Boundary critical behaviour at m-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Abstract

Semi-infinite $d$-dimensional systems with an $m$-axial bulk Lifshitz point are considered whose ($d-1$)-dimensional surface hyper-plane is oriented perpendicular to one of the $m$ modulation axes. An $n$-component ${\varphi }^4$ field theory describing the bulk and boundary critical behaviour when (i) the Hamiltonian can be taken to have O(n) symmetry and (ii) spatial anisotropies breaking its Euclidean symmetry in the $m$-dimensional coordinate subspace of potential modulation directions may be ignored is investigated. The long-distance behaviour at the ordinary surface transition is mapped onto a field theory with the boundary conditions that both the order parameter $\mathbf{\varphi }$ and its normal derivative ${\partial }_n\mathbf{\varphi }$ vanish at the surface plane. The boundary-operator expansion is utilized to study the short-distance behaviour of $\mathbf{\varphi }$ near the surface. Its leading contribution is found to be controlled by the boundary operator ${\partial }_n^2\mathbf{\varphi }$. The field theory is renormalized for dimensions $d$ below the upper critical dimension $d^{\ast }(m)=4+m/2$, with a corresponding surface source term $\propto {\partial }_n^2\mathbf{\varphi }$ added. The anomalous dimension of this boundary operator is computed to first order in $\epsilon =d^{\ast }-d$. The result is used in conjunction with scaling laws to estimate the value of the single independent surface critical exponent ${\beta }_{\mathrm{L}1}^{(\mathrm{ord},\perp )}$ for $d=3$. Our estimate for the case $m=n=1$ of a uniaxial Lifshitz point in Ising systems is in reasonable agreement with published Monte Carlo results.

arXiv commentsubmitted to J. Phys. A: Math and Gen, special issue on Renormalization Group 2005 as featured in the international workshop Renormalization Group 2005, Helsinki, Finland 30 August - 3 September 2005 (http://theory.physics.helsinki.fi/~rg2005/)

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