Conformal invariance and critical phenomena

Critical phenomena occur in diverse physical systems, including liquid-vapor critical points, paramagnetic-ferromagnetic transitions, multicomponent fluids, alloys, superfluids, superconductors, polymers, and fully developed turbulence, extending even to the quark-gluon plasma and the early universe. Early theoretical approaches sought to simplify these complexities using a limited number of degrees of freedom, exemplified by the van der Waals equation and mean field approximations, culminating in Landau's general theory. Today, it is recognized that these phenomena share a common foundation in the strong fluctuations of infinitely many coupled variables. This understanding was first articulated through Onsager's exact solution of the two-dimensional Ising model. Subsequent advancements have led to scaling theories and the renormalization group, which provide precise descriptions near critical points, often aligning well with experimental results. Unlike the general perspective from a century ago, current insights emphasize that fluctuations occur across all length scales at critical points, highlighting the scale invariance of systems in such states. Additionally, conformal invariance allows for non-uniform, local rescaling as long as angles remain unchanged.