## 3 Results
Similarly, for magnetic hyperbolic materials [12], the function from Equation (17) is \(h(\mathbf{k})=\epsilon\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)\), assuming \(\varepsilon\) is a scalar, and the tensor \(\hat{\mu}\) has to have principle values of different signs. The HKP waves have \(Z_{l}=0\) in magnetic hyperbolic materials (see Equations (17) and (19)). It has been theoretically predicted in Reference [7] that in the absence of magnetoelectric coupling (\(\hat{X}=\hat{Y}=\hat{0}\)), if both \(\hat{\varepsilon}\) and \(\hat{\mu}\) have principal values of different signs at the same frequency \(\omega\), then bihyperboloid isofrequency k-surfaces are possible. As can be seen from Equation (14), if \(\hat{X}=\hat{Y}=\hat{0}\), then \(h(\mathbf{k})=\left(\mathbf{k}^{T}\hat{\varepsilon}\mathbf{k}\right)\!\left(\mathbf{k}^{T} \hat{\mu}\mathbf{k}\right)\) and, indeed, two hyperboloids in the k-surface can form corresponding to electric \(Z_{l}^{-1}=0\) branch with \(\left(\mathbf{k}^{T}\hat{\varepsilon}\mathbf{k}\right)=0\) and magnetic branch \(Z_{l}=0\) with \(\left(\mathbf{k}^{T}\hat{\mu}\mathbf{k}\right)=0\) (cf. the discussion of Figure 3). Let us consider anisotropic magnetoelectric coupling and study the drastic changes to the topology of the isofrequency surfaces it leads to. We first study materials which are nonhyperbolic in the absence of magnetoelectric coupling, i.e., the principal values of their \(\hat{\varepsilon}\) and \(\hat{\mu}\) tensors have the same signs. Consider a material with \(\hat{\varepsilon}=\epsilon\hat{1}\), \(\hat{\mu}=\mu\hat{1}\), and \(\hat{Y}=-\hat{X}^{T}=\text{diag}(i\kappa_{1},i\kappa_{1},i\kappa_{2})\). Equation (14) then turns into \(\left(q_{\rho}^{2}\kappa_{1}+q_{z}^{2}\kappa_{2}\right)^{2}=\epsilon\mu\), which shows that HKPs propagate if \(\left(\kappa_{1}^{2}-\epsilon\mu\right)\) and \(\left(\kappa_{2}^{2}-\epsilon\mu\right)\) have opposite signs. Note that \(\kappa=\kappa_{1}=\kappa_{2}=\pm\sqrt{\epsilon\mu}\) does not lead to the formation of HKPs, since in this case a 0/0 indeterminacy forms in Equation (9), which resolves as 0, since for an isotropic chiral material \(n=\sqrt{\epsilon\mu}\pm\kappa\)[13]. The formation of a hyperbolic material here is due to the anisotropic chirality tensor and is shown in Figure 2a. If reciprocity is broken \(\hat{X}^{T}\
eq-\hat{Y}\), then \(h(\mathbf{k})=\epsilon\mu-\big{(}\mathbf{k}^{T}\hat{X}\mathbf{k}\big{)}\big{(}\mathbf{k}^{T}\hat {Y}\mathbf{k}\big{)}\) and a bihyperboloidal k-surface may form. In Figure 2b we plot k-surfaces for a material with \(\hat{e}=\hat{e1}\), \(\hat{\mu}=\hat{\mu1}\), \(\hat{X}=-i\;\text{diag}(\kappa_{1},\kappa_{1},\kappa_{2})\), and \(\hat{Y}=i\;\text{diag}(\kappa_{2},\kappa_{2},\kappa_{1})\) for the same values of \(\kappa_{1}\) and \(\kappa_{2}\) as in Figure 2a and demonstrate the bihyperbolic dispersion. In Figure 2c,d we demonstrate that a material with anisotropic \(\hat{e}=\text{diag}(1,1,5)\), \(\hat{\mu}=\text{diag}(5,1,1)\), which is nonhyperbolic without magnetoelectric coupling, forms bihyperbolic and trihyperbolic phases if anisotropic chirality \(\hat{Y}=-\hat{X}^{T}=\text{diag}(i\kappa_{1},i\kappa_{2},i\kappa_{1})\) is added with \(\kappa_{1}=2.9\) and \(\kappa_{2}=0.9\) for Figure 2c and \(\kappa_{1}=2.3\) and \(\kappa_{2}=1.1\) for Figure 2d. Figure 2: (**–d**) Chirality-inducing hyperbolicity in nonhyperbolic phases (material parameters detailed in text). (**e,f**) Chirality-induced phase modifications in anisotropic materials. (**e**) Plot of the function \(\big{(}\mathbf{k}^{T}\hat{e}\mathbf{k}\big{)}\big{(}\mathbf{k}^{T}\hat{\mu}\mathbf{k}\big{)}\) for the \(\hat{e},\hat{\mu}\) matrices used in (**f**) and Figure 3.  
Figure 3: Topological phase transitions (**f**,**h**) between bihyperbolic (**a**,**e**), trihyperbolic, (**b**,**g**) and tetrahyperbolic (**c**,**i**) phases with material parameters matrix \(\hat{M}=\hat{M}_{1}+i\kappa\hat{M}_{2}\) as \(\kappa\) is changed. Panel (**j**) shows the limit of high \(\kappa\). Matrices \(M_{1},\ M_{2}\) are plotted in (**d**).