Boundary critical behavior atm-axial Lifshitz points for a boundary plane parallel to the modulation axes

Abstract

The critical behavior of semi-infinite d-dimensional systems with n-component order parameter φ and short-range interactions is investigated at an m-axial bulk Lifshitz point whose wave-vector instability is isotropic in an m-dimensional subspace of R d . The associated m modulation axes are presumed to be parallel to the surface, where 0 ≤ m ≤ d − 1. An appropriate semi-infinite |φ| 4 model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term ∝ φ 2 of the Hamiltonian must be supplemented by one of the formλ m α=1 (∂φ/∂xα) 2 involving a dimensionless (renormalized) coupling constant λ. The implied boundary conditions are given, and the general form of the field-theoretic renormalization of the model below the upper critical dimension d * (m) = 4 + m/2 is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value λ * if ǫ ≡ d * (m) − d > 0. The surface critical exponents of the ordinary transition are determined to second order in ǫ. Extrapolations of these ǫ expansions yield values of these exponents for d = 3 in good agreement with recent Monte Carlo results for the case of a uniaxial (m = 1) Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l4 /ν l2 is the anisotropy exponent.

Fluctuation-induced forces in strongly anisotropic critical systems

“…However, when considering FBC, we shall restrict ourselves in two ways: We assume that the BC that result in the large length-scale limit (i) do not break the O(n) symmetry and (ii) are associated with the respective most stable renormalization-group (RG) fixed point. For parallel orientation this simply means that Dirichlet BC φ = 0 hold asymptotically [22,23,24,25]. In the case of perpendicular orientation, two BC hold on either ‡ Note that since ξα and ξ β are bulk correlation lengths, they diverge at the bulk critical point.…”

Critical phenomena at perfect and non-perfect surfaces

“…The discovery of the existence of this kind of multicritical points in 1975 led to numerous theoretical studies of their bulk critical properties. With the exception of an early attempt by Gumbs [114], surface critical phenomena at a bulk Lifshitz point have only been studied very recently [115,116,117,118,119,120]. Most of the results have been obtained for the semi-infinite ANNNI model.…”

“…Very recently Diehl and coworkers [119,120] analysed the surface critical behaviour at bulk Lifshitz points using renormalization group methods. They thereby considered general m-axial Lifshitz points where the wave vector instability takes place in an mdimensional subspace of the d-dimensional space.…”

Relevance of space anisotropy in the critical behavior ofm-axial Lifshitz points

“…The function A (u,wϭ0) has been computed to one-loop order in Ref. 27. Its explicit form will not be needed in the following.…”

“…27, we have included a renormalization function A (u,w) to absorb momentum-independent poles proportional 2 of the two-point vertex function. The fact that the theory must reduce for wϭ0 to the m-isotropic one implies the relations…”