\hat{S}^{(3)}_{i} - \mu \sum_{i} E_{i} \hat{S}^{(3)}_{i}.\end{aligned}$$ This is equivalent to the Blume-Capel model in a transverse magnetic field [@Albayrak:2013]. The third term introduces an anisotropic pseudospin energy. If $J_\mathrm{an} > 0$, this term tends to align dipoles along the (3)-axis, making it an easy axis, which enhances the polarization; if $J_\mathrm{an} < 0$, the term tilts the dipole away from the (3)-axis, creating an easy plane and reducing the polarization.

The TIM is traditionally formulated with a spin-$\frac 12$ pseudospin. In that case, $\hat{S}^{(3)}_{i}$ is written in terms of a Pauli spin matrix, and $(\hat{S}^{(3)}_{i})^2$ is proportional to the identity operator. The new term therefore does not produce the desired anisotropy when $S = \frac{1}{2}$. This problem does not exist for higher spin models, and for this reason we formulate the TIM in terms of a general pseudospin $S$. However, we will show below that at the mean-field level, the model provides nearly the same results for any value of $S$, and for simplicity we revert to $S=1$ when we show results as a way of gaining insight into the general case.

Applying mean-field theory to equation  gives the following self-consistent expression for $S^{(3)}_{i}$: $$\label{S3}
S^{(3)}_i = \frac{S h^{(3)}_i}{h_i} f_{S}(h_i),$$ 