SCHR\"Odinger Invariance and Strongly Anisotropic Critical Systems

TL;DR

This paper explores how local scale invariance, specifically Schrödinger invariance for the case z=2, constrains correlation functions in strongly anisotropic critical systems, with applications to various statistical models.

Contribution

It extends the concept of local scale invariance to anisotropic systems and derives explicit forms of correlation functions under Schrödinger invariance, supported by exact solutions.

Findings

Schrödinger invariance determines two- and three-point functions in the bulk.

Scaling forms near free surfaces are derived for different surface orientations.

Exact solutions in statistical models confirm the theoretical predictions.

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 $\theta=z=2$, the group of local scale transformation considered is the Schr\"odinger group, which can be obtained as the non-relativistic limit of the conformal group. The requirement of Schr\"odinger 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 $\theta$, evidence from higher order Lifshitz points in the spherical model and from directed percolation suggests a simple scaling form of the two-point function.