuremath{\zeta}=\ensuremath{\zeta}(\ensuremath{\omega})$, much smaller than the correlation length. This depends on a constant $\ensuremath{\omega}$, a nonzero value of which indicates deviation from strong scaling and $\ensuremath{\zeta}(\ensuremath{\omega})$ is determined as that length over which an effective strong-scaling correlation in dimension ($d\ensuremath{-}\ensuremath{\omega}$) yields the expected "scaling" relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$. It is found that $\ensuremath{\zeta}(\ensuremath{\omega})$ is much larger than the second length which appears in the weak-scaling theory of Stell. The constant $\ensuremath{\omega}$ enters all relations between critical exponents which depend explicitly on dimensionality and it is shown that a value $\ensuremath{\omega}\ensuremath{\simeq}\frac{1}{12}$ is consistent with the exponents for the three-dimensional Ising model. It is argued that weak scaling should be expected for general pair correlations and it is shown how a further inequality between critical exponents involving energy-density correlations can be derived. The limitation of our scheme to values of $d$ smaller than that at which the mean-field behavior sets in is explicitly pointed out.