arXiv subjects
Publications and source records attributed to V. Schulte-Frohlinde.
We extend existing renormalization group calculations for the exponents describing scaling of star polymers and polymer networks constituted by chains of different species (the so-called copolymer star exponents). Our four loop results find application in the description of various phenomena involving self-avoiding and random walks that interact.
For an anisotropic euclidean $ϕ^4$-theory with two interactions $[u (\sum_{i=1^M ϕ_i^2)^2+v \sum_{i=1}^M ϕ_i^4]$ the $β$-functions are calculated from five-loop perturbation expansions in $d=4-\varepsilon$ dimensions, using the knowledge of the large-order behavior and Borel transformations. For $\varepsilon=1$, an infrared stable cubic fixed point for $M \geq 3$ is found, implying that the critical exponents in the magnetic phase transition of real crystals are of the cubic universality class. There were previous indications of the stability based either on lower-loop expansions or on less reliable Pad\'{e approximations, but only the evidence presented in this work seems to be sufficently convincing to draw this conclusion.
Motivated by the discovery of errors in six of the 135 diagrams in the published five-loop expansions of the $β$-function and the anomalous dimensions of the ${O}(n)$-symmetric $ϕ^4$-theory in $D=4-\ep$ dimensions we present the results of a full analytic reevaluation of all diagrams. The divergences are removed by minimal subtraction and $\ep$-expansions are given for the critical exponents $η$, $ν$, and $ω$ up to order $ε^5$.
The renormalization group functions are calculated in $D=4-ε$ dimensions for the $ϕ^4$-theory with two coupling constants associated with an ${O}(N)$-symmetric and a cubic interaction. Divergences are removed by minimal subtraction. The critical exponents $η$, $ν$, and $ω$ are expanded up to order $ε^5$ for the three nontrivial fixed points O(N)-symmetric, Ising, and cubic. The results suggest the stability of the cubic fixed point for $N\geq3$, implying that the critical exponents seen in the magnetic transition of three-dimensional cubic crystals are of the cubic universality class. This is in contrast to earlier three-loop results which gave $N > 3$, and thus Heisenberg exponents. The numerical differences, however, are less than a percent making an experimental distinction of the universality classes very difficult.