Chern numbers and chiral anomalies in Weyl butterflies

The Hofstadter butterfly of lattice electrons in a strong magnetic field is a cornerstone of condensed matter physics, exploring the competition between periodicities imposed by the lattice and the field. Here, we introduce and characterize the Weyl butterfly, which emerges when a large magnetic field is applied to a three-dimensional Weyl semimetal. Using an experimentally motivated lattice model for cold-atomic systems, we solve this problem numerically. We find that Weyl nodes reemerge at commensurate fluxes and propose using wave-packet dynamics to reveal their chirality and location. Moreover, we show that the chiral anomaly—a hallmark of the topological Weyl semimetal—does not remain proportional to the magnetic field at large fields, but rather inherits a fractal structure of linear regimes as a function of the external field. The slope of each linear regime is determined by the difference of two Chern numbers in the gaps of the Weyl butterfly and can be measured experimentally in time of flight.

Figure 1

Energy spectra of Weyl fermions in a magnetic field, as obtained from Eq. (2). (a)–(c) New Weyl nodes emerge at commensurate fluxes per plaquette with an additional $q$-fold degeneracy (see text). (d)–(f) Hofstadter-like spectrum for the $k_z$ values where the Weyl nodes appear for the corresponding fluxes shown in (a)–(c). The isolated zeroth Landau level occurring around the Weyl nodes is highlighted by the red circles. The colors in the butterflies denote the Chern number $\mathcal{C}$ in the gap.

Figure 2

Hall drift of the center of mass of a wave packet for different values of the electric field $E$ at two representative $k_z$ values, 0 and $\pi$, along with the linear fits used to calculate ${\mathcal{C}}_q$ for (a) $q=1$ and (c) $q=2$. (b) and (d) show the topological transition of ${\mathcal{C}}_q$ at the positions of the Weyl nodes, as obtained from Eq. (4). For the simulations $L=128$ and $L_s=48$.

Figure 3

Chiral anomaly of the Weyl butterfly. (a) The chiral charge counts the number difference of left and right movers ${\mathcal{N}}_5=(N_L-N_R)/L^2$. (b) ${\mathcal{N}}_5$ increases linearly with time when both $E$ and $B$ are applied, as shown for $E=0.1$ and fluxes $\mathrm{\Phi }=n{\mathrm{\Phi }}_0/L$, with $n$ going from 1 to 6 (lighter to darker). (c) The rate of chiral charge production grows linearly with the flux with a slope $q$ in the vicinity of commensurate fluxes, for which the model hosts $q$ pairs of Weyl nodes. Linear fits are shown for the data around $\mathrm{\Phi }/{\mathrm{\Phi }}_0=1,\frac{1}{2},\frac{1}{3},\frac{2}{5}$ which correspond to $q=1,2,3,5$, respectively. Inset: Quantized plateaus corresponding to each slope given by differentiating with respect to $\mathrm{\Phi }$. The simulations are performed on a cubic lattice with linear dimension $L=144$. (d), (e) The magnitude of each plateau can be extracted as described by (7). The slopes in (c) are color coded to the Chern numbers in (d) and (e).

Figure 4

(a) Evolution of occupancies in the bands after an electric field $E=0.1$ is turned on, showing the development of the chiral anomaly for $\mathrm{\Phi }={\mathrm{\Phi }}_0/4$. The dashed horizontal line represents the Fermi level ${\mathcal{E}}_F$. (b) Time-of-flight occupancy profiles in the $(k_y,k_z)$ plane in arbitrary units. The vertical dashed lines serve as a guide to the eye for the initial profile at $t=0$. Simulation parameters are the same as Fig. 3.