Stability of critical behaviour, critical-exponent renormalization and first-order transitions

The critical behaviour of systems with short-range interactions in which the free energy has divergent second derivatives is shown to be unstable under small perturbations. The perturbations can arise from additional terms in the hamiltonian with a long-range nature, but also from hidden variables, subjected to constraints. In general there will be either critical-exponent renormalization or first-order transitions; in special cases one can also have more complicated multicritical behaviour. In the analysis use is made of Legendre transformations and homogeneity properties of the short-range system. The results are rather general and independent of specific properties of operators in the hamiltonian, or the nature and number of hidden variables.