First-order transitions breaking O(n) symmetry: Finite-size scaling

Abstract

The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n≥2) is broken at the bulk transition point. Both ‘‘block,’’ V=$L^d$, and ‘‘cylinder,’’ $L^{d\mathrm{-}1}$×∞, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/$L^{d\mathrm{-}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of Néel’s phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ‘‘degeneracy kernel’’ to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ‘‘spin-wave stiffness’’ or ‘‘superfluid density’’) and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{-}\alpha }$, that is needed for d>$d_{>}$=4. All the results found can be checked in the limit n→∞ against exact results for spherical models (described elsewhere).