four-dimensional representation, which can be described by “angular momentum” operators $\vec{J}$ (which are $j=3/2$ matrices) transforming as the T$_2$ representation of the cubic group. In our minimal model, we assume only these states close to $\Gamma$ are important. Then $k\cdot p$ theory and cubic symmetry determines the band structure in its vicinity to be precisely described by the Luttinger Hamiltonian with three effective mass parameters [@luttinger; @murakami], $$\begin{aligned}
\mathcal{H}_0(k)&=&\frac{k^2}{2 \tilde{M}_0} + \frac{ \frac{5}{4}k^2-(\vec{k} \cdot \vec{J})^2}{2m} - \frac{ (k_x^2 J_x^2 + k_y^2 J_y^2 +k_z^2 J_z^2)}{2 M_c} .\nonumber\end{aligned}$$ This describes doubly degenerate bands with energy $$\begin{aligned}
E_{\pm}(k)= \frac{k^2}{2M_0} \pm \sqrt{\left(\frac{k^2}{2m}\right)^2+ \frac{m+2 M_c}{4 m M_c^2} p_c(k) },\end{aligned}$$ where $ p_c(k) = \sum_{i} k_i^4 -\sum_{i \neq j} k_i^2 k_j^2$ and $M_0 = (4 M_c \tilde{M}_0)/(4M_c-5 \tilde{M}_0)$. Henceforth we assume $M_0 > m$, which describes conduction and valence bands touching quadratically at $E=k=0$, where the chemical potential for the undoped material crosses.

The LAB is obtained by adding to this the long-range Coulomb interaction. We implement the latter by a scalar potential $\varphi$, which in the Euclidean path integral formalism gives the action $$\begin{aligned}
&& {S}_{L} = \int d \tau d^d x \left\{ \psi^{\dagger}  \Big[\partial_{\tau}
-i e \, \varphi +\hat{\mathcal{H}}_0\Big] \psi + \frac{c_0}{2} (\partial_i
\varphi)^2 