Corrections to finite-size-scaling laws and convergence of transfer-matrix methods.

Abstract

We examine the finite-size-scaling laws relating the values of a quantity in a (hypercubic) box of size L, or on a bar of transverse size L (i.e., of cross section ${L}^{D\mathrm{\ensuremath{-}}1}$), to the same quantity in the infinite-volume limit. A field-theoretical argument shows that the corrections to these laws are governed by the bulk correction-to-scaling exponent \ensuremath{\omega} (also denoted ${\ensuremath{\Delta}}_{1}$/\ensuremath{\nu} or -${y}_{3}$). The data of transfer-matrix methods, like those of the phenomenological renormalization, have therefore generally the same asymptotic convergence exponent \ensuremath{\omega}. It is shown explicitly in some examples that other convergence laws may occur. The large-N limit of the O(N) spin model allows for a more refined quantitative study: dependence of corrections to finite-size scaling upon the details of interactions, range of values of L for which the convergence is asymptotic, and nonuniversality of apparent critical exponents in the mean-field case (D>${D}_{c}$).