Orthomagnetic materials

For this hypothetical kind of material $$E=-\boldsymbol{\mu}\cdot\textbf{B}=0,$$ so the question arises how to describe two vectors orthogonal whose magnitudes are proportional?

Which is solved by

$$\textbf{v}\cdot\textbf{w} :=vw e^{i\theta}.$$

Then

$$Z=e^{i\beta\mu_B B}+e^{-i\beta\mu_B B}=2\cos(\beta\mu_B B)$$

Periodic magnetization:

$$M=-\frac{N\mu_B}{V}\tan(\beta\mu_B B)$$