Boundary critical behaviour at $m$-axial Lifshitz points: the special transition for the case of a surface plane parallel to the modulation axes

TL;DR

This paper investigates the surface critical behaviour at $m$-axial Lifshitz points with a focus on the special transition where the surface plane is parallel to the modulation axes, using field-theoretic renormalization group methods.

Contribution

It provides the first-order epsilon expansion of surface critical exponents at $m$-axial Lifshitz points for the special transition with a parallel surface orientation.

Findings

Surface critical exponent $oldsymbol{ ext{eta}_ ext{||}^{ ext{sp}}}$ has a vanishing first-order epsilon term.

Surface crossover exponent $oldsymbol{ ext{Phi}}$ depends on $m$ at first order in epsilon.

Difference between surface order parameter exponent $oldsymbol{eta_1^{sp}}$ and bulk exponent $oldsymbol{eta}$ is of order $oldsymbol{ ext{epsilon}^2}$.

Abstract

The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bm{\phi}$ is studied at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. Field-theoretic renormalization group methods are utilised to examine the special surface transition in the case where the $m$ potential modulation axes, with $0\leq m\leq d-1$, are parallel to the surface. The resulting scaling laws for the surface critical indices are given. The surface critical exponent $\eta_\|^{\rm sp}$, the surface crossover exponent $\Phi$ and related ones are determined to first order in $\epsilon=4+\case{m}{2}-d$. Unlike the bulk critical exponents and the surface critical exponents of the ordinary transition, $\Phi$ is $m$-dependent already at first order in $\epsilon$. The $\Or(\epsilon)$ term of $\eta_\|^{\rm sp}$ is found to vanish, which implies that the difference of $\beta_1^{\rm sp}$ and the bulk exponent $\beta$ is of order $\epsilon^2$.