Condensed Matter Physics, 2010, vol. 13, No. 4, p. 43703:116
DOI:10.5488/CMP.13.43703
Title:
Analysis of the 3d massive renormalization group perturbative expansions: a delicate case
Author(s):

B. Delamotte
(LPTMC, CNRSUMR 7600, Université Pierre et Marie Curie, 75252 Paris Cédex 05, France)
,


M. Dudka
(Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, UA79011 Lviv, Ukraine)
,


Yu. Holovatch
(Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, UA79011 Lviv, Ukraine; Institut für Theoretische Physik, Johannes Kepler Universität Linz, A4040 Linz, Austria)
,


D. Mouhanna
(LPTMC, CNRSUMR 7600, Université Pierre et Marie Curie, 75252 Paris Cédex 05, France)

The effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed. Three models are considered: the O(N) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice. We consider all models at fixed d = 3 and analyze the resummation procedures currently used to compute the critical exponents. We first show that, for the O(N) model, the resummation does not eliminate all nonphysical (spurious) fixed points (FPs). Then the dependence of spurious as well as of the WilsonFisher FPs on the resummation parameters is carefully studied. The critical exponents at the WilsonFisher FP show a weak dependence on the resummation parameters. On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters. For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters. It appears to be spurious, as expected. As for the frustrated models, there are two cases depending on the value of the number of spin components. When N is greater than a critical value N_{c}, the stable FP shows common characteristic with the WilsonFisher FP. On the contrary, for N<N_{c}, the results obtained at the stable FP are similar to those obtained at the spurious FPs of the O(N) and cubic models. We conclude from this analysis that the stable FP found for N_{c} in frustrated models is spurious. Since N_{c}>3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.
Key words:
field theory, renormalization group, critical phenomena, perturbation theory, resummation
PACS:
75.10.Hk, 11.10.Hi, 12.38.Cy
