Critical Dimension - Lower Critical Dimension

Lower Critical Dimension

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 ε. Extracting this energy from other degrees of freedom decreases entropy by ΔS=-ε/T. This entropy change must be compared with the entropy of the domain wall itself. In a system of length L there are L/a positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain ΔS=k_Bln(L/a). For nonzero temperature T and L large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at T>0. Space dimension d=1 thus is a lower bound for the lower critical dimension of such systems.

A stronger lower bound d=2 may 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 d=2 at T>0, and there thus is no phase transition of the usual type at d=2 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.