Large-n expansion for m-axial Lifshitz points

Abstract

The large-n expansion is developed for the study of critical behaviour ofd-dimensionalsystems at m-axial Lifshitz points with an arbitrary numberm of modulation axes. The leading nontrivial contributions toO (1/n) are derived for the two independent correlation exponentsηL2 andηL4, and the relatedanisotropy index θ. Theseries coefficients of these 1/n corrections are given for general values ofm and d with 0≤m≤d and 2+m/2<d<4+m/2 in the form of integrals. For special values ofm andd suchas (m,d) = (1,4), they can be computed analytically, but in general their evaluation requires numerical means. The1/n corrections are shown to reduce in the appropriate limits to those of knownlarge-n expansionsfor the case of d-dimensional isotropic Lifshitz points and critical points, respectively,and to be in conformity with available dimensionality expansions aboutthe upper and lower critical dimensions. Numerical results for the1/n coefficientsof ηL2, ηL4 and θ are presented for the physically interesting case of a uniaxial Lifshitzpoint in three dimensions, as well as for some other choices ofm andd. A universalcoefficient associated with the energy-density pair correlation function is calculated to leading order in1/n for generalvalues of m and d.