Critical Phenomena

F.J.W. Hahne

Barry M. McCoy

Abstract This chapter presents the phenomenological description of critical phenomena. It derives inequalities for critical exponents which follow from thermodynamics. It explains the scaling theory of critical phenomena of Ising-like systems, it gives scaling for Heisenberg-like systems, and introduces the concept of universality.

Javier Brey, Robert B. Jones

Somendra M. Bhattacharjee

Power laws are distinctive features of critical phenomena and can be ascribed to diverging length-scales. One of the most important contributions of the studies of critical phenomena is the shift in point of view to a length-scale based analysis and classifying terms of a Hamiltonian as relevant or irrelevant rather than classifying them as numerically weak or strong. The idea of universality transcends the domain of critical phenomena. Critical points are classified by the number of relevant variables required to describe them. A universality class would be described by the exponents and also the amplitude ratios of the various singulax quantities on both sides of the critical point. In a correlation based approach, we need to compute the critical correlations of all possible combinations of the degrees of freedom. Scaling is the rule for criticality, originating from a diverging length scale, while anomalous dimension is seen for fluctuation dominated cases.

Kyozi Kawasaki, T. Ohta, Akira Ōnuki

Effects of non-uniform external force fields which vary in one direction, say, along the z-axis, on critical phenomena are studied in a general way. Nature of critical phenomena depends crucially on a type of the spectrum of a one-dimensional eigenvalue problem besides the symmetry breaking nature of the force. In particular, if the external potential maintains the symmetry of the order parameter and the lowest eigenvalue is isolated, there is a crossover from the d-dimensional bulk critical phenomena to the d-1-dimentional surface critical phenomena. On the other hand, if the external potential breaks the symmetry, a sharp phase transition disappears as for the liquid-gas transition under the gravitational field. The cases with periodic external potentials are also discussed.