On conjectured local generalizations of anisotropic scale invariance and their implications

Abstract

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 $(t,\mathbf{r})\in \mathbb{R}\times {\mathbb{R}}^{d-1}$. 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 $4+m/2-\epsilon$ 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.

Latex source file with 8 eps files, 45 pages, 7 figures; v2: minor changes, misprints corrected

References in corpus (8)

• Quantum Impurity Problems in Condensed Matter Physics
• A massive Feynman integral and some reduction relations for Appell functions
• On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions
• Large-n expansion for m-axial Lifshitz points
• Relevance of space anisotropy in the critical behavior of m-axial Lifshitz points
• Landau singularities and singularities of holonomic integrals of the Ising class
• Compatibility of 1/n and epsilon expansions for critical exponents at m-axial Lifshitz points
• Boundary critical behavior at m-axial Lifshitz points for a boundary plane parallel to the modulation axes

Cited by in corpus (7)

• 2d Galilean Field Theories with Anisotropic Scaling
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• Schrödinger symmetry: a historical review
• Boundary conformal field theory at the extraordinary transition: The layer susceptibility to $O(\epsilon )$
• Lifshitz-point correlation length exponents from the large-n expansion
• A Feynman integral in Lifshitz-point and Lorentz-violating theories in $R^D\oplus R^m$
• Feynman integral in ${\mathbb{R}}^1\oplus {\mathbb{R}}^m$ and complex expansion of ${}_2{F}_1$