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

Abstract

The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\mathbf{\varphi }$ 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 $0\le m\le d-1$, are parallel to the surface. The resulting scaling laws for the surface critical indices are given. The surface critical exponent ${\eta }_{\Vert }^{\mathrm{sp}}$, the surface crossover exponent $\Phi$ and related ones are determined to first order in $ε=4+\case{m}{2}-d$. 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 $\Or(ε)$ term of ${\eta }_{\Vert }^{\mathrm{sp}}$ is found to vanish, which implies that the difference of ${\beta }_1^{\mathrm{sp}}$ and the bulk exponent $\beta$ is of order ${\epsilon }^2$.

21 pages, one figure included as eps file, uses IOP style files

References in corpus (3)

• Critical phenomena at perfect and non-perfect surfaces
• Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points
• Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes

Cited by in corpus (4)

• Fluctuation-induced forces in strongly anisotropic critical systems
• On conjectured local generalizations of anisotropic scale invariance and their implications
• Boundary critical behaviour at m-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations