Critical Dimension - Upper Critical Dimension in Field Theory

Upper Critical Dimension in Field Theory

Determining the upper critical dimension of a field theory is a matter of linear algebra. It nevertheless is worthwhile to formalize the procedure because it yields the lowest-order approximation for scaling and essential input for the renormalization group. It also reveals conditions to have a critical model in the first place.

A Lagrangian may be written as a sum of terms, each consisting of an integral over a monomial of coordinates x_i and fields φ_i. Examples are the standard φ4-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 b according to

Time isn't singled out here - it is just another coordinate: if the Lagrangian contains a time variable then this variable is to be rescaled as t→tb-z with some constant exponent z=-. The goal is to determine the exponent set N={,}.

One exponent, say, may be chosen arbitrarily, for example =-1. In the language of dimensional analysis this means that the exponents N are counting wave vector factors (a reciprocal length k=1/L₁). Each monomial of the Lagrangian thus leads to a homogeneous linear equation ΣE_i,jN_j=0 for the exponents N. If there are M (inequivalent) coordinates and fields in the Lagrangian, then M such equations constitute a square matrix. If this matrix were invertible then there only would be the trivial solution N=0.

The condition det(E_i,j)=0 for a nontrivial solution gives an equation between the space dimensions, and this determines the upper critical dimension d_u (provided there is only one variable dimension d in the Lagrangian). A redefinition of the coordinates and fields now shows that determining the scaling exponents N is equivalent to a dimensional analysis with respect to the wavevector k, 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 doesn't directly correspond to physical scaling because a cutoff is required to give a meaning to the field theory and the path integral. Changing the length scale also changes the number of degrees of freedom. This complication is taken into account by the renormalization group. The main result at the upper critical dimension is that scale invariance remains valid for large factors b, but with additional ln(b) factors in the scaling of the coordinates and fields.

What happens below or above d_u depends on whether one is interested in long distances (statistical field theory) or short distances (quantum field theory). Quantum field theories are trivial (convergent) below d_u and not renormalizable above d_u. Statistical field theories are trivial (convergent) above d_u and renormalizable below d_u. In the latter case there arise "anomalous" contributions to the naive scaling exponents N. These anomalous contributions to the effective critical exponents vanish at the upper critical dimension.

Naive scaling at d_u thus is important as zeroth order approximation. Naive scaling at the upper critical dimension also classifies terms of the Lagrangian as relevant, irrelevant or marginal. A Lagrangian is compatible with scaling if the x_i- and φ_i -exponents E_i,j lie on a hyperplane, for examples see the figure above. N is a normal vector of this hyperplane.