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

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:

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

  • $\begingroup$ 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: $\beta=1/2$, $\alpha=0$, $\gamma=1$. $\endgroup$ – Christophe Jun 26 at 11:00

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