A. Gerwinski | Searcharxiv

arXiv2003

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

The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bmϕ$ 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 $\mathbb{R}^d$. The associated $m$ modulation axes are presumed to be parallel to the surface, where $0\le m\le d-1$. An appropriate semi-infinite $|\bmϕ|^4$ model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term $\propto \bmϕ^2$ of the Hamiltonian must be supplemented by one of the form $\mathringλ \sum_{α=1}^m(\partial\bmϕ/\partial x_α)^2$ 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 $ε\equiv d^*(m)-d>0$. The surface critical exponents of the ordinary transition are determined to second order in $ε$. Extrapolations of these $ε$ 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.

cond-mat.stat-mech↗

arXiv2003

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

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 $λ$) must be included in addition to the familiar ones $\proptoϕ^2$. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in $d=4+\frac{m}{2}-ε$ dimensions (with $ε>0$) are located at $λ=λ^*=\Or(ε)$. At second order in $ε$, the surface critical exponents of both the ordinary and the special transitions start to deviate from their $m=0$ analogues. Results to order $ε^2$ are presented for the surface critical exponent $β_1^{\rm ord}$ of the ordinary transition. 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 bulk anisotropy exponent.

cond-mat.stat-mech↗