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Anisotropic Long-Range Spin Systems

We consider anisotropic long-range interacting spin systems in $d$ dimensions. The interaction between the spins decays with the distance as a power law with different exponents in different directions: we consider an exponent $d_{1}+σ_1$ in $d_1$ directions and another exponent $d_{2}+σ_2$ in the remaining $d_2\equiv d-d_1$ ones. We introduce a low energy effective action with non analytic power of the momenta. As a function of the two exponents $σ_1$ and $σ_2$ we show the system to have three different regimes, two where it is actually anisotropic and one where the isotropy is finally restored. We determine the phase diagram and provide estimates of the critical exponents as a function of the parameters of the system, in particular considering the case of one of the two $σ$'s fixed and the other varying. A discussion of the physical relevance of our results is also presented.