Universality Classes Different from Directed Percolation
As we have seen in the previous chapters, universality classes of continuous phase transitions are usually characterised by the dimensionality, the type of order parameters, and a set of certain symmetries. For example, in equilibrium statistical mechanics, the hallmark of an Ising transition is a discrete ℤ2-symmetry under spin reversal. Sometimes these symmetries are implemented as exact symmetries on the microscopic level. In many cases, however, they emerge only as asymptotic symmetries. A simple example is directed percolation, which is symmetric under rapidity-reversal (see Sect. 4.1.2) within the corresponding path integral formulation [496, 331]. Generally this symmetry is not present on the level of the microscopic dynamics, instead it emerges only asymptotically on a coarse-grained scale near criticality, where all irrelevant terms of the underlying field theory can be neglected. It is therefore not always possible to determine a system’s universality class just by identifying the symmetries of its microscopic dynamics.
Fortunately, the DP universality class can be characterised by very few properties: According to the DP-conjecture by Janssen and Grassberger [326, 240] (see Sect. 3.2.2), systems with short-range interactions, exhibiting a continuous phase transition into a single absorbing state, belong generically to the DP universality class, provided that they are characterised by a onecomponent order parameter without additional symmetries and without unconventional features such as quenched disorder. Non-DP behaviour is expected to occur in systems where at least one of these requirements is not fulfilled. Therefore, it is interesting to search systematically for other universality classes of non-equilibrium phase transitions.
KeywordsDomain Wall Ising Model Critical Exponent Universality Class Scaling Behaviour
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