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 $L^{D\mathrm{-}1}$), 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 ω (also denoted ${\Delta }_1$/ν or -$y_3$). The data of transfer-matrix methods, like those of the phenomenological renormalization, have therefore generally the same asymptotic convergence exponent ω. 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>$D_c$).