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Stability of a three-dimensional cubic fixed point in two-coupling-constant phi^4-Theory

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Kleinert, H. , Thoms, S. and Schulte-Frohlinde, V. (1997): Stability of a three-dimensional cubic fixed point in two-coupling-constant phi^4-Theory , Physical review b , 56 , pp. 14428-14434 .
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Abstract:

For an anisotropic euclidean phi^4 field theory with two interactions$[u (\sum_{i=1}^M {\phi}_i^2)^2+v \sum_{i=1}^M \phi_i^4]$the $\beta$-functions are calculated from five-loop perturbation expansionsin $d=4-\varepsilon$ dimensions,using the knowledge of the large-order behavior and Borel transformations.For $\varepsilon=1$, an infrared stable cubicfixed point for $M \geq 3$ is found,implying that the critical exponents in the magnetic phase transitionof real crystals are of the cubic universality class.There were previous indications of the stability based either onlower-loop expansions or on less reliable Pad\'{e} approximations,but only the evidence presented in this work seems to besufficently convincing to draw this conclusion.

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