Critical dimension

In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. Above the upper critical dimension the critical exponents of the theory become the same as that in mean field theory. An elegant criterion to obtain the critical dimension within mean field theory is due to V. Ginzburg. In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. Above the upper critical dimension the critical exponents of the theory become the same as that in mean field theory. An elegant criterion to obtain the critical dimension within mean field theory is due to V. Ginzburg. Since the renormalization group sets up a relation between a phase transition and a quantum field theory, this has implications for the latter and for our larger understanding of renormalization in general. Above the upper critical dimension, the quantum field theory which belongs to the model of the phase transition is a free field theory. 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. The precise number may be determined by the required cancellation of conformal anomaly on the worldsheet; it is 26 for the bosonic string theory and 10 for superstring 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 {displaystyle x_{i}} and fields ϕ i {displaystyle phi _{i}} . Examples are the standard ϕ 4 {displaystyle phi ^{4}} -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 thecoordinates and fields with a factor b {displaystyle 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 → t b − z {displaystyle t ightarrow tb^{-z}} with some constant exponent z = − [ t ] {displaystyle z=-} . The goal is to determine the exponent set N = { [ x i ] , [ ϕ i ] } {displaystyle N={,}} . One exponent, say [ x 1 ] {displaystyle } , may be chosen arbitrarily, for example [ x 1 ] = − 1 {displaystyle =-1} . In the language of dimensional analysis this means that the exponents N {displaystyle N} count wave vector factors (a reciprocal length k = 1 / L 1 {displaystyle k=1/L_{1}} ). Each monomial of the Lagrangian thus leads to a homogeneous linear equation ∑ E i , j N j = 0 {displaystyle sum E_{i,j}N_{j}=0} for the exponents N {displaystyle N} . If there are M {displaystyle M} (inequivalent) coordinates and fields in the Lagrangian, then M {displaystyle M} such equations constitute a square matrix. If this matrix were invertible then there only would be the trivial solution N = 0 {displaystyle N=0} . The condition det ( E i , j ) = 0 {displaystyle 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 {displaystyle d_{u}} (provided there is only one variable dimension d {displaystyle d} in the Lagrangian). A redefinition of the coordinates and fields now shows that determining the scaling exponents N {displaystyle N} is equivalent to a dimensional analysis with respect to the wavevector k {displaystyle k} , with all coupling constants occurring in the Lagrangian rendered dimensionless. Dimensionless coupling constants are the technical hallmark for the upper critical dimension.