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

TL;DR

The paper develops a framework of local scale invariance to extend anisotropic and dynamical scaling beyond conformal and Schrödinger invariances, introducing Typ I (static anisotropic scaling) and Typ II (dynamical aging scaling) with a space-time dependent dilatation. It constructs infinitesimal generators using commuting fractional derivatives, derives linear fractional differential equations for two-point functions, and solves them to obtain explicit scaling functions for Typ I ($\Omega(v)$) and Typ II ($\Phi(u)$). The approach is tested against uniaxial Lifshitz points (ANNNI/ANNNS) and aging in ferromagnets (Glauber-Ising, spherical model), where predictions for correlators and autoresponses show remarkable agreement, supporting local scale invariance as a predictive symmetry for non-equilibrium and anisotropic critical phenomena. The results unify known conformal and Schrödinger cases as special limits ($\theta=1,2$) and offer a broad phenomenological testbed for future field-theoretic realizations and broader applications in non-equilibrium statistical physics.

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.

Figures (9)

• Figure 1: Scaling functions $\Phi(u)$ for Typ II as given by eq. (\ref{['4:Phiu']}) with $\beta_1+\gamma_1=1$ for (a) $\theta=1.5$ and (b) $\theta=2.5$, for several values of $\Lambda$ as indicated.
• Figure 2: Scaling functions $\Phi(u)$ for Typ II as given by eq. (\ref{['4:Phiu']}). (a) Several values of $\theta$ as indicated, $\Lambda=1/2$ and $\beta_1+\gamma_1=1$. (b) Several values of $\theta$ and $\gamma_1=-\beta_1=1$.
• Figure 3: Scaling functions $v^{\zeta}\Omega_p(v)$ for Typ I, $N=4$, $\alpha_1=1$ and $p=0,1$. The different curves belong to values of $\zeta=N x_1$ as indicated and the squares are the values of $\Omega_{p,\infty}$.
• Figure 4: Scaling functions $v^{\zeta}\Omega_p(v)$ for Typ I, $N=5$, $\alpha_1=1$ and $p=0,1,2$, for different values of $\zeta=Nx_1$. The circles indicate the values of $\Omega_{p,\infty}$.
• Figure 5: Perturbative scaling functions $\Omega(v)$ eq. (\ref{['4:OmegaStoe']}) for Typ I around $N_0=4$, with $\zeta=1.3$, $\alpha_1=1$ and (a) $\mathfrak{p}=+0.11$ and (b) $\mathfrak{p}=-0.11$, for several values of $\varepsilon$.