Large-n expansion for m-axial Lifshitz points

TL;DR

This paper develops a large-n expansion method to analyze critical behavior at m-axial Lifshitz points in d-dimensional systems, providing analytical and numerical results for correlation exponents and anisotropy indices.

Contribution

It introduces a comprehensive large-n expansion framework for m-axial Lifshitz points, deriving explicit 1/n corrections for correlation exponents and anisotropy indices for general m and d.

Findings

Derived 1/n corrections for ta_{L2}, ta_{L4}, and re in agreement with known expansions.

Provided analytical results for special cases like (m,d)=(1,4).

Numerical results for uniaxial Lifshitz point in three dimensions.

Abstract

The large-n expansion is developed for the study of critical behaviour of d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of modulation axes. The leading non-trivial contributions of O(1/n) are derived for the two independent correlation exponents \eta_{L2} and \eta_{L4}, and the related anisotropy index \theta. The series coefficients of these 1/n corrections are given for general values of m and d with 0<m<d and 2+m/2<d<4+m/2 in the form of integrals. For special values of m and d such as (m,d)=(1,4), they can be computed analytically, but in general their evaluation requires numerical means. The 1/n corrections are shown to reduce in the appropriate limits to those of known large-n expansions for the case of d-dimensional isotropic Lifshitz points and critical points, respectively, and to be in conformity with available dimensionality expansions about the upper and lower critical dimensions. Numerical results for the 1/n coefficients of \eta_{L2}, \eta_{L4} and \theta are presented for the physically interesting case of a uniaxial Lifshitz point in three dimensions, as well as for some other choices of m and d. A universal coefficient associated with the energy-density pair correlation function is calculated to leading order in 1/n for general values of m and d.