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 d₁+σ₁ in d₁ directions and another exponent d₂+σ₂ in the remaining d₂≡d-d₁ ones. We introduce a low energy effective action with nonanalytic power of the momenta. As a function of the two exponents σ₁ and σ₂ we show the system to have three different regimes at criticality, 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 where one of the two σ's is fixed and the other varying. A discussion of the physical relevance of our results is also presented.