Critical dimension
An elegant criterion to obtain the critical dimension within mean field theory is due to V. Ginzburg.
Below the lower critical dimension, there is no field theory corresponding to the model.
In the context of string theory the meaning is more restricted: the critical dimension is the dimension at which string theory is consistent assuming a constant dilaton background without additional confounding permutations from background radiation effects.
Determining the upper critical dimension of a field theory is a matter of linear algebra.
It is worthwhile to formalize the procedure because it yields the lowest-order approximation for scaling and essential input for the renormalization group.
A Lagrangian may be written as a sum of terms, each consisting of an integral over a monomial of coordinates
-model and the isotropic Lifshitz tricritical point with Lagrangians see also the figure on the right.
This simple structure may be compatible with a scale invariance under a rescaling of the coordinates and fields with a factor
count wave vector factors (a reciprocal length
Each monomial of the Lagrangian thus leads to a homogeneous linear equation
A redefinition of the coordinates and fields now shows that determining the scaling exponents
, with all coupling constants occurring in the Lagrangian rendered dimensionless.
Dimensionless coupling constants are the technical hallmark for the upper critical dimension.
Naive scaling at the level of the Lagrangian does not directly correspond to physical scaling because a cutoff is required to give a meaning to the field theory and the path integral.
The main result at the upper critical dimension is that scale invariance remains valid for large factors
In the latter case there arise "anomalous" contributions to the naive scaling exponents
For small external wave vectors the vertex functions
This equation only can be satisfied if the anomalous exponents of the vertex functions cooperate in some way.
Naive scaling at the upper critical dimension also classifies terms of the Lagrangian as relevant, irrelevant or marginal.
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.
Creating a domain wall requires a fixed energy amount
Extracting this energy from other degrees of freedom decreases entropy by
positions for the domain wall, leading (according to Boltzmann's principle) to an entropy gain
large enough the entropy gain always dominates, and thus there is no phase transition in one-dimensional systems with short-range interactions at
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
For systems with quenched disorder a criterion given by Imry and Ma[3] might be relevant.
These authors used the criterion to determine the lower critical dimension of random field magnets.
