Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

TL;DR

This paper develops a framework for local scale invariance in anisotropic and dynamical systems, deriving exact two-point functions and confirming predictions through models like Lifshitz points and ferromagnets.

Contribution

It introduces two types of local scale transformations applicable to static and dynamical anisotropic systems, extending the concept of conformal and Schrödinger invariance.

Findings

Derived linear fractional differential equations for two-point functions.

Exact solutions for equilibrium correlators and out-of-equilibrium response functions.

Confirmed predictions in Lifshitz points and ferromagnetic models.

Abstract

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent theta or a dynamical exponent z. For a given value of theta, we construct local scale transformations which can be viewed as scale transformations with a space-time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of theta, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for theta=1 and Schrodinger invariance for theta=2. The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. Aparticularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber-Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.