Surface critical behaviour at m-axial Lifshitz points: continuum models, boundary conditions and two-loop renormalization group results

TL;DR

This paper investigates the surface critical behavior at m-axial Lifshitz points using continuum models and two-loop renormalization group analysis, revealing new fixed points and surface critical exponents in higher dimensions.

Contribution

It constructs continuum models with boundary derivative terms for surface critical behavior at Lifshitz points and computes two-loop renormalization group results for surface exponents.

Findings

Identification of infrared-stable fixed points at two-loop order.

Surface critical exponents deviate from m=0 case at order ε^2.

Exact scaling dimension of surface energy density derived.

Abstract

The critical behaviour of semi-infinite $d$-dimensional systems with short-range interactions and an O(n) invariant Hamiltonian is investigated at an $m$-axial Lifshitz point with an isotropic wave-vector instability in an $m$-dimensional subspace of $\mathbb{R}^d$ parallel to the surface. Continuum $|\bphi|^4$ models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant $\lambda$) must be included in addition to the familiar ones $\propto\phi^2$. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in $d=4+\frac{m}{2}-\epsilon$ dimensions (with $\epsilon>0$) are located at $\lambda=\lambda^*=\Or(\epsilon)$. At second order in $\epsilon$, the surface critical exponents of both the ordinary and the special transitions start to deviate from their $m=0$ analogues. Results to order $\epsilon^2$ are presented for the surface critical exponent $\beta_1^{\rm ord}$ of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (\theta-1)$, where $\theta=\nu_{l4}/\nu_{l2}$ is the bulk anisotropy exponent.