Finite-size analysis of first-order phase transitions: Discrete and continuous symmetries

First-order phase transitions are characterized by $\delta$-function singularities in thermodynamic quantities. The way in which these singularities develop in taking the thermodynamic limit is qualitatively different for finite systems and systems infinite in one direction only. The corresponding crossover behavior, which we predict in detail with a renormalization-group analysis, is a unique feature of first-order transitions and is suggested to be of considerable utility. Both systems with discrete and continuous symmetries are discussed. For the latter we verify our results for the two geometries within the spin-wave approximation.