Fig. 3. Density of states (DOS) and coefficients of Taylor expansion around saddle point during the topological transition of VHS.

a DOS of Eq. (5) with different $\beta$, where $E_0={\alpha }^3∕{\tilde{\gamma }}^2,{\rho }_0={\alpha }^{-1}$. b Normalized coefficients of Eq. (3) as functions of $g$ when $g^{\prime }=1.2g$. Here ${\alpha }_0=\lambda v,{\gamma }_0={\lambda }^{2}v$ and ${\kappa }_0={\lambda }^{3}v$. Inset of b is zoom-in plot where $\beta$ changes sign near high-order VHS while $\alpha ,\gamma ,\kappa$ do not.