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

Original · EN

The critical behavior of d-dimensional systems with n-component order parameter ϕ is studied at an m-axial Lifshitz point where a wave-vector instability occurs in an m-dimensional subspace Rᵐ (m>1). Field theoretic renormalization group techniques are exploited to examine the effects of terms in the Hamiltonian that break the rotational symmetry of the Euclidean group E(m). The framework for considering general operators of second order in ϕ and fourth order in the derivatives ∂α with respect to the Cartesian coordinates xα of Rᵐ is presented. For the specific case of systems with cubic anisotropy, the effects of having an additional term, ∑α₌₁ᵐ(∂α²ϕ)², are investigated in an ε expansion about the upper critical dimension d*(m)=4+m/2. Its associated crossover exponent is computed to order ε² and found to be positive, so that it is a relevant perturbation on a model isotropic in Rᵐ.