arXiv subjects

S. Rutkevich

Publications and source records attributed to S. Rutkevich.

4 recordsLinked to original sources

arXiv2010

On conjectured local generalizations of anisotropic scale invariance and their implications

The theory of generalized local scale invariance of strongly anisotropic scale invariant systems proposed some time ago by Henkel [Nucl. Phys. B \textbf{641}, 405 (2002)] is examined. The case of so-called type-I systems is considered. This was conjectured to be realized by systems at m-axial Lifshitz points; in support of this claim, scaling functions of two-point cumulants at the uniaxial Lifshitz point of the three-dimensional ANNNI model were predicted on the basis of this theory and found to be in excellent agreement with Monte Carlo results [Phys. Rev. Lett. \textbf{87}, 125702 (2001)]. The consequences of the conjectured invariance equations are investigated. It is shown that fewer solutions than anticipated by Henkel generally exist and contribute to the scaling functions if these equations are assumed to hold for all (positive and negative) values of the d-dimensional space (or space time) coordinates $(t,\bm{r})\in \mathbb{R}\times\mathbb{R}^{d-1}$. Specifically, a single rather than two independent solutions exists in the case relevant for the mentioned fit of Monte Carlo data for the ANNNI model. Renormalization-group improved perturbation theory in $4+m/2-ε$ dimensions is used to determine the scaling functions of the order-parameter and energy-density two-point cumulants in momentum space to two-loop order. The results are mathematically incompatible with Henkel's predictions except in free-field-theory cases. However, the scaling function of the energy-density cumulant we obtain for m=1 upon extrapolation of our two-loop RG results to d=3 differs numerically little from that of an effective free field theory.

cond-mat.stat-mech↗

arXiv2004

Boundary critical behaviour at $m$-axial Lifshitz points: the special transition for the case of a surface plane parallel to the modulation axes

The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bmϕ$ is studied at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. Field-theoretic renormalization group methods are utilised to examine the special surface transition in the case where the $m$ potential modulation axes, with $0\leq m\leq d-1$, are parallel to the surface. The resulting scaling laws for the surface critical indices are given. The surface critical exponent $η_\|^{\rm sp}$, the surface crossover exponent $Φ$ and related ones are determined to first order in $ε=4+\case{m}{2}-d$. Unlike the bulk critical exponents and the surface critical exponents of the ordinary transition, $Φ$ is $m$-dependent already at first order in $ε$. The $\Or(ε)$ term of $η_\|^{\rm sp}$ is found to vanish, which implies that the difference of $β_1^{\rm sp}$ and the bulk exponent $β$ is of order $ε^2$.

cond-mat.stat-mech↗

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↗