# How to get mean field critical exponents for this Hamiltonian?

$$\mathcal{H} = -J \sum_{\langle ij\rangle} \sum_{\alpha=1}^N s_i{}^\alpha s_j{}^\alpha -g \sum_{\langle ij\rangle} \sum_{\alpha\beta} (s_i{}^\alpha s_j{}^\alpha) (s_i{}^\beta s_j{}^\beta)$$

Above Hamiltonian represents coupled Ising spin $$\tfrac{1}{2}$$ model where $$\alpha$$ and $$\beta$$ are flavour indices running from $$1$$ to $$N$$ (denotes no of spin) and $$i$$ and $$j$$ refers to lattice location of infinitely long square lattice whose every point contains $$N$$ Ising $$\tfrac{1}{2}$$ spins. So spins interact within themselves as Ising model and system of spins in our square lattice also interacts in such a way that neighboring vertex only interacts ($$\langle ij\rangle$$ denotes that in both summations). So how does one calculate the critical exponents and prove the relation:

$$\alpha + 2 \beta + \gamma = 2$$

• This model is known as the N-color Ashkin-Teller model. You can find the determination of the phase diagram in the mean-field approximation in this paper : Grest, Widom (1981) PRB 24 6508 at section III. The critical exponents are not computed but I guess that they take the usual mean-field values: $\beta=1/2$, $\alpha=0$, $\gamma=1$. – Christophe Jun 26 at 11:00