of the Lagrangian as relevant, irrelevant or marginal. A Lagrangian is compatible with scaling if the - and -exponents lie on a hyperplane, for examples see the figure above. is a normal vector of this hyperplane. Lower critical dimension The lower critical dimension of a phase transition of a given universality class is the last dimension for which this phase transition does not occur if the dimension is increased starting with . Thermodynamic stability of an ordered phase depends on entropy and energy. Quantitatively this depends on the type of domain walls and their fluctuation modes. There appears to be no generic formal way for deriving the lower critical dimension of a field theory. Lower bounds may be derived with statistical mechanics arguments. Consider first a one-dimensional system with short range interactions. Creating a domain wall requires a fixed energy amount . Extracting this energy from other degrees of freedom decreases entropy by . This entropy change must be compared with the entropy of the domain wall itself. In a system of length there are positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain . For nonzero temperature and large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at . Space dimension thus is a lower bound for the lower critical dimension of such systems. A stronger lower bound can be derived with the help of similar arguments for systems with short range interactions and an order parameter with a continuous symmetry. In this case the Mermin–Wagner Theorem states that the order parameter expectation value vanishes in at , and there thus is no phase transition of the usual type at and below. For systems with quenched disorder a criterion given by Imry and Ma might be relevant. These authors used the criterion to determine the lower critical dimension of random field magnets. References External links Kanon: A free windows program to determine the critical dimension, with examples, online help and mathematical details Critical phenomena Statistical mechanics Phase transitions String theory