Topological van Hove singularities at phase transitions in Weyl metals

We show that in three-dimensional (3D) topological metals, a subset of the van Hove singularities of the density of states sits exactly at the transitions between topological and trivial gapless phases. We may refer to these as topological van Hove singularities. By investigating two minimal models, we show that they originate from energy saddle points located between Weyl points with opposite chiralities, and we illustrate their topological nature through their magnetotransport properties in the ballistic regime. We exemplify the relation between van Hove singularities and topological phase transitions in Weyl systems by analyzing the 3D Hofstadter model, which offers a simple and interesting playground to consider different kinds of Weyl metals and to understand the features of their density of states. In this model, as a function of the magnetic flux, the occurrence of topological van Hove singularities can be explicitly checked.

Figure 1

Dispersion relation of Eq. (1) for $k_z=0$ alongside the associated DOS. We highlight the Weyl point energy $\epsilon =0$ (black dashed line) and the VH singularities at $\epsilon =\pm E_s$ (magenta dashed lines). In this figure and in the following, energies are in units of $t$.

Figure 2

Landau levels ${\epsilon }_n\left(\mathbf{k}\right)$ at fixed $k_y=k_z=0$, up to $n=5$, in the case of negative magnetic field in the two regimes $\left|B\right|<B_c$ (a) and $\left|B\right|>B_c$ (b). The chiral Landau level $(n=0)$ is plotted in blue. For large magnetic fields, we distinguish the three regimes discussed in the main text: ${\mu }_0\in (-\sqrt{\left|B\right|},E_s)$ (brown dashed line), ${\mu }_1\in [E_s,\sqrt{\left|B\right|})$ (gray dashed line), and ${\mu }_2\ge \sqrt{\left|B\right|}$ (magenta dashed line).

Figure 3

Landau levels ${\varepsilon }_n\left(\mathbf{k}\right)$ of the model in Eq. (5) at fixed $k_z=\pi /2$ in the case of positive magnetic field $B\gtrsim B_c$. The chiral Landau levels ($n=0$) are plotted in blue. We show two of the four regimes discussed in the main text: ${\mu }_0\in [\epsilon ,E_s]$ (magenta dashed line), ${\mu }_1\in [E_s,{\varepsilon }_{1,\text{min}}]$ (gray dashed line). The saddle point energy $E_s$ (dotted brown line) is reported, too.

Figure 4

Left panel: second band in Eq. (7) for $k_z=\pi /2$ centered around the point ${\mathbf{k}}_{\mathbf{W}}^{-,-}$, together with the topological (brown dots) and trivial (magenta dots) VH singularities. Right panel: Hofstadter model DOS for $n=2$ and $\epsilon \le 1$. We highlight the topological (brown dashed lines) and trivial (magenta dashed lines) VH singularities. Insets: Fermi surfaces in the MBZ for the different regions of $\mu$ explored in the text, highlighting the main points in momentum space.

Figure 5

Normalized DOS profiles from $n=3$ to 8, obtained with the exact diagonalization of the Hofstadter Hamiltonian on a finite cubic system of linear size $L=120$ (except for $n=7$, where we considered $L=119$). Because the energy spectrum is symmetric with respect to the origin, we plot the profiles for $\epsilon <0$.

Figure 6

Left panel: Weyl node energy ${\epsilon }_0$ for $n\le 6$ (a dotted function ${\epsilon }_0=n^{-2.8}+\kappa$ is depicted as a guide for the eye). Central panel: ${\epsilon }_{\text{min}}$ for $n\le 50$ together with the fit function in Eq. (B1). Right panel: $\mathrm{\Delta }\epsilon$ for $25\le n\le 50$, superimposed with the function $\mathrm{\Delta }\epsilon \propto n^{-1}$. All the estimated values are obtained through a finite-size scaling analysis, up to $L=120$.

Figure 7

Left: energy difference $\delta$ as a function of $n$ for $n\le 8$. We also plot $\delta =\tau n+q$ (see the text) with the estimated parameter values (blue dashed line) and the line $\delta =0$ (black dotted line), for which the energy difference closes. The errorbars in the plot are estimated through a binning analysis on the DOS profiles. Right: two lowest energy bands at fixed $k_x=k_x^{\left(w\right)}$, $k_y=k_y^{\left(w\right)}$ (left panel), alongside the DOS of the model (right panel) for $n=8$. We highlight the VH singularity associated with the minimum of the second band at energy $E_m$ (brown dashed line) and the energy position of the Weyl nodes $E_w>E_m$ (black dashed line).

Figure 8

Plot of the central bands of the Hofstadter model with $n=4$ at fixed $k_x=k_y=0$. We highlight the stationary points $\mathbf{k}=(0,0,\pm \pi )$, (0,0,0) at $\epsilon =0$ (orange dots) together with the Weyl points (black dots).