## 3 Results
Now let us turn to materials that are hyperbolic in the absence of magnetoelectric coupling and consider how magnetoelectric coupling changes their topologies and HKPs. Below we consider a numerical example of matrix \(M_{1}=\left[ \begin{array}{cc}\hat{e}&0\\ 0&\hat{\mu}\end{array} \right]\) shown in the inset Figure 3d (to the left of the black line). This matrix \(M_{1}\) describes an anisotropic material with the dispersion transitional between hyperbolic and bihyperbolic. The HKP propagation directions for \(M_{1}\) are illustrated in Figure 2e,f. In Figure 2e we plot the function \(\left(\mathbf{k}^{T}\hat{e}\mathbf{k}\right)\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)\) (yellow), whose intersection with zero (blue) corresponds to the HKP propagation direction. These directions are also plotted in Figure 2f for the electric branch \(Z_{l}^{-1}=0\), with \(\left(\mathbf{k}^{T}\hat{e}\mathbf{k}\right)=0\) (purple) and magnetic branch \(Z_{l}=0\) with \(\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)=0\) (red), which corresponds to the color scale for \(Z_{l}\) shown to the right. Let us turn to a material characterized by matrix \(\hat{M}=\hat{M}_{1}+i\kappa\hat{M}_{2}\), where \(M_{2}=\left[ \begin{array}{cc}\hat{0}&\hat{1}\\ -\hat{1}&\hat{0}\end{array} \right]\) is responsible for magnetoelectric coupling. The change in the HKP propagation directions with increasing \(\kappa\) from zero corresponds to raising the level of the blue plane in Figure 2e to level \(\kappa^{2}\), which confines the HKP states to the positive region \(\left(\mathbf{k}^{T}\hat{e}\mathbf{k}\right)\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)\). The HKP propagation directions for \(\kappa=1\) are shown in Figure 2f with four disconnected curves, which corresponds to bihyperbolic dispersion. These connectivity curves are color-coded corresponding to \(Z_{l}\) of the HKP states. Let us consider a material described by the matrix \(\hat{M}=\hat{M}_{1}+i\kappa\hat{M}_{2}\) with the individual matrices \(M_{1},\ M_{2}\) plotted on Figure 3d. Changing parameter \(\kappa\) leads to topological transitions between bihyperbolic (Figure 3a,e), trihyperbolic (Figure 3b,g), and tetrahyperbolic (Figure 3c,i) phases. The progression of the phase changes is demonstrated in Figure 3e-j. In these panels the HKP states for \(\kappa=0\) satisfying \(\left(\mathbf{k}^{T}\hat{e}\mathbf{k}\right)\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)=0\) are outlined in black, which are the same curves as in Figure 2f. The directions for which \(\kappa=0\) and \(\left(\mathbf{k}^{T}\hat{X}\mathbf{k}\right)=0\) (brown) satisfy Equations (14) and (18) for all \(\kappa\) and serve as the framework for the topological transitions. These intersection points in Figure 3e-j have invariant impedances \(Z_{l}^{-1}=0\) for the electric branch with \(\left(\mathbf{k}^{T}\hat{e}\mathbf{k}\right)=0\) and \(Z_{l}=0\) for the magnetic branch. The HKP states for the \(\kappa\) noted in panels (e)-(j) are color-coded in accordance with their \(Z_{l}\) as indicated to the right of Figure 3f. The HKP for nonzero \(\kappa\) are positioned in the \(\left(\mathbf{k}^{T}\hat{e}\mathbf{k}\right)\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)>0\) regions between the \(\kappa=0\) lines, in accordance with Equations (14) and (18), and touch these lines at the intersections of \(\kappa=0\) and \(\left(\mathbf{k}^{T}\hat{X}\mathbf{k}\right)=0\). The HKP states group into several disconnected curves, the number of which characterizes the topology of the phase. Note that each curve is split into two parts by the \(\left(\mathbf{k}^{T}\hat{X}\mathbf{k}\right)=0\) curves at their intersections with \(\kappa=0\). The resulting two subcurves of HKP states correspond to two different branches of Equation (18). The addition of a small \(\kappa\) leads to a transition to the bihyperbolic phase (four disconnected curves) as shown in Figure 3e.