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

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=Lᵈ, and ``cylinder,'' L^d-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-2, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel'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, Vt^2-α, 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).