SCHRÖdinger Invariance and Strongly Anisotropic Critical Systems

TL;DR

The paper proposes Schrödinger invariance as a natural local symmetry for strongly anisotropic critical systems with $\theta=2$, deriving explicit two-point and three-point correlation forms and nontrivial surface/initial-state generalizations. The approach mirrors conformal methods but incorporates non-relativistic phase structure and the Bargmann mass rule, yielding testable predictions. Across several exactly solvable models (diffusion, Glauber dynamics, Lifshitz points, spherical relaxation), the bulk correlators conform to the Schrödinger forms, while models with $\theta\neq2$ reveal non-conformal, stretched-exponential scaling, indicating limits of conformal intuition in anisotropic contexts. The work demonstrates the potential of local scale invariance to organize critical dynamics and points to avenues for extending the symmetry to a full infinite-dimensional algebra and to broader $\theta$ values.

Abstract

The extension of strongly anisotropic or dynamical scaling to local scale invariance is investigated. For the special case of an anisotropy or dynamical exponent $θ=z=2$, the group of local scale transformation considered is the Schrödinger group, which can be obtained as the non-relativistic limit of the conformal group. The requirement of Schrödinger invariance determines the two-point function in the bulk and reduces the three-point function to a scaling form of a single variable. Scaling forms are also derived for the two-point function close to a free surface which can be either space-like or time-like. These results are reproduced in several exactly solvable statistical systems, namely the kinetic Ising model with Glauber dynamics, lattice diffusion, Lifshitz points in the spherical model and critical dynamics of the spherical model with a non-conserved order parameter. For generic values of $θ$, evidence from higher order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function.