Continuously varying critical exponents to O(1/n).

Abstract

In the context of a phase transition, the critical singularity is characterized by a set of numbers, the critical exponents. It is widely believed that for transitions in systems with pair interactions on Bravais lattices (as opposed to hierarchical lattices), the critical exponents are determined solely by parameters d and n, representing the dimensionality of space and of the order parameter, respectively, and by \ensuremath{\sigma}, the interaction range. In this paper we consider a certain generalization of the Lifshitz singularity in the n-vector model in which a new parameter \ensuremath{\lambda}, describing a type of anisotropy of the interactions, is introduced. This is described by a model Hamiltonian in k space in which the leading term is of the form [${\mathrm{\ensuremath{\Sigma}}}_{\mathit{i}=1}^{\mathit{d}}$(${\mathit{k}}_{\mathit{i}}^{2}$${)}^{\mathrm{\ensuremath{\sigma}}/(2\ensuremath{\lambda})}$]\ensuremath{\lambda}. It should be noted that, as for the usual Lifshitz case, what would otherwise be the dominant ${\mathit{k}}^{2}$ term is supposed to have already been cancelled. For this model the parameters d, n, and \ensuremath{\sigma} are not sufficient to determine the value of the critical exponents to O(1/n), although they are sufficient for the first order in the \ensuremath{\epsilon} expansion.Calculations exact to O(1/n) show that, for a given set d, n, and \ensuremath{\sigma}, the exponents vary continuously with the parameter \ensuremath{\lambda} and include as a special case (for \ensuremath{\lambda}=1) the critical exponents for an anisotropic Lifshitz point. The critical exponents for this Lifshitz point differ markedly in value from those for the usually studied isotropic Lifshitz point, which is also a special case (for \ensuremath{\lambda}=\ensuremath{\sigma}/2) of our model. For \ensuremath{\sigma}=2, however, the two cases corresponding to \ensuremath{\lambda}=1 and \ensuremath{\sigma}/2 become degenerate. We provide another explicit example of the dependence of the critical exponents on \ensuremath{\lambda} by performing, in addition to the 1/n expansion, an expansion in \ensuremath{\sigma}-2. This helps to shed some light on the approach to the degeneracy at \ensuremath{\sigma}=2. The analysis of the Feynman integrals arising in the O(1/n) calculation provides interesting examples of the applications of hypergeometric functions; these integrals have themselves led to new results (published separately) in the theory of hypergeometric functions.