Calculation of Critical Exponents from Field Theory

D.I. Kazakov, Alexey A. Vladimirov, O.V. Tarasov

C. Mergulhão, C. E. I. Carneiro

We use field theory to calculate the critical exponents ${\ensuremath{\eta}}_{l2},$ ${\ensuremath{\eta}}_{l4},$ ${\ensuremath{\nu}}_{l2},$ and ${\ensuremath{\nu}}_{l4}$ for the Lifshitz point (LP) with $m=2$ and $m=6$. We were motivated by an old controversy on the order ${\ensuremath{\epsilon}}^{2}$ corrections for the exponents of this multicritical point. In a previous paper, we studied the renormalization of the theory that describes the LP and derived expressions for the exponents and scaling relations that hold to all orders in perturbation. In this work we concentrate on the calculation of the Feynman diagrams that are involved in the determination of the exponents and present our results for the exponents to order ${\ensuremath{\epsilon}}^{2}.$ The evaluation of these diagrams, due to a free propagator that contains a quartic term in the momentum and is not rotationally invariant, requires special techniques that are presented in detail in the appendixes.

Daniel J. Amit, Víctor Martín-Mayor

Claude Bervillier, Claude Godrèche

We use a regularized (ϕ2)2 field theory for the determination of the critical exponents γ and ν in three dimensions. The regularization is done by a Gaussian propagator. This enables any order in g for any correlation function to be easily calculated but introduces a second expansion parameter. In spite of the difficulties encountered when dealing with double series, we obtain values for the critical exponents in good agreement with those found with the renormalized theory. This and the discussion we give of the link between the renormalized theory and our model illustrate the concept of universality in critical phenomena.

Jeffrey S. Reeve, Anthony John Guttmann, B W Keck

The methods of renormalized perturbation theory are used to find the critical exponents of the percolation and Yang-Lee edge singularity problems. Both of the above critical phenomena are described by ${\ensuremath{\varphi}}^{3}$ field theories, and the Wilson functions are calculated explicitly in three dimensions to order three loops and summed using the Pad\'e-Borel method. The resulting critical exponent values are consistent with direct series expansion results.

E. Brézin, J. C. Le Guillou, Jean Zinn‐Justin +1 more