Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems

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 $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2/N$, with $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,3\dots{}$. Differential equations for the two-point function are derived and explicitly solved for all values of $N$. Known special cases are conformal invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2$) and Schr\"odinger invariance ( $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$). For $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$ and $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}6$, 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 model at its Lifshitz points of first and second order.