Droplet theory of low-dimensional phase transitions

Abstract

We have developed a theory of the universal configurational physics underlying critical-point phenomena in the Ising universality class. The theory is formally justifiable in d = 1 + [Sgrave] dimensions and may be regarded as the natural continuation of the kink-based theory of one dimension, which it incorporates as a limiting case. In d = 1 + [Sgrave] the configurational building block is the droplet. The typical droplet is not spherical and the many-droplet assembly is not dulute the implied problems are handled with renormalisation group methods. It is found that droplet shape fluctuation effects control the correlation length exponent v, while the nesting of droplets within droplets controls the order parameter exponent β. The exponents v and β thus effectively define, respectively, the fractal dimensions of the droplet surface and the droplet volume. The theory has been extended² to include the effects of an ordering field allowing us to study the effects of fluctuations on the critical droplet controlling nucleation phenomena. It has also been generalised to describe the s-phase (Potts) problem allowing us to obtain an explicit theory of percolation (realised in the s $$ limit) above the threshold.