Local scale invariance and strongly anisotropic equilibrium critical systems

TL;DR

This paper introduces a generalized form of scale invariance for strongly anisotropic critical systems, deriving and solving related equations for the two-point function, unifying known symmetries like conformal and Schrödinger invariance.

Contribution

It proposes a new set of infinitesimal transformations extending scale invariance for anisotropic systems, applicable when the anisotropy exponent is 2/N, and provides explicit solutions for the two-point function.

Findings

Derived differential equations for the two-point function.

Explicit solutions for all N values, including special cases.

Connected results to known models at Lifshitz points.

Abstract

A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent \theta =2/N, with N=1,2,3 ... Differential equations for the two-point function are derived and explicitly solved for all values of N. Known special cases are conformal invariance (N=2) and Schr\"odinger invariance (N=1). For N=4 and N=6, the results contain as special cases the exactly known scaling forms obtained for the spin-spin correlation function in the axial next nearest neighbor spherical (ANNNS) model at its Lifshitz points of first and second order.