Local Scale Invariance and Strongly Anisotropic Equilibrium Critical Systems

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 θ0ex0ex=0ex0ex2/N, with N0ex0ex=0ex0ex1,2,3. Differential equations for the two-point function are derived and explicitly solved for all values of N. Known special cases are conformal invariance ( N0ex0ex=0ex0ex2) and Schr\"odinger invariance ( N0ex0ex=0ex0ex1). For N0ex0ex=0ex0ex4 and N0ex0ex=0ex0ex6, 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.