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

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

2004-06-09

arxiv.org/abs/cond-mat/0406216v2

J.Phys. A37 (2004) 8575-8594

Physics

Condensed Matter

Statistical Mechanics

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

Scientific paper

10.1088/0305-4470/37/36/001

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 $\epsilon^2$.

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